Lecture 1 | Modern Physics: Special Relativity (Stanford)



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Lecture 1 of Leonard Susskind’s Modern Physics course concentrating on Special Relativity. Recorded April 14, 2008 at Stanford University.

This Stanford Continuing Studies course is the third of a six-quarter sequence of classes exploring the essential theoretical foundations of modern physics. The topics covered in this course focus on classical mechanics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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this program is brought to you by Stanford University please visit us at stanford.edu this quarter we're going to learn about field theory classical field theory fields such as the electromagnetic field gravitational field other fields in nature which I won't name right now propagate which means they change according to rules which give them a wave-like character moving through space and one of the fundamental principles of field theory in fact more broadly nature in general is the principle of relativity the principle the special printless the the principle of special relativity in this particular case the principle of special relativity well let's just call it the principle of relativity goes way back there was not an invention of Einstein's I'm not absolutely sure when it was first announced or articulated in the form which I'll spell it out I don't know whether it was Galileo or Newton or those who came after them but those early pioneers certainly had the right idea it begins with the idea of an inertial reference frame now inertia reference frame this is something a bit tautological about an inertial reference frame Newton's equations F equals MA are satisfied in an inertial reference frame what is an inertial reference frame it's a frame of reference in which Newton's equations are satisfied I'm not going to explain any further what an inertial reference frame is except to say that the idea of an inertial reference frame is by no means unique a reference frame first of all was a reference frame in tale of a reference frame first of all entails a set of coordinate axes in ordinary space X Y & Z and you know how to think about those but it also entails the idea that the coordinate system may be moving or not moving relative to whom relative to whomever we sitting here or you sitting here in this classroom here define a frame of reference we can pick the vertical direction to be the z axis the horizontal direction along my arms here to be the x axis X plus that way X my X is minus in that direction and which one have I left out I've left out the y axis which points toward you from me so there are some coordinate axes for space XY and Z and I didn't this in addition to specify a frame of reference one also imagines that this entire coordinate system is moving in some way relative to you sitting there presumably with a uniform velocity in a definite direction if your frame of reference is an inertial frame of reference in other words if when you throw balls around or juggle or do whatever is supposed to do in an inertial frame of reference if you find yourself in an inertial frame of reference then every other frame of reference that's moving with uniform velocity relative to you now remember what uniform velocity means it doesn't just mean with uniform speed it means with uniform speed in an unchanging direction such a frame of reference is also inertial if it's accelerated or if it starts standing still and then suddenly picks up some speed then it's not an inertial frame of reference all inertial frames of reference according to Newton and also I think also Galileo Galileo was often credited with the idea but I never read enough of Galileo to know whether he actually had it or not neither did I read enough of Newtons they both wrote in languages that I don't understand what was I saying oh yes right according to both Newton and anybody else who thought about it very hard the laws of physics are the same in all inertial reference frames laws of physics meaning F equals MA the forces between objects all the things that we would normally call laws of nature or laws of physics don't distinguish between one frame of reference of and another if you want a kind of pictorial example that I like to use a lot when I'm explaining this to the children or to grownups I like to think about the laws of juggling there are very definite procedures that you train your body to do uh in order to be able to juggle balls correctly now you can imagine yourself being in a railroad car moving with perfectly uniform velocity down the x axis and trying to juggle do you have to compensate for the fact that the train is moving and for particular when you throw a ball up into the air that you have to reach over to the right to compensate for the fact that the train is moving to the left my left your right the answer is no you don't the laws of juggling are the same in every reference frame and every inertial reference frame whatever you do in one reference frame you do exactly the same thing and you'll succeed or fail depending on whether you're a good juggler or not but it will not depend on whether you're moving with uniform velocity so the laws of juggling are the same in every inertial reference frame the laws of mechanics are the same in every inertial reference frame the laws Newtonian laws of gravity are the same in every inertial frame according to Newton what about the laws of electrical phenomena well there there was a clash the clash had to do with Maxwell's equations Maxwell's equations were the field equations the field theory that governed the electromagnetic field and the way that it propagated and sent waves electromagnetic waves that we ordinarily call light or radio waves or so forth and the fundamental dilemma as you all know I'm sure you all know the fundamental dilemma was both according to well here was the dilemma Maxwell's equations said light moves with a certain velocity if you take the various constants that appear in Maxwell's equations and put them together in the right way you get the velocity of waves moving down an axis and that velocity comes out to be a certain number out of Maxwell's equations you have two choices one is to believe that Maxwell's equations are true laws of nature as good as any other laws of nature in which case the principle of relativity says they should be the same in every reference frame but if it follows from Maxwell's equations that the speed of light is three times ten to the eighth meters per second which is about what it is if it follows from Maxwell's equations that light moves that fast and if Maxwell's equations are laws of physics fundamental laws of physics and if the laws of physics are the same in every reference frame then the speed of light must be the same in every reference frame but that was very hard to swallow because if a light beam is going down that axis and you chase it and run along with it that lets say three-quarters of the speed of light then you want to see that light ray moving much more slowly than three times ten to the eighth meters per second relative to you on the other hand the light ray going in the other direction since you're sort of running into it you should see going even faster so all these possibilities could not simultaneously be correct that the laws of nature are the same in every reference frame and that Maxwell's equations are laws of physics in the same sense that Newton's laws of physics namely the same in every reference frame something had to give well the point was of course that they were good laws of nature and that they were the same in every reference frame the thing that had to give is our concepts of velocity space and time and how we measure velocity especially velocities were up which are up near the speed of light now I'm not going to spend the full amount of time that I did previously on the special theory of relativity that can be found on lectures from how long ago and there on the Internet I believe relativity and electromagnetism I think that was maybe about three quarters ago I've lost track yeah they're up there they're on the net and they're the lectures on relativity special relativity and electromagnetic theory we're just going to cut through it real fast we're going to cut through the basic ideas of relativity a little more mathematically than I would do if I were teaching it for the first time I teach it the first time I tend to teach it the way Einstein first conceived of it how do you measure distances how do you measure velocities how do how does the propagation of light influence these things instead I'm going to take a more mathematical view of it and think about the properties of various kinds of coordinate transformations coordinates now consists not only of XY and Z but also time T so imagine every event in the world is characterized by just like every particle would be characterized by a position x y&z every event taking place in space-time is characterized by four coordinates X Y Z and T let's suppress for the moment y&z let's just forget I forget them for the moment and concentrate on X and T that would be appropriate if we were mainly interested in motion along one axis let's focus on that motion along the x axis let's suppose there is no motion along y&z then we can forget y&z for the moment momentarily we'll come back to them and think of motion along X and T and the various reference frames that might be moving along the x axis alright here's here's time vertically is space horizontally physicists always draw space horizontally and time vertically I found out that mathematicians are at least certain computer scientists always draw time going horizontally I didn't know that and I got into an enormous argument with a quantum computer scientist which was ultimately resolved by the fact that he had time going horizontally and I had it going vertically these are traditions I guess traditions grow up around subjects but time is north and X is east I guess or at least time is upward yeah yeah yeah that's what that that that's the point that is the point yes they're thinking of time is the independent variable and everybody knows that it's a law of nature that the independent variable should be horizontal ok all right now let's in let's imagine a moving observer moving down the x axis with a velocity V let's take his origin of spatial coordinates his origin of spatial coordinates at time T equals zero is just the same let's assume that my I'll be the moving observer I move down the x-axis I am my own origin there's nobody who was your origin that seat is vacant over there so that absent a human over there is the center of the x-coordinates in your frame I'm the X prime coordinates and of course I being very egocentric will take my x-acto is origin to be where I am there I do I move down the x-axis we pass each other our origins pass each other at t equals 0 so that means at T equals 0 your axis and my axes are the same or your origin in my origin is the same but then as I move down the x axis my core my coordinate center moves to the right most of the right that's supposed to be a straight line that's as good as I can do under the circumstances that's a straight line and it's moving with velocity V which means it's X prime equals SR it means x equals VT but it's also that's the way you describe it in terms of your coordinates my centre you described by saying x equals VT how do I describe it I just say X prime my coordinate X prime is 0 X prime equals 0 is the same as x equals VT all right what's the relationship between X Prime and X and T well it's easy to work out if you believe this picture the X prime coordinate is the distance from my origin the x coordinate is the distance from your origin so one of these is X the other is X prime the upper one here is X prime the low and here is X and the relationship between them is that they differ by an amount VT in particular X is equal to X prime minus VT or X prime is equal to X plus VT will have it wrong yes I do X prime is X minus BT and X is X prime plus VT yeah I think I have that's correct now all right what about time itself well according to Newton and according to Galileo and according to everybody who came afterward up until Einstein time is just time is just time is just time there was no notion that time might be different in different reference frames Newton had the idea of a universal time sort of God's time God upon his cloud ticking off with his with his super accurate watch and that time was universal for everybody no matter how they were moving and so everybody would agree on what on the time of any given event in this map of space and time here and so the other equation that went with this is that T prime is equal to T let's forget the top equation here let's just forget it one might say that this was the Newtonian or the Galilean transformation properties between X and T your coordinates and the coordinates that I ascribe to a point in space-time now let's examine a light ray moving down the plus x axis if it starts at the origin here then it moves along a trajectory which is x equals CT C being the speed of light now shortly I'm going to set C equal to 1 we're going to work in units in which C is equal to 1 but not quite yet incidentally once you understand a bit of relativity working in coordinates in which C is not equal to 1 is about as stupid as using different units for x and y are if we used yards for x and feet for y then we will have all kinds of funny factors in our equations which would be conversion factors from X which is measured in feet to Y which is measured in our yards the cycle has its uses log scale has its uses no long skilling long scale well let common interest yep I'm not sure we good but okay I'm just saying it is quite often in practical circumstances that one uses different scales yeah you sometimes you might there might be a good reason I mean um it wouldn't be totally unreasonable for a sailor to use different units for horizontal direction and vertical direction hmm I mean he's used to moving around horizontally he might use what miles miles versus fathoms or something nautical miles versus paddles yeah Persian is relative but um when you talk about a frame of reference you need to specify a period of time because obviously goes that 15 billion years there is no yeah we're ignoring now the fact that the universe began at some time and we're imagining now as Newton did and as the early Einstein did that the universe has just been here forever and ever and ever unchanging totally static and space and time have properties which don't change with time now of course that's incorrect in the real world and at some point we will take up the subject of cosmology and find that's not right but as long as we're interested in time intervals which are not I suspect this is what you're getting at as long as we're interested in time intervals which are not too long in particular time intervals over which the universe doesn't expand very much and so forth we can mainly say the properties of space don't change over a period of time and so everything just stays the same as always was is that what you're asking it seems that that this assumption if it is made it needs to what you're describing so well so the question is without imagining to some point as it doesn't lead it doesn't lead to what I'm describing where is this this room for different formulas here this is a formula which is based on an assumption the assumption being that time is universal that's what Einstein found was wrong basically what he found is that when you're in a moving frame of reference to different the observers will not agree about what time a particular event takes place this is the culprit here this one and some modifications to this one but in any case to see what's wrong let's go to Maxwell's equations Maxwell's equations say that light always moves with this velocity C being some numbers in meters per second okay 3 times 10 to the 8th meters per second we will later as I said say C equals 1 let's imagine a light beam moving down the x axis let's describe how X prime sees it in other words you see the light move this way to the right how do I see the light well let's see what I see let's just work it out X prime will be X which is CT for that light ray minus VT which is the same as C minus VT all this says is that I see the light moving with a diminished velocity a velocity C minus V why is that because I'm moving along with the light so naturally I see it move slowly the slow compared to what you see it what about the light going in the other direction supposing it was a light beam going in the other direction then how would you describe it you would describe it as x equals minus CT and if I do exactly the same thing I will find that X prime is equal to X that's minus CT – VT which is the same as minus C plus V times T so what this says is that I will see the light moving also in the negative direction that's the minus sign but I'll see it moving with an enhanced velocity C plus V if this were the right story and if these were the right transformation laws for space and time then it could not be the case that Maxwell's equations are laws of physics or laws of nature in the sense that they were true in every reference frame they would have to be corrected in moving frames just like the juggler who had to reach to the right who didn't actually but who thought he had to reach to the right to collect the ball when train is moving the physicist interested in light beams would have to correct things for the motion of his reference frame now it's an experimental fact that this is not the case that you don't have to correct for motion was the famous Michelson Morley experiment Einstein he just rejected he just felt this can't be right Maxwell's equations were much too beautiful to be relegated to the approximate or to the contingent on which reference frame and so he said about to find a framework in which the speed of light would be the same in every reference frame and he basically focused on these equations and after various very very beautiful Gedanken experiments thought experiments about light and about measuring and so forth he came to a set of formulas called the Lorentz transformations I'm going to explain them the Lorentz transformations in a more mathematical way not fancy mathematics but just get we want to get right to the heart of it and not spend the three weeks doing it the best way is to a mathematical problem but before I do let me set up a different mathematical problem which is for most of you you've seen me do this before but nonetheless let's go through it again the problem of rotation of coordinates we're going to do this quickly let's just take spatial coordinates now for the moment two dimensional spatial coordinates let's forget X and T and just concentrate on X&Y two coordinates in space instead of events in space-time concentrate on a point in space a point in space has coordinates and we can determine those coordinates the x and y coordinates just by dropping perpendicular to the x axis in the y axis and we would describe this point as the point at position let's just call it X Y now there's nothing sacred about horizontal and vertical so somebody else may come along some crazy mathematician a really nutty one who wants to use coordinates which are at an angle relative to the vertical maybe a couple of beers and you don't know the difference between vertical and worth worth worth we should give this direction a name oblique yeah all right the oblique observer the blue observer can blue be seen everybody can see blue okay good ah the blue observer also characterizes points by coordinates which he calls X Prime and Y Prime the X Prime and the Y prime coordinates are found by dropping perpendicular to the X Prime and the Y prime axis so here's X prime is y prime and given a point X Y there's a role it must be a role if you know the value of x and y you should be able to deduce the value of X I'm in y-prime if you know the angle between the two coordinates between the x coordinate and the X prime coordinate and the formulas simple we've used it least in these classes many times I'll just remind you what it is that's X prime is equal to x times cosine of the angle between the two frames between the two coordinate systems minus y times sine of the angle and Y prime is equal to minus plus I think X sine of theta plus y cosine theta I just want to remind you about a little bit of trigonometry all of trigonometry is encoded in two very simple formulas I've used them this signs on these signs of are on the right let's Ella and X prime is bigger than X for small theta since ours here are all so it's Auto Expo Rhine is bigger than it is is it yeah let's see if you rotate it to the next so that y is y prime is zero it's further out X prime rook will have it backward yeah what's your gift I'm not gonna fit nobody so let's say just make sure the links take survive is the little perpendicular there no my life primary so that's y prime y prime is this is why I'm here right right that's why I'm in X prime is bigger than X so there has to be a plus sign on the second you know its prime is bigger than X let's see um yeah X prime is bigger than X yeah X prime is bigger than X looks like that's probably right probably sign but then this one must be man negative yeah okay there's an easy way to correct for it another way to correct for it just call this angle minus theta that would also do the trick because cosine of minus theta is the same as cosine of theta and sine changes sign when you change theta 2 minus theta so if instead of calling this angle theta I called it minus theta then my previous formulas would be right it's true true but the it's an excuse all right what do we know about sine and cosine it's important to understand sine and cosine everything you ever learned about trigonometry can be codified in two very simple formulas if you know about complex numbers the two very simple formulas are that cosine of theta is e to the I theta plus e to the minus I theta over 2 and sine of theta is e to the I theta minus e to the minus I theta over 2i those two formulas contain everything about trigonometry you don't have to know any other formulas other than these for example I will assign you the homework problem of using these two formulas to find cosine of the sum of two angles but the way you would do it is just write the sum of two angles in here and then reexpress the Exponential's in terms of cosine and sine that's easy to do e to the I theta is equal to cosine of theta plus I sine theta and e to the minus I theta is cosine of theta minus I sine theta so work through these formulas get familiar with them they're extremely useful formulas once you know them you will never have to remember any trigonometric formulas again the other thing to know is that e to the I theta times e to the minus I theta is 1 all right e to the anything times e to the minus the same thing is one those things characterize all trigonometric formulas in particular as was explained to me by Michael a number of times if we multiply e to the I theta times e to the minus I theta we will get one on this side but on this side we will get cosine squared of theta plus sine squared of theta naught minus sine squared but plus sine squared cosine squared and then ice minus I squared sine squared that gives us cosine squared plus sine squared cosine squared theta plus sine squared theta so that's equivalent to the fact that e to the I theta times e to the minus I theta is 1 all right now the most important fact that again follows from the simple trigonometry is that when you make the change of coordinates from XY to X prime Y prime something is left unchanged namely the distance from the origin to the point XY that's something which is you know you count the number of the molecules along the blackboard from here to here and that doesn't change when I change coordinates so the distance from the origin to the point XY has to be the same independent of which coordinate axes we use well let's take the square of that distance the square of that distance we know what it is let's call it s squared I'm not sure why I use s but s for distance s s for distance s for space I think it must be for space that I'm using it for the spaces for the spatial distance from the origin to the point XY we know what that is it's Pythagoras theorem x squared plus y squared but as I said there's nothing special about the XY axes we also ought to be able to calculate it as X prime squared plus y prime squared well it's not too hard to work out that X prime squared plus y prime squared is x squared plus y squared it's easy to use do X prime squared plus y prime squared will have x squared cosine squared theta it will also have x squared sine squared theta when you add them you'll get x squared plus y squared you know you know the rigmarole so it follows from cosine squared plus sine squared equals 1 that X prime squared plus y prime squared equals also equal is equal to x squared plus y squared work that out make sure that you have this on the control that you understand why from the trigonometry not from the the basic physics of it or the basic geometry of it is clear make sure that you understand that you can see that from the trigonometry okay one last thing about sines and cosines if I plot on the blackboard for every angle if I plot sine or cosine along the horizontal axis supposing I plot cosine of theta along the horizontal axis and sine of theta along the vertical axis then if I plot all possible angles they will correspond to a bunch of points that lie on a unit circle Y on a unit circle because sine squared plus cosine squared equals 1 so one might call the properties of sine and cosine the properties of circular functions circular in that they're convenient for rotating they're convenient for describing unit circles points on unit circles are described in terms of coordinates which are cosines and sines of angles and so forth it's natural to call them circular functions these are these are not the functions that come in to the transformation the new transformation properties first of all these are wrong and I don't want to use X what's X ya ya now just wrong Newton had it wrong Newton or Galileo however it was postulated who postulated it Einstein modified it now we're going to have to make sure that Einstein's modification doesn't change things in situations where Newton knew where Newton's equations were good approximations the situations where I'm Stan's modifications are important is when we're talking about frames of reference moving very rapidly up near the speed of light before the 20th century nobody or nothing had ever moved faster than a hundred miles an hour probably well of course some things did light did but for all practical purposes light didn't travel at all it's just when you turned on the switch the light just went on so light didn't travel nothing and anybody's experienced direct experience traveled faster than 100 or 200 miles an hour and well I should say nothing travels faster than 100 miles an hour and then live to tell about it so all of experience was about very slow velocities on the scale of the speed of light on the scale of such velocities newton's formulas must be correct they work they're they're very useful they work Nutan got away with it so there must be good approximations it better be that whatever einstein did to the equations in particular to these two equations here had been a reduced to newton's equations in the appropriate limit okay let's come back now to light light according to the Newton formulas doesn't always move with the speed of light but let's let's try to figure out what it would mean of a better formula of a replacement for this but light always moves with the speed of light first of all let's set the speed of light equal to one that's a choice of units in particular it's a choice of the relation between space units and time units if we work in our light years for spent for a distance and years for time then light moves one light year per year the speed of light is one if we use seconds and light seconds it's also one whatever whatever scale we use for space if we use for time the time that it takes light to go that distance one unit of space if we use that for time units then the speed of light is equal to one now from the ordinary point of view of very slowly moving things those are odd units but if we were electrons with neutrinos and whizzing around like photons they would be the natural units for us speed of light equals one so let's set the speed of light equal to one as I said it's just the choice of units and then a light ray moving to the right just moves along a trajectory x equals T C is just equal to one a light ray moving to the left is x equals minus T how can we take both of these equations and put them together sorry x equals minus T can I write a single equation which if it's satisfied is a light ray either moving to the left or to the right yes here's an equation x squared equals T squared it has two solutions x equals T and X equals minus T the two square roots or x squared equals T squared is equivalent to either x equals T or x equals minus T in other words this equation here has the necessary and sufficient condition for describing the motion of a light ray either to the right or to the left supposing we found a replacement for this equation which had the following interesting property that whenever let's let's write it this way X square minus T squared equals 0 this is even better for our purposes x squared minus T squared equals 0 that's the necessary and sufficient condition to describe the motion of a light ray supposing we found a new set of rules a new set of transformation properties which which um had the property that if x squared minus T squared is equal to 0 then we will find that X prime squared minus T prime squared is equal to 0 in other words supposing this implied this and vice-versa then it would follow that what the unprimed observer you and your seats see is a light ray the primed observer me moving along also see as a light ray both of us agreeing that light rays move with unit velocity now this doesn't work for Newton's formula here it just doesn't work if X is equal to T it does not follow that X prime is equal to the T prime in fact it says something quite different okay so the form of these equations must be wrong let's look for some better equations now at this point let's in fact let's even be a little bit more ambitious it turns out being a little bit more ambitious actually simplifies things let's not only say that when X square minus T squared is equal to zero then X prime squared minus T prime squared is equal to zero let's say something even bolder let's say the relation between XT and X prime T prime is such that x squared minus T squared is equal to X prime squared minus T prime squared in other words pick any X and any T and calculate X square minus T squared then take the same point except reckoned in the primed coordinates in other words we take a certain event a light bulb goes off someplace you say that corresponds to X and T I say it corresponds to X Prime and T Prime but let's require just to try it out see if we can do it let's look for transformations so that X square minus T squared will always be equal to X prime squared minus T's prime squared that would be enough to ensure that everybody will agree about the speed of light why if x squared minus T squared equals X prime minus T prime squared for all X and T and so forth then when X square minus T squared equals zero X prime minus T prime squared will be zero and then if this is a light ray so is this a light ready everybody get the logic ok good so let's assume now that let's ask can we find transformations which have this particular property now it's not so different from looking for transformations which preserve x squared plus y squared equals x prime squared plus y prime squared it's just a little minus sign other than a minus sign here X square minus T squared look of these two is very similar and the mathematics is quite similar here are the transformations which preserve x squared plus y squared what are the transformations which preserve x squared minus T squared well they are the Lorentz transformations they are the fundamental transformations of the special theory of relativity they're not this but they're closely related or perhaps one should say closely analogous to these equations here but we have to substitute for circular trigonometry hyperbolic trigonometry so let's go back and remember a little bit about hyperbolic functions instead of circular functions well I didn't want to erase that all right these are the basic rules governing circular functions cosine theta this sine theta is equal to this and the e to the I theta in terms of cosine and sine all right let's see if we have a yeah we do have a blank blackboard here let me write whoops what did I do here I erased something I didn't mean to erase incidentally does everybody see how I got this side from the side you just add and subtract the equations appropriately and you isolate it to the I theta e to the minus R theta that's elementary exercise alright hyperbolic functions what are hyperbolic functions alright those are functions of the form hyperbolic cosine cosh hyperbolic cosine first of all the angle theta is replaced by a variable called Omega which I will call Omega Omega is called a hyperbolic angle it doesn't go from zero to two pi and then wind around on a circle it goes from minus infinity to infinity goes from minus infinity to infinity so it's a variable that just extends over the entire real axis but it's defined in a manner fairly similar to cosine and sine cosh Omega is by definition you're not allowed to ask why this is definition e to the Omega plus e to the minus Omega over 2 all we do is substitute for theta or for Omega theta I theta substitute Omega and that gives you hyperbolic functions likewise or similarly there's the hyperbolic sine and that's given by e to the Omega minus e to the minus Omega over 2 essentially you throw away all eyes out of that formula out of the top formulas just throw away all Sun all eyes the equations on the right-hand side become e to the Omega equals hyperbolic cosh Omega plus sin Chi Omega and e to the minus Omega equals cosh so mega- cinch Omega I think that's right is it right gosh – cinch it is yeah it is right okay now what about the analog of cosine squared plus sine squared equals one that simply came by multiplying this one by this one so let's do the same operation multiplying e to the Omega by each by e to the minus Omega gives one and now that gives cosh squared minus cinch squared you see we're getting a minus what we want we want that minus the minus is important we want the well somewhere is under here was a formula with a minus sign yeah we want to get that – into play here that's cos Omega squared knockouts Prakash squared Omega minus sin squared Omega so it's very similar everything you want to know about hyperbolic trigonometry and the theory of these functions is called hyperbolic trigonometry everything you ever want to know is codified in these simple formulas these in these and they're more or less definitions but there are the useful definitions now yeah go ahead yeah not only is it worth mentioning I was just about to mention it so I squared minus y squared is what hyperbola yeah right exactly so if I were to play the same game that I did here namely plot on the horizontal and vertical axis the values not of cosine of theta and sine of theta but cosine cosine cosh of that of Omega and since Omega what's in other words on the x-axis now we're going to plot cos Omega and on the y-axis cinch Omega then this is a hyperbola not a circle but a hyperbola and it's a hyperbola with asymptotes that are at 45 degrees you can see let me show you why why the asymptotes are at 45 degrees when Omega is very large when Omega is very large then e to the minus Omega is very small right when Omega is very large e to the minus Omega is very small and that means both cosh and cinch are both essentially equal to e to the plus Omega in other words when Omega gets very big cosh and cinch become equal to each other and that's this line here cash equals cinch along this line here so when Omega gets very large the curve asymptotes to to a curve which is a 45 degrees it's not hard to see that in the other direction when Omega is very negative that that it asymptotes to the other asymptotic line here so that's why it's called hyperbolic geometry it the hyperbolic angle the hyperbolic angles the caches the cinches play the same role relative to hyperbolas as sines and cosines do two circles any questions No so cosh Omega equals zero how would you plot that hi purple okay show me hmm Oh cos squared minus sin squared equals zero no that's no no cos squared minus sin squared equals one in the same sense that sine squared plus cosine square it never equals zero I think what I think you want to ask a different question I think oh well since Omega equals zero is the horizontal axis the costume a equals zero is the vertical eyebrows right okay well this is the x-intercept yeah it's it's the vertex I just think here's one point on a minute oh man the x-intercept there is one yeah because Kostroma cost of zero is one to see that just plug one r 0 in here 1 plus 1 divided by 2 is 1 at least it was yesterday yeah stores okay so now we we're sort of starting to cook a little bit we're starting to see something that has that nice minus sign in it but what's it got to do with X and T and X Prime and T prime we're now set up to make let's call it a guess but it's a guess which is based on the extreme similarity between hyperbolas and circles cautions and cosines and so forth he is the guess I'm going to make and then we'll check it we'll see if it does the thing we wanted to do my formula instead of being this has gotten with and we're now going to have instead of x and y we're going to have x and t time and x later on we'll put back y&z we're going to have to put back y&z but they're very easy okay so let's start with X prime X prime is the coordinate given to a point of space-time by the moving observer namely me and I'm going to guess that it's some combination of X and T not too different but not the same as where is it X prime equals X minus VT I'm going to try cosh Omega X let's write X cos Omega minus T sin Omega sort of in parallel with this I could put a plus sign here but you can go back and forth between the plus and the minus by changing the sign of Omega just as you did here so this let's do it this way X cos Omega minus T sin Omega and T prime going to look similar but without the extra minus sign here this you know the relation between sines cosines and cautious and cinches is one of just leaving out an eye you go from sines and cosines the clashes and cinches by leaving out the I well if you track it through carefully you'll find that this minus sign was really an I squared it's not going to matter much I will just tell you it was really came from some I squared and if you leave out I I squared just becomes one squared is no minus sign so here's the guess for the formula connecting X prime T Prime with X and T it equals let's say X since Omega – no – plus T cos Omega in this case there are two minus signs in this case there was only one minus sign okay but but let's check what do we want to check we want to check that X prime squared minus T prime squared is equal to x squared minus T squared your ask you're probably asking yourself what is this Omega what does it have to do with moving reference frames I'll tell you right now what Omega is it's a stand-in for the velocity between the frames we're going to find the relationship between Omega and the relative velocity of the reference frames in a moment there has to be a parameter in the lower end these are the lines in these are the Lorentz transformations connecting two frames of reference in the Lorentz transformations as a parameter it's the velocity the relative velocity that parameter has been replaced by Omega it's a kind of angle relating the two frames a hyperbolic angle but we'll we'll come back to that for the moment let's prove that with this transformation law here that X prime squared minus T prime squared is equal to zero ah is equal to X square minus T squared I'm getting to that point in the evening where I'm going to make mistakes all right this is easy you just work it out you use all you have to use is that cosine squared minus sine squared is 1 you can work that out by yourself but we can just see little pieces of it here X prime squared will have x squared cos squared Omega t prime squared will have x squared sin squared Omega if I take the difference between them I'll get a term with an x squared times cos squared minus sin squared but cos squared minus sin squared is one fine so we'll find the term with an x squared when we square take the square of the difference between the squares of this and this and likewise will also find the T squared the cross term when you square X Prime you'll have XT cost cinch when you square T Prime you'll have XT costs inch when you subtract them it'll cancel and it's easy to check that's our basically one liner to show that with this transformation here x prime squared minus T's prime squared is x squared minus T squared which is exactly what we're looking for let me remind you why are we looking for it if we find the transformation for which the left-hand side and the right-hand side are equal then if x squared equals T squared in other words if the right-hand side is 0 the left-hand side will also be 0 but x squared but x equals T that's the same as something moving with the speed of light in the X frame of reference if this being 0 is equivalent to the left hand side being 0 it says that in both frames of reference the light rays move with the same velocity so that's the basic that's the basic tool that we're using here X prime squared minus T prime squared is equal to x squared minus T squared all right that does follow by a couple of lines using cos squared minus N squared equals 1 but what I want to do let's take another couple of minutes now let's take a break for five minutes and then come back and connect these variables Omega with the velocity of the moving frame of reference somebody asked me a question about the ether and what it was that people were thinking somehow Einstein never got trapped into this mode of thinking um well what were they thinking about when they were thinking about the ether what exactly was the michelson-morley experiment well I'll just spend the minute or two mentioning it certainly Maxwell understood that his equations were not consistent with with Newtonian relativity he understood that but his image of what was going on is that the propagation of light was very similar to the propagation of sound in a material or water waves propagating on water and of course it is true that if you move relative to the atmosphere or move relative to the substance that sound is propagating in you'll see sound move with different velocities depending on your motion if you're at rest in a gas of material isn't there's a natural sense in which is a particular rest frame the rest frame is the frame in which on the average the molecules have zero velocity if you're in that reference frame then first of all light has the same velocity that way as that way number one and it has a velocity that's determined by the properties of the fluid that the sound is moving in okay Maxwell more or less thought that light was the same kind of thing that there was a material and the material had a rest frame and that particular rest frame was the frame in which light would move with the same velocity to the left as to the right and he thought that he was working out the mechanics or the behavior of this particular material and that we were pretty much at rest relative to this material and that's why we saw light moving the same way to the left of the right one would have to say then that Maxwell did not believe that his equations were a universal set of laws of physics but that they would change when you moved from frame to frame just happened by some luck we happen to be more or less at rest relative to the ether to this strange material um of course you could do an experiment with sound if you're moving through the sound you can check that the velocity in different directions is different you do let's not worry exactly how you do that that's what the Michelson Morley experiment was Michelson and Morley I suppose said look the earth is going around in an orbit maybe at one season of the year we just happen to be at rest relative to the ether by accident and some other season six months later we're going to be moving in the opposite direction and we won't well we won't be at rest only at one point in the orbit could we be at rest relative the–this or at any other point in the orbit we wouldn't be so if we measure in November that light moves the same than all possible directions then in what's what's the opposite of November May then in May we should find that light is moving with great with the different velocities in different directions and he tried it and a very fancy and sophisticated way of measuring the relative velocity in different directions but he found that there was no discrepancy that the light traveled the same velocity in every direction at every time of year there were all sorts of ways to try to rescue the ether but none of them worked none of them work and the result was one had to somehow get into the heart of space and time and velocity and mid distance and all those things in a much deeper way in a way that didn't involve the idea of a material at rest in some frame of reference that that propagated the light okay oh where are we I forgotten where we were when we stopped somebody remind me whoo-hah Omega yeah what is Omega forgotten Omega Oh how Omega is really metal speed of light but to the velocity of the moving reference frame here we have two reference frames X T and X Prime and T prime what's the relationship between them well the whole goal here was to understand the relationship between frames of reference moving with relative velocity between them Omega is not exactly the relative velocity but it is closely related to it okay let's say let's see if we can work out the relationship we know enough to do it let's see if we can work out the relationship between Omega and the velocity of the moving frame all right again let's go back to this picture there's a frame of reference moving let's redraw it here's my origin moving along okay what does it mean to say that from your perspective my frame of reference so my origin is moving with velocity V well by definition this is not a law now this is a definition and says that this line here has the equation x equals VT that's the definition of this V here my origin moves relative to your origin it moves with a uniform constant velocity that's an assumption that we're talking about two inertial frames of reference and you in your frame of reference will write x equals VT that's the definition of V if you like what will I call it I will call it X prime equals zero all along there I will say X prime is equal to zero it's my origin of coordinates okay now let's come to this transformation law here and see if we can spot how to identify V well X prime equals zero that's this trajectory moving at an angle with a velocity V X prime equals zero is the same as saying X cos Omega equals T sin Omega X prime equals zero set this side equal to zero and that says that X cos Omega equals T sin Omega all right so whatever the connection between velocity and Omega is it must be such that when X prime is equal to zero X cos Omega equals T sin Omega well let's look at that equation it also says that X is equal to sin CH Omega over cos Omega times T well that's interesting because it's also supposed to be equivalent to x equals VT now I know exactly how to identify what the velocity is as a function of Omega the velocity of the moving transformation the moving coordinate system must just be sin Chi Omega over cos Omega that's the only way these two equations can be the same x equals VT x equals sin Chi Omega over cos Omega times T so now we know it we know what the relationship between velocity and Omega is write it down the velocity of the moving frame now this is not the velocity of light it's just the velocity of the moving frame must just be cinch Omega over cos omega well actually i want to invert this relationship i want to find sin and cos omega in terms of the velocity i want to rewrite these Lorentz transformations where are they i want to rewrite these Lorentz transformations in terms of the velocity that's the familiar form in which you learn about it in in elementary relativity books X prime is equal to something with velocities in it to exhibit that all we have to do is to find Cinch and cosh Omega in terms of the velocity that's not very hard let's let's work it out the first step is to square it and to write V squared is equal to cinch Omega squared over cosh Omega squared that was easy next I'm going to get rid of since Omega squared and substitute where is it I lost it one is equal to cos Omega squared minus cinch Omega squared alright so wherever I see cinch Omega squared I can substitute from here namely cosh squared Omega minus one is equal to sine squared Omega so here we are this is just equal to hash of Omega squared minus one divided by cost of Omega squared or let's multiply by what I want to do is solve for cost Omega in terms of velocity I want to get rid of all these cautions and cinches of Omega and rewrite it in terms of velocity so first x cost Omega squared we have cosh squared Omega times V squared equals cosh squared Omega minus one or it looks to me like this is cosh squared Omega times one minus V squared equals one what I've done is transpose yeah cos squared times V squared minus cos squared itself that gives you cos squared 1 minus V squared equals 1 change the sign can everybody see that the second line follows from the first I'll give you a second yeah yeah yeah it's clear ok finally we get that cos Omega is equal to 1 divided by 1 minus V squared but now I have to take the square root cos Omega / one minus V squared and then take the square root and that gives you cos Omega now we've all seen these square roots of 1 minus V squared in relativity formulas here's where it begins the kayne we begin to see it materializing what about sin Chi Omega let's also write down sin Chi Omega well from here we see that sin Chi Omega is just equal to V times cos Omega this is easy since Omega equals V times cos Omega sorrow sin Chi Omega is V divided by square root of 1 minus V squared let's go back to these Lorentz transformations over here and write them getting rid of the trigonometric functions the hyperbolic trigonometric functions and substituting good old familiar velocities let's get rid of this and substitute the good old ordinary velocities ok so we have here X prime equals x times cos Omega and that's divided by square root of 1 minus V squared then this minus T times sin Omega which is V over the square root of 1 minus V squared or if I put the two of them together and combine them over the same denominator it's just X minus VT divided by square root of 1 minus V squared I think most of you have probably seen that before maybe slightly different let's let's clean it up a little bit X prime equals X minus VT divided by the square root of 1 minus V squared what about T prime T Prime is equal to t minus V X over square root of 1 minus V squared T prime is equal to T times cos cost is just 1 over square root and then x times sin CH that gives us the extra V in other words the formulas are more or less symmetrical and those are all good old Lorentz transformations now what's missing is the speed of light let's put back the speed of light the put back the speed of light is an exercise in dimensional analysis there's only one possible way the speed of light can fit into these equations they have to be modified so that they're dimensionally correct first of all one is dimensionless has no dimensions it's just one velocity is not dimensionless unless of course we use dimensionless notation for it but if velocity is measured in meters per second then it's not dimensionless how do we make V squared dimensionless we divide it by the square of the speed of light in other words this V squared which is here which has been defined in units in which the speed of light is 1 has to be replaced by V squared over C squared likewise over here V squared over C squared now velocity times time does have notice first of all the left hand side has units of length the right hand side this is dimensionless X has units of length but so does velocity times time so this is okay this is dimensionally consistent as it is but over here it's not the left hand side has dimensions of time that's all right 1 minus V squared over C square that's dimensionless this has units of time but what about velocity times X velocity times X does not have units of time in order the given units of time you have to divide it by C square okay let's check that velocity is length all the time times length divided by C squared that's length square R which gets correct but it's correct all right this is probably familiar to most of you who've seen relativity once or twice before these are the equations relating to different moving coordinate systems moving relative to the x axis but you see the deep mathematics or the mathematical structure of it in many ways is best reflected by this kind of hyperbolic geometry here and you know most physicists by now never write down the Lorentz transformations in this form much more likely to write them in this form easier to manipulate easier to use trigonometry or or hyperbolic trigonometry it's a little exercise it's a nice little exercise to use this the hyperbolic trigonometry to compute their to compute the compounding of two Lorentz transformations if frame two is moving relative to frame one with velocity V and frame three Israel moving relative to two with velocity V Prime how is three moving relative to one the answer is very simple in terms of hyperbolic angles you add the hyperbolic angles not the velocities but the hyperbolic angles the hyperbolic angle of three moving relative to one is the hyperbolic angle of three moving relative to two plus two moving relative to one and then you use a bit of trigonometry or hyperbolic trigonometry to figure out how you do the inches and kosh's of the sum of 2 hyperbolic angles very straightforward and I'll leave it as an exercise to see if you can work that out much easier than anything else ok so there there we have the Lorentz transformations yeah oh oh absolutely yes that's that's that's a good point yeah when we that's right if we have frame 1 let's call this x1 and y1 x2 and y2 and finally x3 and y3 well then the angle of – let's call F of 3 relative to 1 let's call it theta 1 3 is just equal to theta 1 2 plus theta 2 3 the angle connecting frame one with frame 3 is just the sum of the angle theta 1 2 plus theta 2 3 so in that respect the Lorentz transformations are much simpler in terms of the Omegas it's the Omegas which combined together to add when you add velocities now how different is omega from the velocity let's work in units in which the speed of light is equal to 1 where is our formula for velocity all right let's take this formula over here what a cinch Omega 4 small Omega let's put the C squared there a let's not put the C square there or not put the C square there since Omega is essentially Omega when Omega is small just like sine is omega where is theta when theta is small the cinch function the cost function looks like like this the cinch function looks like this but it but it crosses the axis with a slope of 1 for small Omega cinch Omega is proportional to Omega for small velocity one minus V squared is very close to 1 if the velocity is a hundredth of the speed of light then this to within one ten-thousandth is just 1 if we're talking about velocities a millionth of the speed of light then this is very close to 1 and so since Omega and velocity are very close to each other it's what's going on here Thanks okay so for small velocities Omega and velocity are the same the actual correct statement is that V over C is like Omega the dimensionless velocity over the speed of light is like Omega for small Omega and small velocity so for small velocity adding velocities and adding omegas are the same things but when the velocities get large the right way to combine them to find relationships between different frames is by adding Omega and not adding velocities when you add Omega like compounding velocities as you've got it there I guess you won't go greater than 45 degrees that guess because that would be faster than light no but Omega no more you see this bit the speed of light is V equals one that corresponds to Omega equals infinity yeah yeah so Omega Omega runs over the whole range from minus infinity to infinity but when it does V goes from minus the speed of light to the speed of light so you can add any omegas and still add any omegas Omega that's right there's no there's no speed limit on Omega is this like we just go on that diagram it looks like it's greater than 45 degrees if here where where I make a and I guess they use the definition of state along the hyperbola yeah that's right sorry where are we right there today I guess that's theta though isn't it this is Theta that's a good oh god yeah right right yeah Omega is the distance along hyperbola that's right distances that's right Omega is a kind of distance along the hyperbola all right now let's let's talk about that a little bit all right now that we've established the basic mathematics structure of the transformations I think we should go back and talk about some simple relativity phenomena and derive them oh one thing which is important which I yeah well let's see we're here are my Lorentz transformations over here I said we should we ought to at the end make sure that our transformations are not too dissimilar from Newton's in particular when the velocities are small they should reduce to Newton that's all we really know that's or at least that's all that Newton really had a right to assume that when the velocities are smaller than something or other that his equations should be good approximations isn't adding velocity good enough isn't velocities adding good enough in fact you're right in fact you're right but let's just look at the transformations themselves all right as long as the velocity is a small percentage of the speed of light an ordinary velocities are what a hundred miles an hour versus 186,000 miles an hour what is that it's small right and it's doubly small when you square it so for typical ordinary velocities even the velocities of the earth around the Sun and so forth fairly large velocities what 60 kilometers per second or something like that 60 kilometers per second is pretty fast that's the that's the orbital earth around the Sun it's pretty fast but it's nowhere near 300,000 kilometers per No yeah looks here on a thousand meters per second we're I'm sorry three times ten to the eighth no three times three hundred thousand kilometers per second right 60 kilometers per second three hundred thousand kilometers per second small fraction and then square it so for ordinary motions this is so close to one that the deviation from one is negligible so let's start with the top equation for the top equation this is negligible and it's just x prime equals X minus VT the bottom equation here you have a C squared in the denominator whenever you have a C squared in the denominator that's a very very large thing in the denominator this is negligible compared to T so here the speed of light is also in the denominator just forget this and it's just T but it's just T prime equals T it's just D prime equals T so in fact Newton's formulas are essentially correct for slow velocities no no significant departure from Newton until the velocities get up to be some some appreciable fraction of the speed of light okay let's talk about proper time proper time and then let's do a couple of relativity examples yeah question the bottom equation when X is very large yes that's right when X is exceedingly large you get a correction but that correction that X has to be very large look let's let's discuss before we do anything else let's let's let's talk about that a little bit X minus VT one minus V squared over C squared yeah let's alright in my drawings I'm going to sitt C equal to one but in the equations you can leave the C there okay this equation we understand apart from this one minus V squared over C squared in the denominator it's just this x equals V T or X minus V X minus X minus VT that's Newton let's look at this one over here okay let's look at the surface T prime equals zero T prime equals zero is the set of points that I in my moving reference frame call T call time equals zero it's what I call the set of points which are all simultaneous with the origin T prime equals zero is just everyplace in space-time which has exactly the same time according to my frame of reference and I will therefore call all those points synchronous at the same time what do you say about them if T prime is equal to zero that says that T is equal to V over C squared X now let's set C equal to one for the purpose of drawing just for the purpose of drawing I don't want this huge number C squared to distort my drawings too much it says the T equals V X what does the surface T equals V X look like it looks like this T equals V X which is also X is equal to 1 over V T so it's just a uniform line like that all of these points are at different times from your reckoning this ones later this ones later this ones later and so forth according to my reckoning all these points are at the same time so we disagree about what's simultaneous this was this was the hang-up incidentally this was the basic hang-up that took so long to overcome that took Einstein to overcome it the idea that simultaneity was the same in every reference frame nobody in fact it was so obvious that nobody even thought to ask a question is simultaneous does it mean the same thing in every reference frame no it doesn't in more in your reference frame the horizontal points are all simultaneous with respect to each other in my reference frame what I call horizontal what I call simultaneous you do not okay so simultaneity had to go let me point out one more thing about these equations I'm not going to solve them for you but I will tell you the solution anyway how do you solve for X and T in terms of X Prime and T Prime well think about it in the case of angles supposing I have a relationship like X prime is equal to X cosine theta what is it plus plus y sine theta and y prime is equal to X minus X sine theta plus or Y cosine theta and supposing I want to solve for x and y in terms of X Prime and Y Prime you know what the solution is just change theta 2 minus theta and write that X is equal to X prime cosine of minus theta but what's cosine of minus theta right cosine theta plus y sine of minus theta what's sine of minus theta minus sine theta times y and likewise for y prime Y prime is equal to minus x times sine of minus theta so that becomes plus X sine theta plus y cosine of minus theta which is cosine theta you don't have to go through the business of solving the equations you know that if one set of axes is related to the other by rotation by angle theta the second one is related to the first one or vice versa the first one is related to the second one by the negative of the angle if to go from one frame to another you rotate by angle theta and to go from the second frame back to the first you rotate by angle minus theta so you just write down exactly the same equations interchange Prime and unprimed and substitute for theta minus theta same thing for the Lorentz transformations exactly the same thing if you want to solve these for X and T write down the same equations replace primed by unprimed and change the sign of omegas to minus the sines of omegas change sinus rgn of all the sign all the cinches okay in other words just send Omega 2 minus Omega and that will solve the equations in the other direction yeah yes it's also the same as changing V 2 minus V yes the way to see that is to go right what was it what do we have cosh Omega yep yeah that's right via sign yes that was correct yeah you just well you change Omega 2 minus Omega it has the action of changing V 2 minus V you can just check that from the equations good alright let's let's talk about proper time a little bit proper time if you're doing ordinary geometry you can measure the length along a curve for example and the way you do it is you take a tape measure and you you know sort of take off you take off equal intervals equal equal little separations you can think of these separations as differential distances DS squared small little differential distances and that differential distance is d x squared plus dy squared with the x squared and the y squared are just the differential increments in x and y DX and dy this is d s alright so that's the way and you add them up you add them up that's the way you compute distances along curves it's quite obvious that if you take two points the distance between those two points depends on what curve not the same for every curve so I'll measure the longer curve you have to know not only the two points but you have to know the curve in order to say what the distance between those points are of course the distance between its longer straight line that's that's well-defined but the distance along a curve depends on the curve in any case D s squared equals the x squared plus dy squared is the basic defining notion of distance between two neighboring points if you know the distance between any two neighboring points in a geometry you basically know that geometry almost essentially completely so given this formula for the distance between two points you can compute if you like the distance along a curve because you've got to take the square root of this and then add them up don't anhedonia the squares add the differential distances all right the important thing is here that square root of DX squared plus dy squared which is the distance between neighboring points doesn't depend on your choice of axes I could choose X Y axes I could choose X prime y prime axes if I take a little differential displacement the X and the y or I just take two points two neighboring points don't even give them labels and measure the distance between them the distance between them should not depend on conventions such as which axes are used and so when I make rotational transformations the X square plus dy squared doesn't change the X and the y may change but the x squared plus dy squared does not change the same thing is true in relativity or the analogous thing we don't measure distances along the paths of particles let's say now that this curve here is the path of a particle moving through space-time there's a particle moving through space-time and we want some notion of the distance along it the notion of distance along it another example would just be a particle standing still as a particle standing still particle standing still is still in some sense moving in time I wouldn't want to say that the distance between these two points and space-time is zero they're not the same point I wouldn't like to say it's zero I would like to say there's some kind of notion of distance between them but it's quite clear that that distance is not measured with a tape measure this point and this point are the same point of space boom here at this point of space and that at a later time boom again at the same point of space two events at the same point of space how do I characterize and some nice way the distance between those two events that occurred in the same place you don't do it with a tape measure all right what do you do with a clock a clock you take a clock and you start it at this point tic tic tic tic tic tic tic a stopwatch you press it at this point tic tic tic tic tic it picks off intervals and then you stop it at that point and you see how much time has evolved that's a notion of distance along a particle trajectory it's not the distance the particle moves in space it's a kind of distance that it's moved through space-time and it's not zero even if the particle is moving standing perfectly still in fact what it is is it's the time along the trajectory what about a moving particle well you can imagine that a moving particle carries a clock with it of course not all particles carry clocks but we can imagine they carry clocks with them as they move and we can start the clock over here and then the clock over here what is the time read off by this moving clock the time read off by a moving clock is much like the distance along a curve measured by a tape measure in particular it should not depend on the choice of coordinates why not this is a question that has nothing to do with coordinates I have a clock made in the standard clock Factory the standard clock Factory and I don't know we're in Switzerland someplace makes a certain kind of clock that clock gets carried along with a particle and we ask how much time evolves or how much time elapses or how much the clock changes between here and here that should not depend on a choice of coordinates it shouldn't depend on a choice of coordinates because it's a physical question that only involves looking at the hands of the clock in fact we can ask it for little intervals along along the trajectory we could ask how much time elapses according to the clock between here and here well the answer again should not depend on what coordinates you use which Lorentz frame you use and there's only one invariant quantity that you can make out of the D X's and DTS describing this point describing these two points there's a little interval DT and there's a little interval DX now we're in space and time not ordinary not ordinary space and the quantity which is invariant there's really only one invariant quantity that you can make out of it it is DT squared minus DX squared it's the same quantity x squared minus T squared for a whole you know for a whole interval the T squared minus DX squared that's the quantity which is invariant it's minus D it's the negative of what I wrote over here x squared minus T squared okay this quantity is equal to the X prime squared minus DT power sorry DT prime squared minus the X prime squared the same algebra goes into this as goes into showing that X prime squared minus T prime squared equals x squared minus T squared incidentally this is the same as saying T prime squared minus X prime squared equals T squared minus x squared doesn't matter which way you write it all right so that suggests that suggests that the time read off the invariant time read off along a trajectory between two points separated by DX and DT is just the square root of DT squared minus DX squared why the square-root incidentally okay you're going to integrate in detail I can integrate DT yeah well alright why not just DT square minus the x squared for the time between here and here is it here's an answer supposing we go to you two intervals exactly the same as the first one we go an interval over here DX and DT and then we go another DX in DT what happens when we double the interval to DT squared minus DX squared it gets multiplied by four because everything is squared well I wouldn't expect a clock when it goes along you know when it goes along a trajectory for twice the the interval here to measure four times the the time I expected to measure twice the time so for that reason the square root is the appropriate thing here okay that's called D tau squared the tau squared the proper time along the trajectory of an object you're right that's just the towel or D tau squared being the x squared minus DT squared the Tau is called the proper time let's go I think we'll let's see the towel is called the proper time and it is the time read by a clock moving along a trajectory it's not just DT that's the important thing it's not just DT the T squared minus the x squared let's do one last thing let's just do the twin paradox in this language I think I think I've had it I'm going to finish you can do the twin paradox in this language all you have to do is to compute the proper time along two trajectories one that goes out with a uniform velocity turns around and comes back with the same uniform velocity versa a trajectory which just goes from one point to the st. the another point along a straight line and it's no more weird it's no weirder really from this perspective than saying the distance from one point to another along two different curves do not have to agree the proper time along two different curves in general will not agree what is a little bit weird is that because of this minus sign the proper time this way is less than the proper time this way that's the consequence of this minus sign here moving with some DX decreases the proper time all right we'll do a little bit more next time but then I want to get to the principles of field theory and and connect some of this with field equations for interesting wave fields the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Statistics Lecture 4.2: Introduction to Probability



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Statistics Lecture 4.2: Introduction to Probability

all right so talking about probability again this is our transition between from descriptive statistics into inferential so we're talking about probabilities is chapter four like I said is the basis for making decisions about our data that it's really what we're doing and it's based on this idea if something has a low probability that means its occurrence is going to be rare so if something has a low chance of happening you say it's probably not going to happen that that's really the whole idea here I mean it might be an obvious statement but what we're going to think of it is low probability means rare or unusual occurrences occurrences two hours one hour I never know I think it's too like that one is this occurrence is okay one hour you are right and here's some vocabulary that we do need to talk about speaking of vocabulary there's some vocabulary that we need to talk about as well before we get going on exactly what probability is going to be in relation to this class the first we're going to talk about is a word called an event event in statistics doesn't mean event like in English in English an event means you're having like a party or something that's already vent here that's not what it means what event means for us is a collection of outcomes of a procedure so an event is what you get from a procedure I'll explain this in give you examples in just a minute so event it's a collection of outcomes of a procedure so something you're doing we also have another couple words we have something called a simple prevent a simple event is a single outcome one specific single outcome don't worry I'm going to flesh all of this out with an example in about two or three minutes last word we have something called a sample space the sample space is everything that could possibly happen all the simple events so all simple events in other words every possible outcome so any event if that is something that could happen when you do a procedure procedure such as willing to die flipping a coin so they'd involve some chance in their a simple event is one specific outcome that you can get the sample space is everything that you can possibly get from flipping the coin or willing to die so let me give you example we'll talk about flipping a coin and we will identify some events and some simple events and then the sample space would you like to do that we kind of get an idea but what this up actually means so let's go ahead and our example would be flipping a coin so our procedure we're going to flip the coin one time so you can take a coin out of your pocket you're going to flip it and you're done the others have the procedure right here's an example of an event an event could be what can you get out of the coin can you get edge easily if you would it be cool if you just flipped it in land on the edge sure happening before has ever happened I wonder if it's happened to somebody before well they've been like a edge of a coin it's like that why they can probably happen but like with a quarter or something it's really rare that that's ever going to happen that I don't know that ever has anyway that's just four possibility I guess but if you're flipping a coin there's only two things that can happen right you can get heads or tails when you're talking about an event you're specifying one thing that could happen and what you're looking for so an example of an event in this case would be you're looking to four how many heads you get so an event an example of event would be head look at it which coincidentally that's also a supplement one thing that can happen our sample space includes every possible outcome you can get when you do your procedure now our procedure to slip in a coin how many times just once so you flip the coin what could you possibly get from flipping the coin one time okay so that's what our simple our sample space is as you either get a head or you get a tail can you get anything else besides a head or a tail when you flip the coin one time now we're not going to get through the edge thing that's really not realistic and we put these funny little brackets around there if you're trying for the first time that might end up like that that's okay it's a nice curvy bracket metal takes you a lot of years to master that master's-level stuff you know five brackets so our sample space is a collection of simple events so here's what we're talking about procedure is what your you're doing event is what you're looking for simple events are what could happen and your sample space is a collection of all those things you kind understand more the idea of events simple events and sample spaces the next example will really make it even more clear for you so the procedures what you're doing the event is one outcome that you're looking for so we're looking to get for a head here or you could append tails there what you can get a flip by flip those are called your simple events one out one specific outcome so here we can only get a head or tail if you combine all of those simple events together you get what's called to our sample space now if you're so with me on this let's do one more example to really illustrate this the procedure now is we're going to take that coin back in our pocket we're going to flip it three times so we're going to flip over that be thrice on foot price three times if you flip the coin three times what could you get well you get heads or tails for the first one right but then you flip it again what could you get it that one okay then you flip it again we should get up so an event is like this and that says what what possible combinations could you have those would be considered our events so one event would be I'm looking for one head and two tails that's an example of an event one specific outcome of your your procedure does that make sense to you so this says okay one thing that could happen here is I get a head and two tails now we're going to find out each of the simple events so what are the simple events that could happen what could you get when you flip your coin three times you could get three heads that's a good place to start so you get all three heads Hey what else could you get but you get tell sometimes okay that thing but that down here that could happen right these right now we're finding these are simple events they're single outcomes that we can get from flipping the coin three times can we get just a single head not if you well I'm sorry can you get just a single head and no tails not even flipping it three times right you're going to get three distinct things that happen a head and a tail tail a tail what else could I get okay so I heard two kids at would we need two heads in one table sounds a little weird I mean in order like you can get two heads in a tail several different ways one way could be if you go head and tail right give me another way head tail head that's another Google okay what about another one anything else that we could do starting with heads well you know what that was mean certainly heads anything we could do I'm going to race excited starting with heads we have Edvin head head tail head tail head head so these are all this all simple let's start with heads we can do head head head head head tail head tail head head tail tail let's do the tail ones we could have tail tail tail give me some other things that we could have here tail tail one up you could have tail head head you could have tail tail head tail head tail and we only have a tail tail tail would you look that over did I miss any did I miss any possible outcomes we could get are you seeing how we're getting these things we're just imagining flipping a coin right imagine flipping the coin three times you can get head head head you get ahead ahead in the tail it's easiest to do it that way look for me to sit there and think of all the heads and to think of all the ways you have a head first we head head head great head and then tail or head and then the tail in the head or a head and then two tails and that takes care of everything it starts a page then you go tail tail tail tail tail with an H then tail and a head tail tail and that way you know you have all of them up there you with me okay so what we've just done we've listed out all the possible single outcomes up here you with me on that all the possible single outcomes those are all the simple events simple events mean a single outcome so what we've done here that's a simple event that's a simple event and so are the rest of these all eight of these things are called simple events if I group them all together like I've done what we have is called the sample space the sample space is a collection of everything that could possibly happen bridging out you're with me on that now let me because the biggest thing for people is like well what's the difference parents implement and an event isn't event the same thing the answer's no an event is it doesn't matter how you get it that's what I'm looking for so for instance our procedures flipping ahead or flipping a coin three times our event we're looking for is getting one head and two tails that's one of the things that we could get flipping the coin three times right this is one of the things that you get here is all of our possible simple events or our sample space or each individual outcome those things are kind of synonymous but the collection of simple events is the sample space how many ways can this happen is there more than one way you can get one head and two tails how many ways hey six wins one head two tails does this have one head and two tails two nails does this have one head and two tails oh this one's one head to get out of this one that's one of them a little star there this one this one one head two tails this this one has two dads one time so here's the difference between an event and simplement an event says overall what are you looking to have happen but you're looking for ahead and two days simple events are the way that you can accomplish that event are you seeing the difference the event is what you're looking for simple events are ways you could accomplish that or not accomplish that they're all the specific outcomes so how many ways can we accomplish our event there's three ways three simple events will compass our main event does that make sense to you so it's a little little tricked up gs have any question at all it's a little tricky sometimes if you really don't get the the whole concept so are there any questions on what we just talked about procedure that's kind of basic that's just what you're doing events are what you're looking for simple events are how you accomplish your events there are individual outcomes some of them are going to accomplish your events some of them obviously are not going to accomplish your events let's try one more I want you to do it give me another event that I can add with flipping a coin three times it's another bad what could happen or you can anything else happened besides one head in two tails okay give you one what no one tail one tail two heads okay is there any other events that I could have what's that three days how many tails well not because all it is and the last one we could have is what so these are all examples of events we have here's an event here's another event these are the last two events there's really nothing else could happen right notice how many individual items we have so there's more individual outcomes than we have total events because some of these overlap this right here that I starred that's three ways to accomplish this one event how many ways can you accomplish this event can you see it how many ways where are we finding that out look over there how many how many times do you get one tail and two heads here's one tail and two heads that's extreme one here's one hit Taylor two heads there's another one that's one two three sweetly things that's Kali things so the three sweetly things that accomplishes this event three single outcomes three simple events would accomplish this event true okay how many ways can you accomplish this event there's only one property date how do you think does this one and circle name is that one I'm just rounding I mean this this is the relationship between simple events and events events are the overall thing you're looking for okay that's it simple events are the individual outcomes that you could get from your procedure some of those simple events are going to satisfy your event maybe only one may be up to three maybe more than that if we were flipping a coin four times you have lots of outcomes that satisfy your event do you understand relationship between procedures events and simple events and sample space sample space another problems to collect all individual civil outcomes and that's or supplemented so now that we understand that we can really use those words to kind of describe some probabilities so let's do that right now when we say probability in this class we save possibility we're talking about the likelihood of an event occurring the likelihood of an event occurring notice I'm not saying the likelihood of a simple event although sometimes those might be one of the same if there's only one possible outcome that satisfies your event then the probability is one the same but when we talk about probability we're saying the probability that your event happens or the likelihood the likelihood of an event occurring we're going to use what letter do you think we use for probability geniuses every one of you what it was like our one should be confused no be you're exactly right so probability is P events are usually listed with capital letters so if we're talking about event a we're just going to say a so a could be flipping a coin three times and then whatever you're talking about so we can list it you can even list it in words you don't have to use the letters but if we're talking about an event so for instance event a or you could write flipping a coin three times over anything like this BC etc if we're talking about the probability of an event occurring the way we write that as we say probability of a that doesn't mean multiplication it's not like algebra it says probability of a it's more like a function notation if you want to consider into something you're finding the probability of this event happening basically and so this means the probability of event a actually occurring now when you talk about probability we actually have types of probability we deal with and you deal with this on a day to day basis I really you do when you think about it you'll probably notice this when I'm going through it but there are three types of probability the first type of probability is what you get when you actually perform an experiment it's called observed probability so observe probability happens when like you took your coin and you flipped it a hundred times and you calculate how many heads you got and cousin how many tails you got from there you can actually mathematically figure out what's the probability of getting a head cuz maybe your coins waiting a little funny do you see the Olympics right there that's observed probability when you actually do something and you get a probability from that observation you follow so observe probability its probability that is estimated based on your observation probability that is estimated an estimated wait a second why estimated why isn't the exact can you ever do a procedure for so long that you that you've accomplished all empresa different way that is emitting is inflow right can you ever perform a procedure so many times that you've exhausted all the possible times you can do it for instance could you flip a coin until you can stop flipping it claiming work can you do that forever so can you calculate the probability if you can't do it forever exactly the answer's no you can't flip a coin enough for you to have an exact probability all you can do is say maybe I flip it 100,000 times is that enough to get the probability of flipping the coin answers pretty close but no not exactly I mean you're not going to get the exact probability of flipping a coin by doing observations one that's what's called observed it's estimated you observe it for a certain number that you decide on say I will flip the coin 100 times and after that I'm going to calculate the probability to be estimated it's not gonna be exact because I can't flip that coin forever I don't want to trip up in a certain number of times to make sure that I have at least a good sample of outcomes there does that make sense so we can do it forever that's why it's estimated and fortunately it's not too hard to figure out if we want to find the probability of a here all we do is we take the number of times a occur divided by the number of times you perform that procedure so number of times a occurred over a is your event number of times they occurred just divided by the number of times your procedure was repeated so the number of times you did that thing I'm going to give you all three of these and then we're going to give some examples so we can calculate these things so first what observe its you're actually doing something you're actually going out there and flipping the coin or going out there and taking a poll or going out there observing what someone's doing and that's your basic probability off of that okay a perfect example for this if you really want to right now so you're not really quite clear is too much baseball do you know what baseball is okay so you know these guys up there with the sticks they swing right if this is a white thing coming ten all right sometimes it hits them and they get mad they got hustle hustle haven't earned that while they tussle a little bit so baseball is all about statistics right I'm here statistics on baseball players all the time if you're into sports or you watch sports center for like five minutes are always talking about baseball at person handling baseball but if you play whatever that's cool so but they're always talking about the statistics and so if someone has a batting average of 100 would you expect them to go through the ball do you think that an average of 100 means one out of every ten ten times they're going to hit the ball is that good is that bad a batting average of 400 is excellent okay bad average of 100 really sucks it's they're not getting them all but that that right there with they're getting that batting average of 400 or 450 or 333 any of those those decimals that you see on the back of a baseball court if they're talking about that that is an absurd thing right what they did is they said oh how many times have you hit the ball eight times how many times are you up to bat 24 that means that eight times out of 24 times you hit the ball that's 33% four point three three three that's how they're calculating that that would be an observed probability because later on they're gonna say oh you usually hit the ball eight times out of every twenty four times right are you automatically going to be coming that's huge success in it every single time maybe but probably not probably you're going to stick with those odds that's observed probability and how you use it does that make sense to you so it's what someone's actually done and then you take that and you estimate it and you apply it towards towards their future say if you hit the ball eight times out of every twenty four times chances are you're going to probably continue that statistic so when you come up with that next you get a one-third chance of hitting the ball that's how you use observed probability letter if you're armed with that so observed is something actually happened you measured it the next one is classful probability the next one is what I say to you and you answer me this question I say what's the probability of flipping a coin and getting ahead okay you have to answer to play along here what's the what's the probability of you flipping a coin and getting ahead obviously right there's two choices one of those switches is ahead so you 50% right what's the probability of willing to die at one time and getting a to fly one out of six and how many choices are toons that's how you didn't want on six right that is classical probability are you actually rolling the die to figure that out your head you're just thinking about it right you're thinking oh obviously there's six sides only one of them's a two so a one in six chance you're doing classical probability there notice the difference between observed where they actually calculated how many times you hit the ball divided by how many times is up to classical classical is a theory observed probability is the actuality the classical is what should been observed is what did happen do you see the difference theory classical is what should happen when you flip the coin you should get half heads half tails if you flip a coin ten times are you gonna for sure get five heads and five tails if you think so I'll make you a bet right now and make a lot of money with you that I can flip the coin rarely is it going to be exactly five heads rarely you're rarely going to get that I mean well not very many figures of the time a lot of money from you if we make that bet every single time over it over again so you're not going to get exactly five heads every single time the second happen sometimes you'll get six heads out and sometimes you get nine Spears equal ten sometimes get one but that's okay that's the that's a classical probability as opposed to the observed classical is what should happen every time observe is if you actually do the experiment what does happen every time so let's talk about classical we can pretty much just discuss dude this is the probability based on the chance of something occurring this is this is a theory like the theory aspect of photo by the way for classical probability to work each event has to have an equal chance of occurring inch simple event has to have an equal chance of occurring make an example about this okay let's say that you had because this statement people are like well why why does it have to have an equal chance memory think about that so let me give you a die and I'll tell you it's a way to die okay it's a way to die what's the problem zero weighted diets but die in a corner so it comes out certain numbers differently using the Vegas sometime of the use to his seventh all the time I haven't ever do anything like that the simple event you must have an equal chance McCurry means that if I give you a way to die and I say what's probably going to you can't say one sixth anymore because well you don't know you don't know what the weight is so in order for you to do the theory approach to something that has a chance of occurring you have to have an equal chance there right the only way you were able to figure out one sixth earlier when I said what's probably going to is because you thought that every side has an equal chance of happening right that's why you did that that's why when you said well it gets ahead fifty percent of time when you put the coin once because you figure heads and tails has an equal shot don't you that's what classical is based on it's based on every simple bit that's an equal chance Maternity now the way that we did this you've already done it you know classical probability intuitively that's when we talk about most of time looks really similar it's just that instead of number of times a occurred we say the number of times a could occur or a number of ways I guess divided by the total number of possible outcomes again it's a number of simple events and we just mean how comes there because we've kind of covered that at length right now I need to recap this a little bit before we go any further so you really need to understand the difference between observed probability and classical I'm going to ask you on your test would give you a problem saying what is this calculating probability tell me if it's observed for class well that's going to be like three or four files on your test so you need to be able to identify are you doing something or are you just thinking about it that's the difference if you're observing something or someone has observed something that's observed probability if you're thinking about how many times could you get a – if I'm willing to die if it's something like that where you're actually not doing anything you're just thinking about doing something that's the classical so what right up here is observed and classically this is what could happen this is what did happen you know what let me replace could would should this is what should happen not good this is what should happen this is what you did happen let's not only use example but if you flip the coin 10 times what should you get you should get five heads five tails if you actually did it are you going to get five head spot tails maybe maybe not if you do the observation you might get six heads and for tails that's what did happen so that's the difference you can do the same you can think about the the probability it should be five out of ten you can do the probability it might not be private of ten those things could line up but they don't have to put the act of doing that procedure that's observed accompanying it that you're just thinking about it and figuring out what should what should be events the class will humble understand the difference okay the last thing we have to have to talk about is called subjective probability now before you say well that has no place in statistics why real and subjective probability that's subjective when you've been talking this voice electives some very well subjective puggly is something we do every single day you go to your doctor and you go doctor what are the chances I'm going to make it and he goes 80% does that mean out of every 10 people that he's worked on two of them died you know it just means his best guess for your particular situation is get a pretty good shot to make it it don't worry about it 8% pretty good right 20% only one shot on the fight that you're gonna clever you know we take chances but anyway that's subjective probability how about this one one of the chances right now that I'm going to walk out that door you might want this to happen but my walk out the door get hit by meteor no you won't want that on me with you because you probably take it with me because we're a safe building so if I walk out the door or the chances I'm gonna get hit by a meteor 90 percent probably not um how many to lose it huh I mean is it zero is there a chance any point zero zero zero forever in the little one and maybe but the point is that it's neither classical notes of jet subjected I'm not thinking in my head how many possible ways could I walk outside New Year right now well I'm not thinking I'm going to calculate how many ways I've walked out of this classroom and then how many times I've got hit by meteor and figure out what the percentage is right that's not what I'm doing this is not an observation I haven't walked out this time a million a room a million times and calculated all I've got in my meteor zero therefore the probability is zero there is a chance it's a very small chance but it's a subjective chance I'm just kind of making it up right best on based on my past experience and based on my educated guess it hasn't happened to me before I know that meteor circle around but none of them has ever even come close to me so it's probably close to zero but it's not based on any map it's not classical it's not sort of sheet of difference the doctor thinks probably the best one he's not basing that on map he's not doing the calculations you saying out there you got like a 95% chance of being okay or you get a 20% chance this is going to turn into cancer or something I mean that that happens all the time but people say that so that's the subjective type of probability it's someone's estimate based on an educated guess now let's go ahead and do some examples here and see what we can find out about these things whether they're classical or observe and then we'll calculate the probabilities as we go okay so first one the probability of selecting a part art like that like that not like the feeding heart under the heart shape from a standard deck of cards if they're shuffled up and everything random selecting so someone holds up genome cards some people invited it on and familiar with the cards cards have more suits diamonds spades clubs hearts there's 13 of each suit ok so there's 13 cards of certain clubs that can spades 13 whatever I did say and there's 52 total cards right cards are labeled 2 through 10 then you have Jack Queen King and ace making up 13 individual numbers for each suit of cards if you're not familiar with cards do some packet cards because I'm going to use that some of our tests to illustrate this so probably select a heart from a standard deck of cards so we want the probability of art let's find u symbols like that that's okay we don't have to call it event a in this class we say we want the probability of finding a heart I don't mean true love just kidding just kidding filter love congratulations I better not have a girlfriend watch this video so anyway we're going to count the number of parts there are divided by the number of total cards there are so how many parts do you have in the deck Hardison okay and how many Bogart's calculate probability how much is that would you get 0.25 cool which is actually the probability of finding true love in the real world that's courtesy I mean weird even the thing anyway so yeah there's a 25% chance of finding true love or a heart in a deck now is this classical or observed probability what you think did you actually go pull the card out of the deck then it's not served okay did you actually pull the card out of the deck did anyone pull the card out of the deck then you would talk about pulling several cards other deck can calculate no what you did is you said how many is there divided by how many total cards there are what should happen you should have a 25% shot but pulling out a heart from that day that's classical probability are you always saying that this is classful okay this is what should happen now let's say this last example for today you take a coin you flip a coin 100 times you happen to get 64 tails what I want to know is what's the probability of getting a tail find the positive in your tail here's how we do this with with observed or classically the one you find out the number of possible things you had so in our case how many times did we actually flip the coin how many tails did we find well this shouldn't be too hard to figure out what's probability there once you say here's you have a 64 percent chance of getting a tail now is this classical or just observed did anyone actually flip the coin yes they did that's absolutely definitely absurd because look at the difference here here it says you flip the coin 100 times you get 64 tails someone actually did that okay so it did something here didn't even do anything that's theory this is this is observed this is what actually happens so what this is the observed it's what did happen first objective use the doctor one yeah eh percent chance of being okay how many wanna show we've talked to us man okay I'll just show you my true love it does exist it's making a funny okay so we go home let's start crying or anything it's okay so as we're talking about last time we did some examples of how to do observe classical and subjective probability let's continue that so if you didn't know my favorite quarterback – unfortunately playing anymore I guess is Peyton Manning you know pignetti yeah right his neck or some digging out to see if he was just a real man just play with a neck injury and it's smart to do right yeah yeah yeah no I'm just kidding you never want to mess with nature so I've heard I guess it's important so next couple things anyway Pigman II when he first started elf I'm making a statistic up but he's pretty good so it's probably true completed 385 out of his first 528 passes what I want to know is find the probability that Peyton Manning's going to complete the pass using this information okay let's talk a little bit about the vocabulary of the statistic stuff is probability that we were talking about firstly can you tell me what the event is here what's the event what are we looking to have happen because that's our effect not just a pass or what about the pass okay completing the pass would be the event we're looking for what's the procedure listen what's happening here yes that's right that's the procedure he's actually throwing the ball to somebody that's a procedure the event is we're looking to see if he's going to complete a pass that's what we want to find out you guys okay on those those two things so procedures what's happening event is what we're looking to see find the probability of action encouraged so our men is concluding a pass by the way what letter stands for probability is where you use that's pretty clear so probability we're just going to write the event completing a pass now a lot of people if I ask them it or if you went on the street you said um can you tell me what's the probability of Peyton Manning completed paths this would be the same idea as if you win outside and ask some people really don't understand probability once a project is going to rain today and they say oh well 50% it's going to be 50% with this you completely completely pass or not because either he's going to complete it or he's not either it's going to rain or it's not that type of logic you see how that's kind of like false logic for what we're talking about that there's a whole bunch that goes into calculating whether it's going to rain or not today probably it's not gonna rain I'm not I'm thinking it's probably not fifty-fifty like is in a brain like half the time all the time that happen to make sense we know if it's like July 20th what's the probability it's gonna rain on July 20th in the Central Valley pretty close to zero yes what's the probability Peyton Manning's going to complete a pass well it's not 50/50 because he doesn't complete exactly 50 percent of his passes it ordered makes such a judgment you actually have to consider his past practices what he's been doing so that's where this information is going to come in you can't just arbitrarily say a percentage that would then not be classical or observed probability that would be subjective probability based on actually not an educated guess based on you not understanding the probability so I need to to kind of get away from the thought of if it happens or it doesn't happen that's automatically 50/50 to see how that's not always the case you sure okay well it'd be like this what's the probability I'm going to wear a dress tomorrow it's not fifty fifty bucks it is zero I mean 100 I mean zero it's Tuesday you don't know what I do on Tuesdays so what so I'm just joking I don't wear dresses only on Halloween once yeah that's the last day yeah take them over okay so instead of just going fall it's 5050 we're going to use information than I've given us and how again the probability is we calculated the number of times something actually happened successfully the number of times our event occurs that's what the more specific way to say that the number of times our event occurred divided by the number of times the procedure was repeated so how many times did our event occur here which was completed a pass right how many times did was the procedure repeated right when you're doing the probability give me three decimal places because we like to translate that to a percentage often and we want to make sure we have like 35 points something percent that's comedy what is it like that route it trickling yes only 70 knots point 79 then so 72 point nine percent so this or or that is that good now that's a judgment call right and when this was actually just calculating probability saying whether that's good or not that's a judgement call you say oh well that's good or that's not good what if someone completes 100% pass all the time then relatively he would be as good but 73 percent is pretty good for completing passes so lately are you guys all okay with them with actually calculating this probability now the question I have for you what you also need to know this is the problem like this is going to be directly under test just like that but then there's going to be a Part B and you have to answer whether this is don't say it out loud and what the people think about it whether this is classical or observed probability so think on that for a second is it classical or observed or subjective and here's the differences again to show you you kind of get this in your head subjective means there's no data whatsoever you just are making something up but it's based on educated guess like a doctor would when a doctor says you have a 90% chance of pulling through that would be subject subjected classical would be based on the theory like what should happen in this procedure or this for our outcomes observed is something actually happened you calculated it based on past incidents incidences of occurrence or past procedures so using that information is this observed is this classical or is this subjective definitely observe he actually threw the ball right if she did something that someone just wrote down every time what happen that is observed there'd be no way to do this classically because well really I mean if you think about it in order for you to do a classical probability the outcome has to have the same chance of success every time right every single time and we Peyton Manning throws the football sometimes it's like from you and sometimes it's from here 280 yards down the field this guy's got a rocket laser rocket arm I've seen commercial nobody this is like eight years ago I'm older than I get so anyway he's got a laser rocket on so you know so anyway he throws the ball there's there's less chest of that actually succeeding you can't calculate the chances every single time we can't even do this classically it's the only way we can do it is observed let's look at a couple more let's say I give you a deck card to help you guys loop through the deck of cards yet gets removed with the deck of cards hopefully your so given a standard random deck of cards let's find the probability of randomly selecting a two so we want to find here's how you would write yourself the probability of – what's our event in this case see that letter that's the event what's the procedure procedure what are you actually doing or you pretend you to do I guess your guinea pig in the card that's procedure pick out one card the event would be we're looking to see you what's probability find the to doesn't get between a procedure and it event okay so if we're going to do this we need to have the number of choices that are going to make our event successful or we're going to over the number of choices that we have total so what are the total number of choices we have four cards in a standard deck of cards now how many ways can we accomplish our event four ways or what is yeah because there's one two in each suit so that's four and we calculate 4 out of 52 now which is 4 out of 52 point 0 7 how much is that as a percent yeah is that good or bad let's get subjective right here I mean for you that might be low that's a fairly low probability of getting the two randomly out of 52 cards not like 50% searching about 73% like peyton manning doing football it's like a sure thing but random deck of cards you're selecting the tube now is this subjective probability I'll be just guessing you so definitely up that is it observed probability or is it classical what do you think why isn't it observed we can actually go through the motion of taking the cards out and say no we got it to put it aside I'll keep going up you got another two right you didn't you've yet you didn't do it at all it's not like big many right he actually threw the ball and you calculated that you didn't say oh I drew a card out and put it back eighty-three times and out of those eighty-three times twenty one of them or twos or something like that or five of them or two you didn't actually do a procedure here you just calculated the what should happen in your procedure did you see the difference here between the big man example where he actually did something in this example now could you turn a card example into an actual observed probability answer sure you could if you just took a deck of cards and did it you know so if I gave you this on a test and said okay a person drew out five cards with replacement from a deck of cards he got one two a jack a king another to ten ace what's the probability they're going to that you are going to pull out a 2 from this deck of cards you know you had two 2s and pulled out out of 52 cards that would be I'm sorry out of five tries so that would be your probability is the two out of five so it'd be how much you got out of how many cards you drew does that make sense to you okay so that's that's the difference here you can talk about the same question it depends on how this was actually accomplished whether they did the procedure over the top of the theory of it so this for sure for us this is going to be classical you know a while back someone did a poll on cloning back when stem-cell research was just kind of coming out of this a few years ago and stem cells people thought they're going to be using those from cloning and so they did this this poll on whether people thought cloning cloning people was good or bad so here's the results of that huh-huh so when they did this poll 91 percent I'm sorry about 9 1 % 91 people said cloning was really good idea because they wanted this extra person I mean she seen the I have you seen the movie the island great movie I'll kinda bout the cloning idea all I don't want to ruin it for you but it's about this ruin it for you 91 people said cloning was a really good thing Tony good 901 people say you know why does not run so sure about this cloning thing say cloning back the rest of the people had no opinion and because you're always going to get some no opinions in a group like in whatever little cares open the video games and you know so 20 people now maybe they should know they really don't have the information haven't really thought about that then you have the opinion if this was a random poll this should give us some indication about the general public whether you can go outside right now and ask somebody about cloning whether they think it's a good or a bad idea this was collected randomly in the methods that we've used or in his class remember talking about those like a systematic sampling or the stratified or the cluster sampling all that good stuff so let's pretend this was done that way maybe it was I really don't remember where this came from let's say it was it should give us some indication about everyday people so let's go ahead and find the probability that we can go outside right now and randomly select a person who thinks cloning is a good idea so we want to find we're going to use appropriate symbols here we want to find a probability that someone thinks cloning is good how in the world are we gonna figure this out how in the world firstly before being talk about that can we determine whether this is classical or observed or subjective is a subjective its objective now it's based on some data here so is this going to be classical or is it going to be the serve what do you think it's based on some something that actually happened right it's people went out there click the data polls hopefully this is polls open up polls are always observed because you're always collecting data right you're always talking to somebody that's that's observed you're observing what they're they're doing it's not classical is not based on theory it's what actually you collected so a poll is definitely always going to be inserted so write that down this is certainly observed probability holding the attention observed at classical is becoming really clear to you I hope that's happening now how do we calculate observed probability well it certainly is still division because it's our saw village are calculated we calculate the number of people a number of things that accomplish our event divided by the number of times a procedure was repeated so number of times that we accomplished our event which was cloning was good honey is that how much zidler anymore very good Cloney good feeling good now you want people many more people out of how many people wouldn't have to do find how many people is out of 91 is out 901 add them up add up these two good because even though nobody opinion people they still took that poll right they just didn't categorize themselves so we add all that up what up I got sweet that's how many people were involved in this poll in this procedure so we calculate 91 divided by our 1012 and to the third decimal place we get what is it point zero eight nine nine good the nine moves that nine up to a ten but it okay good it's basically nine percent so right now going out there is what this suggests is that randomly picking out a person you should have a nine percent probability of being somebody thinks calling this good thing so maybe that's higher now who really knows but this is oh this is an old poll that's how you would calculate such things which originally feel good what we talked about so far okay good that's fantastic are there any questions before you go on I have to race this side here yelling can't understand the whole peyton manning thing and some server because he actually did the passes the deck of cards we're not really drawn cards just kind of thinking about what should happen here that's our classical we have another observe any time you did a poll man but if they're doing the research that's definitely observed they're taking that information in hmm there you go find them find the probability that bird will poop on your car today if you wash your car and it happens right out there right up in the frigging are birds which other BB gun everybody anyway so find probably a virgin poop on your car today is that going to be a classical probability we're going to find it we think is there a way to tell how many ways this event can happen to me how many ways can spur poop on your car know everything flying it could like land the car get hit the windshield mule oh crap okay is it observed I mean you could you could talk about observed right if you had calculated how many times Birds had pooped on your car over the past whatever amount of days divided by number of days you have a probability there that would work have we done that so it's definitely a classical that's impossible because they all have an equal chance of the bird poop in your car every single day does it happen but what's it in the garage bird fruits on there I mean the unlucky that's really crazy it's happened me before actually bird was in my garage bird is on a berry anyway it's definitely not observed because we haven't really calculated this so the probability of a bird poop in your car is what is it for you whether it is subjected by me what's your probability for your car today would you say like ten percent sure that good thing I just put words in your mouth okay nobody ok what's the probably million Korean assumption not someone like that some some birds gonna undercard what would you think see for me it'd be like 50% we've done that's another thing about subjective right it can change person to person so if you can think of the probabilities and you say well for me that's 20% maybe your car never gets pooped on just like 5% for me it's like 50 to 70% always get some I'd park and retreat so I mean duh it's going to happen but subjective probabilities can do that right they can change can classical observe change now this is based on hard evidence this one was based on complete theory with jiton is not going to change okay so that's another way to kind of do this as well so find it probability this is stupid birds you know picked up in a car it's a it depends on who you are but this is certainly going to be subjective and it probably depends on where you are if you're partnered by the beach and there's lots of seagulls that yeah okay let's go ahead and do one more I'll give you a couple couple notes that are important for us and we'll continue to talk about some complimentary events what that even means let's find the probability that if a couple has three kids two of them are going to be voice now I also have to tell you that we're going to assume that the probability of a boy or girl coming out is 50% all right this is equal that's not always the case actually you actually do the observed probability girls right now have a higher chance of being born so that's I think there's like 51 percent women born 50 49 percent or something like that it's not exactly 50/50 but we're going to consider it for this exercise to be even is that make sense for you so assuming equal chance of moral some people chance for you to roll hey what firstly is our event having a kid first thing how many babies have you had we haven't just one can we ask the question if you have one baby how many ways can you get two boys if you have one baby and you cut them in half you guys are sick okay so firstly what is our procedure even what's our procedure what's happening here but now you can say it if you're wrong doesn't matter just you video recorded that when the world's going to hear it that's here so what is our procedure what are these people doing then making babies having babies making babies would be a different class so how many babies are there having I think one thing is only three that's our procedure the procedure is having three children deposition for cheezer's not just having babies it's having specific number of babies do you see the difference there you can't even talk about this if you only have one baby because you can't say out of having three children how many ways you have two voids if you're only having one baby you go within the category so our procedure right here you want to write that down the procedure is having three children now the event is based on that procedure what's the event the event is what you're looking for what are you looking for – two widow two bullets two boys and what else hopefully hopefully you get a girl and the f3 key I mean you're not just gonna get two boys or and nothing right ghouls count two guys well if you have two boys what's the other one primero a girl we hope it's going to be a girl so we have two boys and we have one girl we're not going to get three boys that would not be our bet so right here I guess I will write this down for you the procedure is having three children congratulations adding three children the event is getting two boys if I say two boys that means out of three children one has to be girls so we want to find the procedure I'm sorry the probability I'm sorry of our event two boys one girl oh my we have some other other words that we haven't talked about in a couple days now before we do that I do want to figure out whether this is going to be subjective observed or classical probability is it's going to be subjective probability we're going to be calculating stuff over here we're not just going 30% now we're not doing that right we're not basing it on it guess we're not a doctor we're going to be doing the actual either theory or observations here have we observed some people have we have served some people is this what we're doing do I have some data on the board or unit says here are 100 couples who have had three kids thirty of them have two boys have I done that so is this observed or classical do you think also this gives it away equal chance of being a boy-girl because in order to calculate the probability if it's classical you have to have the equal chance of something happening you can't do classical probability if you don't have that case okay if girls had a 51% chance of being born and boys only on 29% chance being born you couldn't do this classically okay you would have to do observe they have to have the equal chance like willing to die member talking about willing to die last time said it's a way to die all bets are off you can't do classical probability because it's not even you don't have an even chance of getting a one two three four five or six one reason you were able to come up with the last time as I said I said what's the probability of rolling a two you said L bones one six there's one two there's 6 sides therefore 1/6 that assumes that every side has an equal chance of coming up if that doesn't happen ie if this does not take place not equal you can't do this classically not going to understand that good now so we have our procedure we have our event we know this is going to be classical probability write that down if you want to this is certainly classical it is not observed we need to find out something called O or to find out what could happen what could happen is a whole group of outcomes is called our this is called fill so a full group outcomes everything that could happen all put together is called are see the low sample space does that weird ring a bell to you the sample space is every possible outcome you could get we need to list our sample space in order to do this classically because they have to know what can happen if you're willing to die your sample space is just easy it's one two three four five or six for this case though we're going to have some different different pens we have so the couple has three kids sample spaces have those funny-looking brackets let's list out what you get for three kids what's the person you could get or what's one thing you what should we start with I should soon all three boys okay great good boy then a boy then will with us okay good luck with that one what do you think it'd be tougher three boys two girls I think the girls would yeah personally boys are are just nasty gross people but girls can eat meat okay so three boys three girls what else could we get by the way I like to start off with this way and this way and then list out everything that starts with B everything certainty that way you don't forget anything so let's start with the two boys we got a bet we read that up there what's the next thing we could do that someone else give me another one okay give me another one boy girl girl anything else starting with a B no that's all okay so girls we need to do the girl girl boy we could do G B G and we can do gtp what have we missed it now before you start saying well mr. Leonard I mean aren't some of these the same like isn't wouldn't this be the same having two girls in one boy and then having a girl and boy a girl and no people I mean we have individual personalities right these are different people so if you had a girl first and then another girl and then a boy you'd have a different family than if you had your girl and then your boy then your girl wouldn't you completely family so these are eight different families that could happen with your situation if you're going to have three kids you with me on this okay so there is a difference there so if we have these eight different choices what we need to find out there's only eight different choices eight possible ways you can have three kids you agree with that right there's only eight possible ways all three boys and then all three girls and whatever you have permutation of those how many ways would accomplish our event which ones that's what I asked for oh this one very good yeah anything that has the two boys and the one girl because we didn't say what order right we just said ultimately to bottom row there's three ways that make it happen notice how it's certain I observed right we didn't make a family have eight sets of three children and then calculate which ones came out with two boys that just be crazy to be like on the kids oh my gosh I'm wondering about that 24 kids so eight eight ways we can accomplish our event out of eight possible outcomes remember these are called these things each individual one are called what what are these now these are probabilities Pugliese what we're calculating here this is our sample space the sample space is made up of every individual what type of event this is an event that's our main event made of end of the engine boat is right here and then we have mini events called sorts of s rhymes with ipil ipil i'm snipping my Apple there's simple events there they're an individual outcome so we need to get this down folks you know what a procedure is what's going on an event what your ultimate ly looking for simple events how your procedure can be accomplished and the way we find our probabilities take the things that accomplish our event divided by the total number of simple events that's what we have written and now give us a probability sound spaces this everything we could happen the sample space is made up of simple events quizzes so 37.5% so you know right now if boys and girls have an equal chance of occurring which they paid them it's really close so this is going to be very accurate for us if you ever go out there right now and have three kids don't do that without thinking about it you're going to go there have three kids you're going to have a 37.5% chance of having two boys and one girl you also have a 37.5 chance for some chance of getting two girls and one boy because there's three more of those probabilities can you find the probability of getting all three boys what's that one out of one at eight or all four-year-olds one and eight thankfully that that's a lower chance than two boys and one girl or two girls in one boy okay couple of milks for us before we go any further first even just a sense for you probabilities always have to be between zero and one you can't ever have a probability less than zero a negative chance of something happening what's the probability the rule of three negative to make sense all right so probabilities are between zero and one notes every probability calculated before we change it to a percentage was between zero and one can't be over one can't have more than a hundred said chance of something happening I know we kind of use that loosely in real life you go how much attention are you focusing on focused 110 percent or just a liar I think you focus hundred ten percent mathematically only you focus on our sentence makes sense yeah it's between zero one though so probabilities are always between zero and one and you can be zero what would a probability of zero be what does that imply about your event if you have a possibility of zero that would say that your event is impossible it's it'd be like this roll a die for me one time what's the probability of rolling a die and getting a rabbit go that's not that's not gonna happen right I'm not a magician but you just I am a musician I'm not a magician can't just make a rabbit appear from a dice I doesn't make sense so something that cannot happen is an impossible event with a probability with that in mind what's the probability equal to one imply we're done well this is possible that's not equal to one that's certainly possible right you can get two girls and a boy I say two boys and a girl you can get that so probability of one means it's more than possible it's certainly more than impossible Oda probability of one means more than just possible what's it mean it will happen it's certain it's certain if I say there's a hundred percent probability that you're going to have homework tonight that sucks huh that means that it's certain you were going to have homework tonight five percent probability P equals one means a certain event if you like this roll a die what's the probability in one two three four five or six you were going to give one of those numbers is certain also one other thing it's called the law of large numbers if you want to write down law of large numbers fill for this is what this makes I want you to think on this number flipping the coin well actually she's a contender flip the coin if we if you took a coin out and you flipped it ten times are you for sure going to get let's say it's a weight a nice even leeway to die so the probability of getting heads and tails is fifty-fifty if you flip it ten times are you for sure going to get five heads and five tails it's possible you get only three heads and seven tails right that's that's quite possible if you flip it a million times you're probably not going to get exactly 500,000 ads in five hundred thousand taels you're probably not going to going to get that but as you increase the number the observed probability is going to get very close to the classical probability for instance if you flip it ten times you might not get five and five if you flip it a million times it's going to be pretty close to 5050 you might get five four hundred and ninety thousand and five hundred ten thousand that ratio if you increase it to infinity observe probability will actually approach which means it's going to become classical probability so those two things will increase does that make sense to you the more you repeat a procedure the closer observed will be to classical theory you can see this in a poll under the polling that we did like the Dozen survey if you go out there to start only five people are they going to be very representative of the population of the United States of America it increases to a thousand is it more representative the increase of two three hundred million is it more representative that's like almost everybody you're like three hundred seven million people here so as you keep increasing your observed probability your observed results it's going to approach classical probability so that's a lot of large numbers as you increase our sorry the more procedures repeated the closer observed will be to classical publishing so the more procedures repeated the closer observe bundling we'll get to classical probability just swallowed large numbers more you do something the more your observations will mimic the theory or the more that what does happen will look like what should happen let me let me show what we talked about today it's any questions on pollinated stuff the law of large numbers or why probabilities are between 0 & 1 or why probably the 0 is impossible or 1 is it's definite that's going to happen or the difference between subjective classical or observed Pugliese you have any questions on those things or those ring a bell in your head does it make sense for you so when we say complimentary events what we're talking about in this class our events which are mutually exclusive have you ever heard that that phrase mutually exclusive you're heard of it number is this idea like it's a quickly mutually exclusive words are hard it says if you're in one group you're automatically discounted for being in another group you can't be in both at the same time have to be either here or you have to be here unless you really unless you're a strange dressing person you're either going to wear shoes or you're going to wear sandals right you're not going to wear both shoes and sandals at the same time I hope because that would just look ridiculous unless you deal with those kind of Teva look at things are kind of sandals – sandals chef would it be shadows at sandals shoes whatever anyway so you're not going to wear both the shoes and sandals at the same time right you're either wearing shoes or you're in sandals those groups or generally mutually exclusive so that's what that term means it means that you're either in one group or another there's no crossover basically so when we talk about complementary events but complementary events are our two events which are mutually exclusive I'll give you some better examples that relate to this classrooms a second by the way when you say complimentary I didn't spell it wrong it's not with the hi it's not like complement like you look nice today so these events are not saying they're going you're such a good looking event oh thank you event I feel like a good looking event today so I appreciate that compliment it's not that type of compliment it's it's this is the definition of their mutually exclusive one doesn't happen while the other one happens so they cannot happen the same time so complementary events these are events which are you to you Julie X will have a hard time I mutually I have sighs promise it's mutually I can't say that word just not today mutually exclusive the most basic definition I can give you from you to exclusive is to events which can't happen at the same time let's talk about just a basic example that okay let's bring back our dice the six-sided your substandard standard I okay I'd say okay I want you to roll the die can you get both a two and a five when your mother died once one time can you get both a two out of five those would be mutually exclusive events one of it would be willing to to the other that would be rolling five they've obviously cannot have at the same time when you're willing to die one time that would be mutually exclusive okay same thing like drawing out some cards drawing out the heart and drawn out me diamond if those are your events would be mutually exclusive events they won't happen the same time remember we talked about one event one procedure at a time not like draw three cards you can you get both the heart and a done yet you could in that case but for one card those would be mutually exclusive others you can finish in that they concept okay so what is a compliment for some notation if we have some event so let's say we have event paid the compliment of event a couplet of n a is denoted it looks a whole lot like me it's not but that's how we write the compliment you shall say this if we're talking about the compliment the compliment of something is a complement of an event is all the outcomes that occur that don't accomplish your event I'll repeat that for you so if we have a vente over here and we want to talk about the complement this is called the complement of a what this says is this is all the outcomes which don't satisfy this event does that make sense to you it's pretty much everything else that's what the complement is so the complement of event a is is denoted complement of a and is all the outcomes when a when event a does not occur does not occur for some reason this helps me to remember I don't know why the assessment memory but maybe this will help you remember it when you see this it's kind of like a minus sign – to me means not or bad not so if this is our event a this means not a so everything else besides a all the outcomes that don't make de you with me on that that's how I remember it don't know if that helps you hopefully that does so let's do an example let's say that my event let's go back to the the dice rolling thing okay the event is we're going to look to see if we can roll five so rolling a 5 that's orbit so if we call this event a so battles are a the complement would be a route that line on top of it or the complement of it what is the complement of rolling a 5 on a diet what do you think compliment of rolling a 5 what else couldn't happen to my answers question what else could happen you will die that doesn't make a 5 what else could you get basically could you get a 7 that's one time did you say what else could you get insights of 5 so anything besides the 5 in combat y'all stated 1 2 3 4 6 perfect so the complement of going to 5 is not going 5 or rolling not 5 for instance when yo people y'all stated here 1 2 3 4 & 6 that's a compliment so the compliment the complimentary events here work so that they add together to create the whole sample space so if you're talking about two complementary events it's got to be either one or the other the mutually exclusive but together they make it the whole thing can you get anything else besides a 5 or a 1 through 6 at some other complementary because together they make up the whole sample space right you can't get a 0 you can't get a 7 or anything else this is everything it could possibly happen they're just in two groups complementary events you have the five you have everything else that's the compliment of going to five we understand the compliment feel okay about that so far good now let's talk about the probability of these things so what's the probability let's say when I say five I'm a rolling a 5 ok what's the probability of rolling in 5 how many outcomes are going to let us accomplish our event of rolling a 5 how many outcomes let us roll with 5 how many files are on the die so there's only one specific outcome using a lot accomplished this particular event how many choices do we have so our probability is going to be one out of six you have millions can you tell me let's think about this if you have two events which are complimentary which means you're either in one event or the other and that takes care of everything that could possibly happen through what does the probability what is the probability of the complement of five or not rolling five have to be without even looking at how many choices you can people to figure this out can't we because you're either going to be here or you're going to be here so once you tell me if this is one six what does this one have to be for sure great how much do you think that a new probability of an event plus the probability of the complement of that event has to add up to all the time we think what is it going to add to the sum sure what's that some have to be he think was one yes you add those probabilities should you get one which stands for a 100% of everything right because you're either here either here or here you're not you're not anywhere else so if you add those probabilities together of the event plus the complement that accounts for everything that could possibly happen so there's your 100% certainly going to be in one of those two places does this make sense to you so probability of not going to five and you can see it I mean there's one two three there's five choices that you could have for not only two five or six possible choices we get five six seven five and we'll write that little note the probability of an event plus the probability of the complement of that event it has to equal warn all the time the probability of an event plus the probability of the complement must equal work it more basic terminology if you have a probability of some event plus the probability of its complement but you got one you we're going to kind of revisit this towards the Latin the latter part of section 4.3 this isn't kind of going to come back at you but if you understand it now that you're headed game do you understand why this takes place here if their view too exclusive you have to be either here here you can't be anywhere else so you add those probabilities together that accounts for everything you have a 1% probability that you're going to be in that net range do you guys feel good about the section 4.2 that we've talked about so far fill right with that again fun yet just lines I guess is it awesome so glad I'm here on Wednesday aren't you well let's see see our four point two we're going to go ahead and start four point three now

How Not to Be Wrong: The Power of Mathematical Thinking – with Jordan Ellenberg



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The maths we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. Jordan Ellenberg shows how wrong this view is through stories that show the power of mathematical thinking.

Buy Jordan’s book “How Not to Be Wrong: The Hidden Maths of Everyday Life” –

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Maths touches everything we do, allowing us to see the hidden structures beneath the messy and chaotic surface of our daily lives. Maths is the science of not being wrong, worked out through centuries of hard work and argument.

Jordan Ellenberg is a professor of Mathematics at University of Wisconsin, and the ‘Do the Math’ columnist at Slate. His book ‘How not to be wrong: The hidden mathematics of everyday life’ was be published in June 2015.

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Well, thank you so much everybody for coming out and being here. As a math professor I'm… it's always a treat when people are coming to listen to me talk about math And they're not required to, as they are… as they are in my class. So, let's see, um The book "How not to be wrong […]", and in some sense it is a book of stories — stories about math And I'm gonna just tell a couple of stories tonight, one short and one long. And let me start with the short story It's just one slide long It's a story about a mathematician named Abraham Wald who was a mathematician — he was born in Hungary, Works in Austria, has to flee Austria when the Germans take over and ends up finally uh in New York City, a professor at Columbia. And during the war he participates in an outfit called the Statistical Research Group, the SRG… I actually had never heard of this before I learned about it for writing this book, but it was a top-secret installation where some of the top Mathematicians and statisticians in the United States Were working in the neighborhood of Columbia University in New York were working on problems related to the conduct of the war, problems of a mathematical nature So it was kind of like the Manhattan project except it was actually in Manhattan and one day One day a group of generals came to the SRG and they came to Abraham Wald with a question They said "We have a math problem for you We have noticed that when the planes, when our bombers and our fighters come back from flying missions over Germany They kind of look like this. They're riddled with bullet holes But the bullet holes are not arrayed evenly over the planes that are coming back. That's what we've noticed There's more holes in the fuselage of the plane, there's fewer holes in the engine, different parts of the plane have different number of bullets and Here's what we want to know: We need to figure out how to put the armor on the planes." This is a serious issue, right? Because if you put too much armor, the plane doesn't fly, and that's bad. If you put too little armor, the plane gets shot down, and that's bad. So they came to the SRG, the generals did, and said "We need to know How much more armor should we be putting on the parts of the plane that are getting hit more? Is there some kind of formula for this? Is there some kind of math equation that you math guys can do for us to solve this problem?" And here's how the story ends. Wald says "No." He says "You have it completely wrong, You have to put the armor where the bullet holes are not. You have to armor the engines, not the fuselage. Why? Because it's not that the Germans can't hit your planes on the engines; It's that the planes that got hit on the engines are not the ones that are coming back from the missions." So is this a math story or not? I say that it is, not just because Abraham Wald was a mathematician But because I think in some fundamental way he was really thinking mathematically And the story is there to tell a lesson, a lesson that to be a mathematician is Not just to compute a formula, right? A machine can do that, even in the 40s, like, machines could do that… The purpose of the mathematical way of thinking is not only to answer numerical questions, but also to ask the right questions to interrogate a question that we're faced with and ask what assumptions underlie it and if those assumptions are reasonable. And sometimes to overturn the question entirely, as Abraham Wald did That's part of the case for mathematical thinking Now that story it's kind of like a Little tiny story with a punch, right? I could sort of tell it in five minutes I tell it in a few pages in the book. One of the great things about writing a long book And it is kind of long– Maybe I shouldn't admit that… Penguin did an amazing job actually. I was just talking with Eddie about this of, like, if anybody has it They made it look like it's not very long. It's very, it's a very impressive piece of book production. It is kind of long actually Anyway, one of the good things about writing a long book as opposed to Writing, let's say, a thousand words in a newspaper or magazine is that it allows you to stretch out a little bit and draw out connections because The fact is that's the way mathematics is Right, I mean, some things in mathematics are, like, these little instant punches, a moment of insight, an a-ha moment But most of the body of mathematics, what's exciting about it is the way that a lot of different ideas are connected under the skin and if you really want to follow The connections out wherever they lead You need a little more space, so I'm going to tell a longer story, one that Actually, I'm going to probably take about 40 minutes to tell a st–, to tell maybe, like, a third of a very long story That's uh, that's in the book And the story has to do with the lottery in the U.S. state of Massachusetts What you're looking at here is a picture from the very last drawing of a lottery game called Cash Windfall in Massachusetts This is a picture from January 2012, um, and the point of this story is to explain Why this was the last drawing. in order, but in order to do that, let me start with a little kind of a starter about how lotteries work in general. Okay, so this is how a lottery works: You pay a small amount of money Let's say $2 and I'm going to give a little simplified version to start with You pay a small amount of money for a small chance of winning a large amount of money, so for instance Let's say a lottery ticket cost $2 and maybe there's a one in two hundred chance that you win $300 back or should I have changed the currency for this? That would have been Thoughtful… Sorry, I didn't think to do that. Okay. Well the story takes place in the United States So if you play this game a thousand times… And people who play the lottery do play a thousand times Oh people really like to play the lottery a lot. If you play this game a lot, if you play it a thousand times Well, how many times are you going to win? Well, of course, There's variation in that, it's random but if there is a one in two hundred chance of winning in your thousand plays you'll probably win about five times, right? So that means you win five of those $300 prizes or $1,500 Sounds pretty good… Until you think about the fact that you spent two thousand dollars in tickets To get that $1500 in winning… So the sort of term of art that mathematicians use to talk about this kind of computation is "expected value" so We would say the expected value of this ticket is a dollar and fifty cents. That is how much you're going to win, on average, per play So that's the mathematical term. I got to say though, that It's sort of a terrible term, it's one of those that we wish we could take back We can't, the notation is what it is, but It's a bad piece of terminology because the expected value Whatever it means, it certainly does not mean the value we expect that ticket to have. In fact that's not even a possible value for the ticket to have, right? That ticket is either worth nothing or it's worth $300 But it is definitely not worth a dollar and fifty cents, so somehow if we had to do it all over again We would probably call this the average value But it's a much more reasonable summation of what we're actually trying to describe. A dollar fifty is how much the average ticket is worth. I'm going to have the average ticket cost two dollars. In fact, all the tickets cost two dollars and a fundamental rule of thumb Is that you shouldn't pay two dollars for something that's worth a dollar fifty and here you have, in a nutshell, the mathematical case against playing the lottery. And now I'll complicate that a little bit so what you're looking at here is an actual list of payoffs for a lottery game in Massachusetts, the regular state lottery. The exact numbers are not important but I want you to look at this computation at the bottom you don't have to check it for yourself but what I want to point out is that The example I gave at the beginning, the simplified example was unrealistic in a couple of ways: One way it was unrealistic is that there was only two classes of prize, real lotteries are not like this, real lotteries Have a lot of different prizes. They have a big jackpot that you get if you get all the numbers correct But that jackpot is really hard to win and it's kind of demoralizing for people If there were only the jackpot people probably wouldn't play, right? Because they probably wouldn't feel like they could win so Jack said real lotteries have, like, a whole sequence of Lower-tier prizes, some of which, like matching three out of the six numbers in the Massachusetts lottery, are really not that hard to win at all, the payoff is kind of low, only $5 But you have a one in forty-seven chance of winning. That means if you play a lot, if you play every day, probably, every so often, you're going to win And it's quite common for like a friend of yours to win or somebody you know to win So it keeps people playing, right? Running a lottery has a lot to do with psychology So that's one way that my simplified example was unrealistic: too few tiers of prizes; and the other way my My simplified model was unrealistic is that it was incredibly generous to the players No real lottery pays back a dollar fifty For every… as the average value of a $2 ticket. In this Massachusetts lottery the expected value of a $2 ticket was just 80 cents That's a lot lower. In fact, I was looking, actually, at some lotteries that are played here. Has anybody played EuroMillions? That's a 40 cent expected value on a 2… Rough, well 0.4 of a Euro expected value on a 2 euro ticket. That's insanity! That's what, this would never be tolerated the United States. I just want to tell you guys that. Even American lottery players would be, like, "That is a stinker of a game." people who, like, play this every day would, like, not play EuroMillions. Ok, so as I said, the the reason you don't have a jackpot– don't have just a jackpot is because it's demoralizing when nobody wins; if people are not winning the jackpot, people start to get depressed and people stop playing. And this is what happened in the state of Massachusetts in the year of of about 2004 / 2005… A whole year went by without anybody winning the jackpot and they could see at the lottery commission that people were stopping playing the game People were depressed, they didn't feel like there was a chance, and it wasn't working. So they said we got to make a change, we got to do something to goose interest in our game, so They instituted a new rule, a rule called the roll down rule. Let me explain how it works They said instead of just letting that money pile up in the jackpot, right? Because if nobody wins the Jackpot the Jackpot pool gets bigger. That money that is not given out in prizes. just kind of makes the jackpot bigger and bigger and bigger They said okay that is not satisfying people, because they feel like that jackpot may get bigger and bigger and bigger But I'm never going to win. They said let's make a new rule if that jackpot goes over two million dollars, and nobody wins the Jackpot that drawing then it's going to roll down All that money is going to roll down into the lower tier prizes and make them bigger That's exciting. That's maybe a good way to get people interested So they were trying to design a game that looked like a better deal for the player And in fact they did their job a little bit too well I can always tell how Mathy an audience is by how big a laugh I can get with a table. That's always um So this is what the payoff Matrix for um for the Massachusetts cash Windfall lottery looked like on February 7th 2005 um so for instance that four of six prize… There's a one in eight hundred chance of winning– you remember in a usual drawing that was a doll $150.00 prize… On this roll down day, in which no one won the jackpot, that prize was actually worth almost $2,400 so stop to think about that There's a one in eight hundred chance of winning and the prize is worth twenty-four hundred dollars So if you bought eight hundred tickets you were probably going to win twenty-four hundred dollars right there For the sixteen hundred dollars you spent buying eight hundred tickets, and that's just the four out of six prizes right there's other prizes too, each of which has some value and when you add it all up You find that the average value of a $2.00 lottery ticket sold on this day Was five dollars and 53 cents So that is not a bad investment! So how do I know, by the way, exactly what the payoffs were, for this– why do I know what payoffs were for this particular day of the Massachusetts State Lottery? I know it because I read about it in the following document. Which you shouldn't be able to like read from where you sit, but let me tell you what this is This is a 25 page letter from the inspector general of the state of Massachusetts to the state treasurer Trying to explain what had happened to the state lottery. And I got to tell you guys… I feel safe and saying this is the only Fiscal oversight document by a municipal official that you will ever read that makes you wonder if somebody has the movie rights to it! It really is kind of a crazy story, which, again, in the book I tell at length; here I'm going to tell it to you somewhat briefly um What happened? Well, what happened is that On February 7th 2005 um The state lottery started getting phone calls, they got a phone call from a Star market in Cambridge, Massachusetts. (which was like a convenience store) Saying some college kids just came in and want to buy 5000 lottery tickets. Is that okay? So there's a rule, you know… if somebody who as a single buyer wants to buy a lot of lottery tickets They have to call the state lottery and make and get a special waiver But this this is granted and by the way this was not the only place there was similar large buys In several places around the state, but what was going on? What was going on is that There are a lot of people who could make a table like the one that I just showed you… And some people did. so for instance one of the main players of the story are two– It's a guy from MIT called James Harvey. He was a senior at MIT at the time and And as it happened He was doing an independent study project in January 2005 on the expected value of State lottery tickets… He was a lucky guy! And as part of his project, um he computed the value of like all currently running, Massachusetts lottery games and Presumably he drew a table very much like the one I just showed you, and the first thing he did was go around to all of his friends in his dorm at MIT and Say look you should really give me all the money that you have right now so I can go buy lottery tickets with it um and if you go to MIT I think Everybody in your dorm can also compute that table and see that that's actually a wise idea and so they sort of coalesced their money and and bought– and bought all those tickets, in Cambridge. There was another group, called the Dr. John Lottery Club, which was based around biomedical researchers at Northeastern University, which is also in Boston and then maybe my favorite guy Was a guy called Jerry Selbee, who was a retired engineer, in Michigan… How did he ever hear about it? well… Where did Massachusetts get the idea for this roll down rule? They got it from a roll down game, in Michigan, that had just closed I'm not sure they asked Michigan why it closed… But Jerry Selbee knew why it closed, because he had made about two million dollars off the Michigan game Over the previous seven years or so So he was– he could not believe his ears when he saw that Massachusetts was opening the gates again So he immediately got in the car with his wife and drove like the nearest point in the state of Massachusetts to Michigan But that's– that's pretty far, guys I don't know if everybody like knows the location of all the US states, but that's probably about a 14 hour drive. I'd say And he made a big buy up in the Northwest corner of the state and kept on doing this; so um The story that is outlined in this long 25 white page document is the way that these three groups of high-volume players Continued to buy more and more lottery tickets taking their winnings and plowing them back into the investment scheme and buying yet more until, by the time this reached some kind of equilibrium, just to give you some sense the Inspector General estimates that, on a given roll down day, somewhere between 80 and 90 percent of all tickets sold in Cash Windfall were being sold to a member of one of these three groups um So how does this story end? um Well, it ends like this. This is the front page of the Boston Globe in summer 2011… At some point somebody figures out this is going on, the Globe gets tipped off. They run this story explaining what's going on with the state lottery; and at this point the game is up. Right? Once people perceive that the game is not what it seems [uh] then people stop playing and then it doesn't work anymore, then no more money flows into the system. So… in a way, that's the end of the story but if you read it from a mathematical point of view the chronological end of the story Is not really the end, right? Because, as mathematicians, there are some puzzles that remain… At least they remained for me when I was like reading this story and I was trying to read it with a mathematician's eye and try to understand what really went on here So I'm gonna spend the rest of our time together talking about two mathematical puzzles That I think we're left with, having told the bare-bones version of this story… One is easy, one is hard… Let's start with the easy one um the first puzzle is How could you actually get away with this? This is a little weird right let me remind you that the state knows who's winning the lottery Right? Because they have to give you the money, so it's not a secret The state knows that all the winning numbers are coming from the same three convenience stores again and again Let me remind you something else; if you guys that were paying attention to the dates When's that first roll down? February 2005. When's this article in the Boston Globe? July 2011. So there was time to figure out, but something was amiss. This is six years, we're talking about um So this is puzzle one. How did the state not figure out what was going on? well… This is the reason this puzzle is easy– No, I like bureaucrats, okay… um So the reason this puzzle is easy is is the following Here's the answer: the state did figure it out, and how do I know this I know it because it's in the inspector General's report and in fact I slightly lied to you I said that when James Harvey figured out the new Massachusetts lottery game had a positive expected value, I said the first thing he did was Get money from all of his friends in his dorm and go buy a lot of tickets But no, that is the second thing that he did. The first thing he did because kids who go to MIT are like Good kids who like play by the rules, and get good grades, right? The first thing he did was get on the subway and go to Braintree, Massachusetts And go to the state lottery headquarters and have a meeting with them And he said "Look, your new game has a positive expected value… I'm planning to buy thousands and thousands of tickets and make a lot of money… Is that legal?" And the Inspector general does not record exactly what response he got to this query But it must have been something like "Sure, knock yourself out" because the next thing that happened was what I just told you and then It went on happening for another six years Okay, so that's the answer to that puzzle, but that answer kind of spawns another question as so often happens so the ques– the– it spawns the question why didn't the state do anything about it? if the state knew from day one that this was going on? okay, so to answer this question I need to use a very sophisticated mathematical diagram which represents the limit of my powerpoint skills… so what's actually going on here? "Random Strategies", I should say, is the name of James Harvey and Yuran — James Harvey's team It's the name of his group of people um You might object that their strategy was really not very random at all. What's ac– what's actually going on here is that the dorm they lived in was called Random Hall Which is a place at MIT and that was where the money was coming from. So, um… How should you think of what's going on here? Well I want to remind you one more very important thing about how the lottery works, which is this: when a lottery ticket is sold for $2 Massachusetts takes 80 cents of that money and that's state revenue, right? That's what goes to pay police officers and pave the streets and keep the lights on and do all the things that the lottery is intended to do And then the rest of that money is eventually going to get disbursed in prizes in one form or another So what does that mean? That means that from the point of view of the state The amount of money it makes is 80 cents times the number of tickets sold; the state does not care who wins the lottery! The state only cares how many tickets are sold. so this is a crucial point because when this story came out of the newspaper, I think was presented as That these folks had somehow Cheated the state out of a lot of money; in fact the inspector general estimates that the state of Massachusetts took in somewhere between ten and fifteen million Dollars extra revenue above what they would have, had these three large groups of betters not existed… So I think it's safe to say that somehow if you come away with an eight-figure win You are not the person who got scammed So what's going on? Where was this money coming from? Well, of course, the money was coming from the people who were playing the lottery on the non roll-down days so that's what this [uh] That's what this figure is meant to emphasize You should think of what was happening as a movement of money from all the regular players To these groups of people who were playing only on the roll down days um With Massachusetts getting 8– 80 cents every time a ticket is sold Maybe a good analogy is like this… um Again when this story came out of the newspapers it was sort of at the same time actually that there was a big story about MIT students um winning a lot of money at blackjack (does anybody remember the story?) in Las Vegas casinos And so it was sort of– that these two stories were talked about in the breath They said how did the kids at MIT figure out how to beat the house? Okay, let me explain why that's wrong. What were the kids at MIT doing? They were making a lot of bets, right? They were buying just to give you the scale, about 200,000 tickets every roll down day They were making a lot of bets, each one of which had a small positive expected value I should also say by the way that once a lot of people were playing It didn't stay like five dollars and fifty cents for $2 ticket was more like a 15% profit on average. That's still pretty good So they're making a small– and so some of those bets are going to win, some of them are going to lose But if each one is slightly tilted towards the MIT kids then on the whole they're very likely going to make money so if that's your strategy You're making a lot of bets, which are slightly tilted in your favor… You are not beating the house, you are the house! I mean that is what the house does. And so I think a productive way to think about what was actually going on, from a mathematical point of view, is to compare it to the following diagram Which is exactly the same. The kids from MIT and the other high volume betters were playing the role of the casino the regular lottery players were playing the role of the regular betters who come to the casino and bet and they in the aggregate make Lots of bets which overall have a slightly negative expected value and money is flowing away from them; and every time it does every time the money flows in Las Vegas the state of Nevada reaches in and takes a cut Right? Because States don't like to gamble, States like to collect taxes That's their skill set. That's what they're good at, and that's what they do And that's what they were doing in Massachusetts. In other words what you should think of as having happened is that the state of Massachusetts… (i still don't know whether on purpose or sort of stumbling into it) had licensed a gigantic under-publicized virtual casino on which they collected lots of taxes and made a good profit And which carried on until people found out about it… So that, in the end, I think, is the answer. I think that's a satisfying answer to the first question, of How did this go on for so long? But now I want to turn to the second question, which turns out to have quite a bit more mathematical heft. You see, I've been talking about these three groups of betters as if they were all the same, but that's not quite true There's one very interesting difference between the three groups, that the Inspector General's pointed out; which is that Jerry Selbee and Dr. John used what's called the quick pick machine. I don't know if there's an analogue to that in the UK… So, what is this? This is a machine that picks random numbers for you to play And that seems like a good idea, right? Because we all know that you can't predict what numbers the lottery's going to come up with; any number is as good as other… if you're going to buy two hundred thousand tickets, It certainly seems like it would save a lot of time and cost you nothing to have those tickets printed out randomly for you by machine, but Random Strategies, contrary to their name, did not do this! They filled out their tickets by hand, 200 thousand of them… Why?! Why do this? This is a humongous pain, and you know, the inspector general's report mentions that they did this, but didn't say why. And I became kind of obsessed with this, because I was like these people are smart, they know what they're doing, they know math They know that the expected value of each ticket is the same, why would they care which tickets they had? That's what I want to spend the rest of our time together talking about so as mathematicians When we're faced with a problem we don't understand, the first thing we do is we try to make it simpler. We try to replace our problem with a simpler problem. Hopefully which has the same features Enough of the same features the original one that we can use it to gain some insight, so that's exactly what I'm going to do… Let me replace this lottery with a smaller game which, for reasons that are lost in history, is sometimes called the Transylvanian lottery… And here, instead of having 46 different numbers like in the actual Massachusetts lottery There are only 7. Only 7 balls in the cage. And instead of picking six of those, you're only going to pick three. And the reason I do that is because it means that the number of Jackpots is now so small that I can list them all on one slide Here they are, all the different ways of picking three numbers out of seven. For the combinatorics fans in the audience The number of these is 35, which is "7 choose 3" and that's called 7 choose 3 because it's the number of ways of choosing 3 things out of 7 But it doesn't matter if you know that… just matters that you believe me that these are all the possible combinations. And now we can start to say well, what if the game were this small? Let's see if we can understand what would be the benefit of choosing the numbers yourself as opposed to picking them randomly. Well, first of all I have to sort of tell you what the rules of the game are, now that I've shrunk it a little bit… Again, let's simplify. Let's not have like six different tiers of prize. Let's only have two so in this simplified game There's two kinds of prizes: you get a jackpot, which is worth six dollars if you get all three numbers right… Order doesn't matter by the way I can emphasize that; if you get two out of the three numbers right you have a smaller prize which let's call it deuce which is worth two dollars; and if you get one or zero of the numbers right then you get nothing. So, okay, this is very simple But it has some of the features of the original game, right? it has multiple tiers of prizes… Well, only two… And it sort of has the same form, but at a much smaller scale. And… Now in this game… what is a high volume better? You don't have to buy 200,000 tickets to be a high volume better in this game I mean there are only 35 different tickets to buy! So for us let's discuss the problem of buying seven tickets. That's a lot, that's a fifth of all possible tickets, right? So that's a pretty big purchase. And what I want to show you is What happens if you pick seven tickets at random? And I've computed this for you, showing you– I haven't actually shown you the jackpot probabilities But I've shown you how many deuces you can expect to get… It turns out that the expected number of deuces, if you buy seven tickets, is 2.4 And so it's not surprising… that the most likely number of deuces to get are two or three… There's a 30% chance of getting two deuces, a tw– about a 26 percent chance of getting three… and there's some probability of getting fewer, and there's some probability of getting more. In fact, you can easily– you can, given this, you can compute the expected value of this ticket Of your 7 tickets you expect to get two point four deuces The expected number of Jackpots… that's 1/5th. Why is it 1/5th? Because well, you've got one fifth of the tickets, so there's a 1/5– a one in five chance that the Jackpot is among them… So let's see: 2.4 times $2 is 4.80 and then one fifth of that six dollar prize is a dollar twenty… put that together, and the expected value of your seven tickets is $6. And again, I want to remind you that um That every ticket has the same expected value, so it actually doesn't matter which seven tickets. I've written down… (this is sort of a fun exercise to do yourself) any seven tickets I chose would have an expected value of six dollars This is the source of our intuition that it shouldn't matter which seven you pick so Here's what we're going to do… Ready? You are going to think and write down seven tickets that you want to buy… So a ticket is just a choice of three numbers from 1 through 7 And here's the game we're going to play: I told you that on average We can expect to win six dollars every time, so what we're going to do is we're going to pick some random numbers We're going to play the game every one of you who has done it is gonna like see how much money you won on your seven tickets… You got to be honest; and here's how it's gonna work It's like an elimination game: if you get less than six dollars, you're out. And I'm playing this game too, by the way, so I'm gonna show you my tickets… Everybody picked? Okay, no copying! All right um Okay, and there's– and here's mine Okay, now I do need one more thing, which is I need somebody to be the random ball cage Is there anybody really random here? Okay, this kid. okay, so um Okay, so try, if you can, not to look at mine, and not to look at your friends' there, and just like, call out three random numbers between one and seven… They should be three different numbers, right? cuz when they come out of the ball cage they're all different… And then we're all gonna score. Okay… Okay, so five– so I'm going to order that as five six seven just so we'll have them in order, okay? So everybody gets what we're doing. The jackpot is five six seven you guys are scoring your seven tickets… if you have five six seven you get six dollars just for that, if you'd– and, if you have a five and a seven you get two dollars for that, if you have a five and a six you get two dollars for that If you have a 6 and a 7 you get two dollars for that. Let's see how I did. Maybe that– so So I've got 167 that gets me $2 I don't have a jackpot there. I have two five six that gets me another $2 And I have three five seven and that gives me another $2, right? because I have the five and the seven. So I have six, so I'm still in, I have six. Okay Okay, all right. Let's do it again. Ball cage man, are you ready? Okay? One six seven. Okay. I've got a jackpot. I already have six… Let me see if I've got anything else. uh… five… six… no, and no Everybody count those up. Okay. Who's still in? Okay… fewer, but still a fair number, okay? I'm ready. We're gonna– two four five Okay, I do not have a jackpot. Okay. I'm gonna look I've got one four five that's a deuce for me. I've got two four seven that's a deuce for me, and I've got two five six And that's it. So I got three deuces so I'm– so I got six dollars, so I'm still in. Alright. Let's do– let's do one more Three one four okay, otherwise known as one three four, okay… so one three four are the numbers when I've got two deuces on my first two right there… and then… and then I've got three four six okay, so I've got Three deuces okay, so I'm in with six dollars Okay, so let's– let's end it at four rounds I kind of like this game, but we can only do it for so long… okay so who– so I got– so I'm still in, I– I made my six every time, and it– who else got 6 or above every time? Okay, so a decent number… probably about like What do you guys think? Maybe like one in five of the people who are playing like some some a handful of people okay, so What did you guys notice, especially a kid about my winnings? Anybody notice anything? besides the fact that I'm still in, because I'm great at this game. I always got six, right? Not only did I never go under, I never went over… And that is no coincidence. In fact, the miraculous thing about these numbers that I've chosen Is that although it looks like I'm playing a gambling game I am not, because no matter what the jackpot is I will win six dollars, on the nose. So this is how the payoff matrix looks for my seven numbers Instead of this wide range of possibilities there are in fact only two things that can happen: four out of five times I will get three deuces and I will win six dollars And one out of five times I will have the jackpot, and I will win six dollars. And what that means is I cannot be eliminated from this game So these two bets have the same expected value, but they're not the same bet, they are rather different; in the language of finance we would say that they have the same return, but the second one has a lot less risk Right? There's no chance of getting one of these, like, very poor results like only two deuces are one deuce or no prize at all… That cannot happen. And most people Given the choice between two investments that have the same return will choose the one with the less risk I mean, don't get me wrong. There's downside to that too. I'm also giving up the possibility of winning a lot more, right? But that is a choice most people will make, especially If your business plan involves borrowing money from people… In finance we would call this "leverage" If the way you're running your business is borrowing money from all your friends to buy lottery tickets and then you buy a lot of lottery tickets, and then you lose all their– all of their money And then if you come back and say, but by the expected value computation, if you can somehow find more money to give me, in the long run we'll come out ahead That is usually not such a successful business strategy, right? If you're playing with other people's money you really don't want to lose, and so a hedging strategy like this one… um where you eliminate the possibility of Loss, is quite attractive. The question is How would you come up with these numbers? How would you come up with these numbers that are somehow so perfectly arrayed so that you eliminate all possibility of Loss? well… They come from at least what to me was a rather surprising source… They come from Geometry So, what are we looking at here? We're looking at very many things One thing we're looking at is a kind of diagram of those seven tickets I just showed you if you see here there's seven points here corresponding to the seven numbers, and there's also seven little curves, right? There's… and each of the Curves contains three of the points So if you look at that little edge along the bottom is one two three… That's one of my tickets The circular thing is two five six. That's another one of my tickets There's a line That's a line second that's one four five is one that's three five seven and if you go back to the previous slide you could those are exactly the tickets that I have. So this is a kind of diagram by which I keep track of what my seven tickets were So one thing it is this is a picture of my magic set of tickets, but another thing it is is I would say it's a picture of the plane… A somewhat weird thing to say Doesn't look like a plane. It looks like some kind of a funny triangle with a circle drawn in it But I'm going to say that this is a plane and the points are points and those little curved things are lines okay, that's a weird thing to– that's an even weirder thing to say, right? because For those taking Geometry even the ones that are lines they don't look like lines They're line segments, right? everybody remember this distinction Because they're finite in extent And then there's that one in the middle that doesn't look like a line at all, looks like a circle! And yet, I demand the right to say that these are lines. Why? because to a modern mathematician we're not tied to our usual geometric notions of points and lines when we talk about points and lines… To us, a point is a thing that behaves like a point and a line is a thing that behaves like a line Okay What does that mean?! well, what are our rules for how points and lines behave? They were given to us by Euclid, right? I mean, this is the rules of geometry and what I want to point out is that with this definition of points and lines the rules of geometry are obeyed It's kind of fun to check this for yourself on the picture, that… every two lines intersect in a single point, and any two points determine a unique line. Just like they're supposed to according to the rules of geometry Is anybody taking geometry right now by the way? I think everybody in the room is either too young or too old Because there is one way in which these are actually not Euclid's rules… Everything I say here is true about this picture, but it's a bit different from regular Geometry because in regular Geometry Lines can be parallel So you can have two lines that don't intersect at a point. See, I consider that a problem Rules with exceptions are bad, rules without exceptions are much nicer so this geometry is actually much better than euclidian geometry it's what's called a projective geometry, in which there are no parallel lines… and if I had another hour to tell you this story I would tell you all about how the basic ideas of projective geometry were first developed not actually by Mathematicians but by painters. Because they were the ones who had to figure out how to depict a three-dimensional world on a two-dimensional canvas and if you've seen sort of one-point perspective painting you're familiar with the fact that there are a lot fewer parallel lines in paintings than there are in the real world, right? the railroad tracks Okay, I know there weren't railroad tracks in the 15th century but I mean that's the idea… A parallel thing is like sort of seemed to come to a point on the canvas exactly for this reason that in projective geometry there's no such thing as parallel lines… any two lines meet. So maybe I've sort of made some kind of convincing case that it's not ridiculous to call these curvy looking things lines and to call these pointy looking things points What– what connection does it have with what I'm actually talking about? Well, here's the deal: Why is it that this set of tickets has the properties I said it had? How did I know that when my friend over here said 1 4 5… oh, that was one that was the jackpot. Let me ch– pick a different one 3– 5 6 7 was another one that he said. Look– let's look at 5, 6, and 7; that was the first one, right? How did I know I was going to have three deuces? well How did I know I was going to have a ticket with a six and a seven? I knew that because there is a line through the points six and seven; there it is: one six seven, that's one of my tickets How did I know I was going to have a ticket that contained five and seven, giving me another deuce prize? because I know there's a line through the points five and seven, and there it is: Three five seven; and finally, five and six… What goes through them? is the line two five six So the fact… the geometric fact that through any two points there's exactly one line that Is exactly what is required to guarantee– I mean in sort of lottery language, that says that given any jackpot each pair of numbers in that jackpot will be on one of– exactly one of my tickets, and so I'll get exactly three deuces. The only thing that wait, that can get a little messed up is if the three points, that my friend chose, were actually collinear, like one four five. Then I do have a ticket containing one and four, and I do have a ticket containing one and five, and I do have a ticket containing four and five as the Geometry guarantees, but they're all the same ticket. And there you have it, those are the two things that can happen: the three points can either be collinear or not; I will either get the jackpot or I get three deuces. so This is quite mysterious this thing by the way, I should I should give credit where it's due This is called the Fano plane It was developed at the very outset of axiomatic geometry, by Gino Fano, who was an Italian Geometer of late– of the late 19th century There's one problem though. I mean I really feel like I've given a nice solution to how you should buy your seven tickets, to eliminate risk… The problem is that, of course, let's not forget that I simplified the problem… There are not seven numbers in Cash Windfall. There are 46; you don't pick three of them, you pick six of them, so maybe I was sort of clever and sort of found some nice picture I could draw from classical Geometry to handle this small problem, but what about the big one, so Here we have our toe sort of on the edge of a very beautiful large Old area of mathematics called the theory of combinatorial designs I'm not going to have the time to tell the whole story, though it's a great story. Let me just say that there exists an entire long mathematical story of how to develop configurations like this in all manner of situations and actually I'll just say because it's kind of UK related This is Peter Keevash. He's a professor at Oxford and In some sense last year he proved a truly remarkable theorem that essentially puts the cap on about 150 years of work in this area, proving that a design like the one I just showed you that those designs are ubiquitous; basically in every context in which you would expect there to exist such a thing, there really does exist such a thing. And this was quite– I think this was quite unexpected actually, that this problem was going to be solved so soon. it's a really miraculous piece of work by Peter That being said, when I was thinking about this problem of Cash Windfall, I wasn't thinking about the general case of good designs appearing in every possible context I was thinking in one– I was thinking about this one particular context of… of, um, six numbers chosen out of 46 balls Moreover, Peter's proof is what's called non-constructive, right? It doesn't necessarily give you an easy quick way to actually write down such a design it sort of just proves that they exist That being said though there is as I said this very long literature about problems like this, and so I became kind of obsessive trying to figure out what these kids at MIT actually did… I became certain that they use some kind of combinatorial design strategy and then minimizing risk was the reason that they were choosing their own tickets… And I wanted to kind of reverse engineer to figure out what it was; and after some searching I found the following paper, also a UK product by the way. This is RHF Denniston for the University of Leicester who wrote a paper in 1976 Generating some kind of combinatorial designs actually these guys were actually on 48 balls not 46 but I tweaked it a little bit and I was able to come up with a configuration of about 230,000 tickets [um] which gives you a 2% chance of winning the jackpot, a 72 chance– percent chance of hitting six of those five out of six prizes, and a twenty four percent chance of winning five of those those five out of six prizes, so essentially guaranteeing that you would win at least five of those big prizes and essentially hedging away all risk in a way that actually looks rather different the risk of Loss would be much greater if you picked your 230,000 tickets randomly… And people always ask if I can draw a picture of this one, I cannot I'm sorry I wish there was like a beautiful picture like the picture of the Fano plane there– there is not… But it's, in some sense, similar in spirit. In that it does have to do with finite projective geometry just like the Fano plane does But in a much more intricate way, which is why it was only developed about 100 years later So, I'll just close by saying that I don't actually know, because I was not able to get these guys to tell me if this is actually what they did. But if they didn't, I think it's what they should have done. Okay, I'll stop there, and I'll take questions. Thank you so much.

Taylor series | Essence of calculus, chapter 11



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Taylor polynomials are incredibly powerful for approximations, and Taylor series can give new ways to express functions. Brought to you by you: …

Braess's Paradox – Equilibria Gone Wild



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Lecture 2 | Machine Learning (Stanford)



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Lecture by Professor Andrew Ng for Machine Learning (CS 229) in the Stanford Computer Science department. Professor Ng lectures on linear regression, gradient descent, and normal equations and discusses how they relate to machine learning.

This course provides a broad introduction to machine learning and statistical pattern recognition. Topics include supervised learning, unsupervised learning, learning theory, reinforcement learning and adaptive control. Recent applications of machine learning, such as to robotic control, data mining, autonomous navigation, bioinformatics, speech recognition, and text and web data processing are also discussed.

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this presentation is delivered by the Stanford center for professional development okay so let's get started with today's material so um welcome back to the second lecture what I want to do today is talk about um linear regression gradient descent and the normal equations um and I should also say lecture notes have been posted online and so you know if some of the math I go over today might go over rather quickly if you want to see every equation written out and work for the details more slowly yourself um go to the course home page and then download the detailed lecture notes that are pretty much described all the mathematical technical content so I'm going to go over today um today I'm also going to delve into a fair amount some amount of linear algebra and so if you would like to see a refresher on linear algebra on this week's discussion section would be taught by the TAS and will be a refresher on linear algebra so so some of the linear algebra talked about today so seems to be going by bit quickly or if you just want to see some of the things I'm claiming today without proof if you want to just see some of those things retail in detail um can come to this week's discussion section um so actually one start by showing you a fun video um remember at the last lecture the initial lecture I talked about supervised learning and supervised learning was this machine learning problem where I said um we're going to tell the algorithm what the quotes right answer is for um you know ever ever for a number of examples and then we want the algorithm to replicate more of the same so the example I had that the first lecture was the problem of predicting housing prices where you may have a training set and we tell the algorithm what quotes right housing price was for every house in the training set anyone the algorithm to learn the relationship between you know sizes of houses in the prices and essentially produce more of the quote right answer so let me play for you a video now below the big-screen news so in favor video now um there was from Dean Pomerleau on some work he did at Carnegie Mellon on applying supervised learning to get a car to drive itself um this is this work on a vehicle known as Alvin was done about this it was done to about fifteen years ago um and it was uh I think was very elegant example of the sorts of things you can get supervised learning algorithms to do um on the video you hear Dean Tom Lewis voice mentioned an algorithm called neural networks will say a little bit about that later but um the the essential learning algorithm for this is something called gradient descent which which I actually talk about later in today's lecture let's watch on the video Alvin is a system of artificial neural networks that learns to steer by watching a person drive Alvin is designed to control the nav lab to a modified army Humvee equipped with sensors computers and actuators for autonomous navigation experiments the initial step in configuring Alvin is training a network to steer during training a person drives the vehicle while Alvin watches once every two seconds Alvin digitizes a video image of the road ahead and records the person's steering direction this training image is reduced in resolution to 30 by 32 pixels and provided as input to Alvin's three-layered network so two comments right one is um this is supervised learning because is learning from a human driver in which a human driver shows that you know we're on dis segment of the road I would steer at this angle of it when the segment of RO is D at this angle and so the human provides a number of quotes correct steering directions to the car and then it's the job of the car to try to learn to produce more of these you know quote correct steering directions that keeps the car following the road um on the monitor display up here I'm going to tell you a little bit about what this display means so on the upper left where where the mouse pointer is moving on this horizontal line actually shows a human steering direction and this you know white bar or this white area here shows that the human shows the steering direction chosen by the human driver by moving the steering wheel so the human is steering a little bit to the left here indicated by you know the position of this white region um this second line here where my mouse is pointing the second line here is on the outputs of the learning algorithm and where the learning algorithms things currently things is a very steering direction and right now what you're seeing is the learning algorithm just at the very beginning of training and so that she has no idea where to steer and so it's out put this little white smear over the entire range of steering directions and as the algorithm collects more examples and learns over time you see it you know you see it start to more confidently choose the steering direction so let's keep watching the video using the back propagation learning algorithm Alvin is trained to output the same steering direction as the human driver for that image initially the network steering response is random after about two minutes of training the network learns to accurately imitate the steering reactions of the human driver this same training procedure is repeated for other road types after the networks have been trained the operator pushes the run switch and Alvin begins driving 12 times per second Alvin digitizes an image and feeds it to its neural networks each Network running in parallel produces a steering direction and a measure of its confidence in its response the steering direction from the most confident network in this case the network train for the one lane road is used to control the vehicle suddenly an intersection appears ahead of the vehicle as the vehicle approaches the intersection the confidence of the one-lane network decreases as it crosses the intersection and the two-lane road ahead comes into view the confidence of the two-lane Network Rises when it's confidence Rises the two-lane network is selected to steer safely guiding the vehicle into its lane on the two-lane road all right so who thought driving could be that dramatic right I switch back to the to the trophies um I should say um this work was done about 15 years ago and autonomous driving has come a long ways so many of you were heard of the DARPA Grand Challenge where one my colleague Sebastian Thrun that the winning team the winning team to drive a car across the desert by itself so Alvin was I think absolutely amazing work for his time but you know states of autonomous driving has also come a long way since then um but so what she just saw was an example um again of supervised learning and in particular it was an example of what they call the regression problem because the vehicle is trying to predict a continuous value variable so if a continuous value steering directions we call these so called the called a regression problem um and what I want to do today is talk about to our first supervised learning algorithm and it will also be two regression tasks um so for the running example I'm going to use um throughout today's lecture she's going to return to the example trying to predict housing prices um so here's actually a data set arm collected by our TA Dan Ramage on housing prices in a Portland Oregon um so so here's a data set of a number of houses of sort of different sizes and here are their asking prices in thousands of dollars this 22,000 and so um you can take this data and plot it square feet this price and so you may get the data set like that and the question is given the data set like this so given a trait what we call a training set like this how do you learn to predict the relationship between the size of house and the price of a house um so I'm actually come back and modify this chart a little bit more later but um we're going to introduce some notation which I'll be using actually throughout the rest of this course um first piece of notation is on I'm going to let the lowercase alphabet M denote the number of training examples that just means a number of roles or the number of examples houses and prices would happen you know in this particular data set we have we actually happen 247 training examples I'm old I wrote down only fine um so throughout this quarter um I'm going to use the alphabet M to denote the number of training examples um I'm going to use what the lowercase um alphabet X to denote all the input variables which are which I often also call the features and so in this case X would denote the size of a house that we're looking at um I'm going to use Y to denote the whole output variable which which is sometimes also called the target target variable and so um one pair X comma Y is what comprises one training example in other words one row on the table I draw just now would be what I call one training example and and the I've training example in other words the I've row in that table I'm going to write as um X comma Y I okay and so um for the in this notation I'm going to use this superscript I is not exponentiation so this is not X ^ iy ^ I in this notation the superscript I in parentheses is just 7 index into the I fro of of my list of training examples so um then supervised learning this is how so this is this is the what we're going to do it is we give the training set um and we're going to feed our training set comprising our M training examples of 47 training examples into learning algorithm okay and our algorithm then has to output a function that slip by tradition for historical reasons um is usually denoted lowercase alphabet H and is called a hypothesis don't worry too much about whether the term hypothesis is a deep meaning is a more term this use of historical reasons and the hypothesis job is to take as input you know if there's some new host in whose price want to estimate what the hypothesis does is it takes us input on a new living area in square feet say and output the estimated price of this house so the hypothesis H maps from n plus x2 outputs y um so in order to design the learning algorithm the first thing we have to decide is how we want to represent the hypothesis right and just for the purposes of this lecture for the purposes of a first learning algorithm I'm going to use a linear representation for the hypothesis so I'm going to represent my hypothesis as H of X equals theta 0 plus theta 1 X where X here is an input feature and so that's the size of the house we're considering um and more generally you can come back to this um more generally for many classification for many regression problems we may have more than one input feature so for example if instead of just knowing the size of the houses if we know also the number of bedrooms in these houses on v2 let's say then so if our if our training set also has a second feature of the number of bedrooms in the house then um you may let's say x1 denote the size and square feet on let X subscript to denote the number of bedrooms and then um I would write the hypothesis H of X as theta Rho plus theta 1 x1 plus theta 2 x2 okay and sometimes when so when I want to take the hypothesis H and when I want to make us dependence on the theta is explicit I'll sometimes write this as a true subscript theta of X and so this is the price that my hypothesis predicts a house with features x cost so given a given the new house of features X with a certain size and so the number of bedrooms this is going to be the price that my hypothesis predicts this house is going to cause um lastly one last one ones with one final piece of notation on simple conciseness um just to write this a bit more compactly I'm I'm going to take the convention of defining X 0 to be equal to 1 and so I can now write H of X to be equal to sum from I equals 1 to 2 of theta I I'm sorry 0 to 2 theta I X I and if you think of Thetas and XS as vectors and this is just say they travel is X um and and the very final piece of notation is um I'm also going to let lowercase alphabet n define let lowercase n be the number of features in my learning problem and so this actually becomes a sum oh I'm just a sum from I equals 0 to n where in this example if you have two features and would be equal to two okay all right I realize that was a fair amount of notation um and as I proceed through the rest of lecture today or in future weeks as well if you know if someday you're looking at me write a symbol and you're wondering gee what was that simple lowercase n again or what was that lowercase X again or whatever please raise your hand and also this is fair mountain notation we'll probably I'll get used to it um you know in a few days and we've standardized notation and make a lot of our descriptions of loading office much easier ok put again if if you see me write some symbol and you don't quite remember what it means chances are there are others in this class of forgotten too so please raise your hand and awesome if you're ever wondering what some symbol means um what questions you have about any of this it can be anything uh let's see what else and again o T da doo da 0 0 1 yes right so yeah so well duh this was not an expert the theatres on all the theatre eyes are called the parameters um the Thetas are called the parameters of our learning algorithm and theta 0 theta 1 theta 2 are just real numbers and then is a job of a learning algorithm to use the training set to choose or to learn appropriate parameters theta ok there other questions yeah um Oh transpose right Korea right so just come on we're right here theta 2 and theta transpose X in the product whatever function our hypothesis function or we have in higher orders or theta all great questions um the answers the questions of what is this a typical hypothesis or tan theta be a be a function of other variables and so on and the answer is sort of yes um for now just just for this first um you know learning algorithm will talk about using a linear hypothesis cause um a little bit actually later this quarter we'll talk about much more complicated hypothesis classes um and why she talked about higher-order functions as well a little bit later today okay so um so for the learning problem then um how do we choose the parameters theta so that our hypothesis H will make accurate predictions about housing X's right so you know one reasonable thing to do seems to be that what we have a training set so and just on the training set our hypothesis will you know make some prediction predictions of the housing prices of the of the prices of the houses in the training set so one thing we do is just try to make um the predictions of a learning algorithm accurate on the training set that leads right so given some features eggs and some correct prices why we might want to make let's say the squared difference between the prediction of the algorithm and the actual price small um so to choose the parameters theta plus I want to minimize over the parameters theta of the sort of squared error between the predicted price in the actual price um and so going to folders in we have M training examples so when sum from I equals 1 through m of my M training examples the price predicted on the I polls in my training set are – the you know actual target variable – the actual price on the I train example um and by convention instead of minimizing this sum of squares differences I'm just going to put a 1/2 there which which will simplify some um some of the math we do later ok and so let me go ahead and define J of theta to be equal to just the step 1 home sum from I equals 1 through m on the number of training examples of the value predicted by my hypothesis – the actual value and so what we'll do let's say is minimize as a function of the parameters of theta this quantity J of theta um I say – delta T they've taken the linear algebra classes or maybe those basic statistics sources some of you may have seen things like these before um in the scenic route you know least squares regression ordering of these squares um many of you will not have seen this before I think some of you may have seen it before but either way regardless of what they've seen it before let's keep going and but we just don't see they have seen it before I should say eventually will actually show that this algorithm is a special case of a much broader class of algorithms but let's keep going or we'll get there eventually um so so I'm going to talk about a couple of different algorithms for performing that minimization over theta of J of theta first I'm gonna talk about is a search algorithm where the basic idea is we will start with some um value of my parameter vector theta um maybe maybe initialize my parameter vector theta to be the vector of all zeros um and excuse me let's break that I write also write zero of an arrow on top to denote the vector of all zeros and then um you know when we keep changing my parameter vector theta to reduce um J of theta a little bit until we hopefully end up at the minimum with respect to theta of J of theta okay so um touch the laptop display these and load a big screen so let me go ahead and show you an animation of this first algorithm for minimizing J of theta which is an algorithm called gradient descent so um here's the idea you see on the display a plot arm and the axes so the the horizontal axis are theta zero and theta one is usually minimize J of theta which is represented by the by the height of this plot so the surface represents a function J of theta and the axis of this function or the inputs is function or the parameters theta 0 and theta 1 written down here below so here's the Umbrian descent algorithm we're going to choose some initial point it could be no vector of all zeros or some randomly chosen points let's say we start from that point denoted by the by the crop idea by the star but across um and now one should imagine that um this display actually shows a 3d landscape mentions of you know all in the holy park or something and this is the 3d shape of like a hill in some park and um so imagine they're actually standing physically at the position of that star of that cross and imagine they're going stand on that hill right and look all 360 360 degrees around you and also if i were to take a small step what would allow me to go down hill dimille's it's imagine that this is physically a hill and you're standing there you look around and also if i take a small step what is the direction of steepest descent that would take me down hill as quickly as possible so the gradient descent algorithm does exactly that gonna you know take a small step in this direction of steepest descent or in the direction of the gradient it turns out to be and then you take a small step you end up in the new point um showing up there and then we keep going you know the new point on this whole and again you're going to look around you look all the agencies agree look all 360 degrees around you and ask what is the direction that would take me downhill you know as quickly as possible you want to go downhill as quickly as possible because we want to find a minimum of j data so you do that again you can take another set okay and you sort of keep going on until you end up at a local minimum of this function J of theta um one property of gradient descent is that um where you end up in this case we ended up at this point on the lower left hand corner of this plot um but you know let's let's try running great in the center game from different position also that was where I started grading descent just now let's rerun grading descent but using a slightly different initial starting point so the point slightly to the further to the up and further to the right so it turns out if you run gradient descent from that point then if you take a steepest descent direction again that's the first step and if you keep going um turns out that with a slightly different initial starting point you can actually end up at a completely different local Optima okay so this is a property of grading descent we'll come back to in a second but so be aware that gradient descent can sometimes depend on where you initialize your parameters theta0 and theta1 but um i should switch back to the trapeze um let's go ahead and work out the math of the grand descent algorithm then we'll come back and revisit this issue of local optima so here's the gradient descent algorithm um we're going to repeatedly take a step instead of this direction of steepest descent and it turns out that you can write down as follows which is we're going to update the parameters theta as um theta I minus the partial derivative respect to theta I J of theta okay so this is how we're going to update the I've your parameter theta I how we're going to update theta I on each iteration or very interesting um just a point of notes a notation I use this colon equals notation to denote um so setting a variable on the left hand side to be equal to the variable on the right hand side so if I write a colon equals B then what I'm saying is this is part of a computer program on this part of an algorithm where we take the value of beale the value on the right hand side and use that to overwrite the value on the left hand side um in contrast if I write a equals B then this is an assertion of a true but this is I'm claiming that the value of a is equal to the value of B okay and so whereas this is a computer operation where we overwrite the value of a if I write a equals B then I'm self ascertain to the values of a and B are equal um so let's see this algorithm sort of makes sense um um well actually let's just move on let's go ahead and take this algorithm and apply it to our problem and to work out gradient descent um let's take green descent and just apply to our problem um and this being the you know first somewhat mathematical lecture I'm going to step through derivations much more slowly and carefully than I will later in this course or losses you know work for the steps of these in in much more detailed and then I will later in the school term let's actually work out what this green December was um so and I'll do this just for the case of if we have only one training example okay so in this case we need to work out what the partial derivative with respect to the parameter theta is of J of theta o if we have only one training example then J of theta is going to be one half if subscript theta of X minus y squared right so if you have only one training example comprising one pair X comma Y then this is what J of theta is going to be um and so taking derivatives um you know you have 1/2 something squared so the two comes down your two times 100 M 60 script theta of X 9 um and then by the chamber of derivatives um we also need to do multiply this by the derivative of what's inside the square right arm 2 2 and 1/2 cancel so this usage times that Steve is your X 0 plus the dawn stay the red ok and if you look inside this sum excuse me we're taking the partial derivative of this sum with respect to the parameter theta I but all the terms in this sum except for one do not depend on theta arrive at the own of in the sum the only term that depends on theta I will be some term here of theta I X I and so we take the partial derivative respect to theta I X I um take the partial derivative respect to theta I of this term theta right X I and so you get x excited okay and so this gives us our learning rule eight of later I gets updated as theta I minus alpha times how's that okay um and this Greek alphabet alpha here is a parameter of the algorithm called the learning rate and this parameter alpha controls how how large a step you take those of you you're standing on the hill you've decided on what direction to take a step in and so this parameter alpha controls how aggressively use how large a step you take in this direction of steepest descent okay um in serve you well if alpha and this is a parameter the algorithm that's often set by hand um maybe choose alpha to be too small then your steepest descent algorithms a very tiny steps and take a long time to converge if alpha is too large then the steepest ascent may may actually end up overshooting the the minimum of your if you're taking too aggressive the step um okay yeah oh say that again these are their final vertical mixing somewhere um Sarah you a one-half missing edge oh goody cool I do I do make lots of errors and maps is good too any questions about this okay so so let me just sum well wrap this up properly into an algorithm so over there I derived the algorithm but if you have just one training example um more generally for M training examples gradient descent becomes the following um we're going to repeat until convergence um the following step theta I guess updated us later I and I'm just writing out you know the appropriate equation for M examples rather than one example um theta I guess of the SAR I minus alpha times something like this one to M okay and I won't so bother to show it but you can go home and so verify for yourself that this summation here this is indeed the partial derivative with respect to theta I of J of theta where when you if you use the original definition of J of theta for when you have M training examples okay um so in research I switch back to laptop display I'm going to show you what this looks like when you run the algorithm um so it turns out that um for the specific problem of linear regression or ordinary least-squares which is what we're doing today um the function J of theta actually does not look like this nasty one does showing you just now with multiple local optima in particular it turns out for ordinary least-squares the function J of theta is is just a quadratic function and so we'll always have a nice bell-shaped nice bow shape like what you see up here and I only have one global minimum with no other local optimum so when you're in very descent here here actually the contours of a function J so the contours of a bow shape function like that are going to be ellipses and if you run gradient descent on andhe's album here's what you might get so let's see I initialize the parameters you know so let's say randomly at the position of that cross over there right that cross on the on the upper right and so after one iteration of gradient descent as you change in the space of parameters so that's that the result of one step of Granger sent two steps resets four steps five steps and so on and it you know converges reasonably rapidly to the global minimum of this function J of theta okay um and this is a property of these squares of ordering these squares regression with with the linear hypothesis calls it that the function J of theta has no local Optima this question I see yeah okay um so turns out that um yes it turns out this was done with I just did this with a fixed value of alpha um and one of the properties of green descent is that as the approach to local minimum it actually takes smaller and smaller steps so they will converge and the reason is the update is due your update theta by subtracting for you know alpha times the gradient and so as you approach a local minimum the gradient also goes to zero right so and as you approach local minimum at the local minimum the gradient is zero and as you approach the local minimum the gradient also gets smaller and smaller and so grand descent will automatically take smaller and smaller steps as you approach a local as you approach the local minimum so this um and here's the same plot of and so yeah here's here's a plot so the housing prices data so here let's you initialize the parameters to the vector of all zeros and so this blue line at the bottom shows the hypothesis with the parameters at initialization right so initially theta zero and theta one above zero and so your hypothesis predicts that you know all prices all all prices are equal to zero after one iteration of gradient descent that's the blue line you get after two iterations three four five and after a few more durations um excuse me it converges and you've now found the least squares fit to the data okay um cool let's switch back to the twelve point um other questions about this yeah sit in that we run each sample give all the sample cases ones update available he probably doesn't run it again with the new values yes Ram and converge means that the value will remain same of the two diagram doesn't remain roughly the same yeah so yeah so another question how do you test the convergence right and there are different ways of testing for convergence one is you can look at two different iterations and see if theta has changed a lot and if it hasn't changed much within two iterations you may say is sort of more or less converged something that's done maybe slightly more often is look at the value of J of theta and if J of theta so if the optimization the quantity you're trying to minimize is not changing much anymore then you might be inclined to believe is converge so these are sort of standard heuristics or standard rules of thumb that are often used to decide if gradient descent is conversions all directions 51 and choose add additional Louis gain so every one one one feature educator to the curve about yeah the math I mean I understand where that comes in when you use with your left we go this way or that way fussy okay engine turns out that um so question is that you know how is grading descent looking 360 around J choosing the direction of C business and also oh so it actually turns out I'm not sure I understood the second part it turns out that if you are if you stand on the hill and if you are turns out that we compute the gradient of a function we compute the derivative of function then it just turns out that that is indeed the direction of steepest descent um but there's no point out you would never want to go in the opposite direction because the opposite direction would actually be this direction of steepest ascent right um so it turns out maybe I hope maybe maybe maybe tears and talk bit more about this on on the dissection of this interest um since I'm going to take the derivative of a function the derivative of a function so turns out give you the direction of steepest descent um and so you don't explicitly you know local 306 degrees around you you sort of just compute the derivative and that turns out to be the direction of steepest descent now maybe I maybe I tease this has been asked minute easing talk a bit more about this on Friday um okay let's see um so let me go ahead and give this algorithm on a specific name so this out room here is actually called on batch gradient descent and the term batch isn't a great term but the term batch refers to the fact that on every step of Granderson you're going to look at the entire training set you're going to you know perform a sum over your M training examples um since oh so Bactrian descent often works very well so if I use it very often um and it turns out that sometimes if you have a really really large training set so imagine that instead of having 47 houses from Portland Oregon the training set if you had say the u.s. sends this database of something with us synthesized databases you can often have you know hundreds of thousands of millions of training examples um so if M is you know a few million then if you're running batch gradient descent then this means that to perform every step of gradient descent you need to perform a sum from J equals 1 to a million which is that's there's sort of a lot of training examples for your computer programs have to look at before you can even take you know one step downhill on the function J of theta so it turns out that when you are when you have very large training sets um just let me write down an alternative algorithm that's called on stochastic reinvestment sometimes also called incremental gradient descent but the algorithm is as follows again will repeat until convergence and will iterate for J equals 1 to M um and we'll perform one of these stuff gradients and updates using just the Jade training example okay and as usual this is that really you perform you update all the parameters data rocks you perform this update you know for all values of I rights meaning for I indexes in the parameter vectors you just perform this update all all of your parameters simultaneously um and the advantage of this algorithm is that um in order to in order to start learning in order to start modifying the parameters um you only need to look at your first training examples use it look your first training example and perform an update using you know the derivative of the error with respect to just your first training example and then you look at the second link training example and perform another update and use of keep adapting parameters much much more quickly without needing to take a scan over your entire near us sensors database before you can even start adapting your parameters um so let's see for large datasets on stochastic gradient descent is often much faster and what happens is the constant variant descent is that it won't actually converge to the global minimum exactly but on one that these are the contours are your function then as you run circles degree in the sense you sort of tend to wander around and you may actually end up going uphill occasionally but your parameters will sort of tend to wonder to the region close to the global minimum but sort of keep wandering around a little bit you and then often that's just fine to have a parameter you know that wanders a little around a little bit the global minimum and so the and in practice this often works much faster than bacteria in descent especially if you have a large training set I'm going to clean the copper balls why do that why don't you take a look at the equations and after I'm done cleaning the balls out also a question okay so what questions you have about is gradient descent is it true that are you just sort of like rearranging the order that you that you do the computation like so do you just use the first training sample and update all of the theta eyes and then step and then upgrade with the second finding example and a bit on the theta eyes and then step and is that why you get sort of this really yeah let's see right so so I'm just look at my first training example and then I'm going to take a step and then I'm going to UM perform the second gradient descent updates using my new parameter vector that has already been modified using my first training example and then I keep going that make sense yeah updatable the theta eyes are only using one Chinese on one training example ask for them back to the dam critical um let's see it's definitely recall um I believe this theory that solar supports that as well knows ego yeah this theory that supports that the the how clean the statement of the theorem is I don't remember okay cool so what I've done so far I've talked about an iterative algorithm um for performing this minimization in terms of J of theta um and it turns out that there's another way for this specific problem of these squares regression of ordinary least-squares turns out there's another way to perform this minimization of J of theta that allows you to you know solve for the parameters theta in closed form without needing to run in iterative algorithm um and I know some of you may have seen some of what I'm about to do before in like an undergraduate linear algebra course and the way is typically done so requires you know messy orthogonal projections or taking lots of derivatives and writing lots of algebra what I'd like to do is show you a way to derive you know the closed form solution of theta in just a few lines of algebra um but to do that I'll need to introduce a new notation for matrix derivatives um it turns out that the notation about to UM define here just in my own personal work has turned out to be one of the most useful things that you know I actually use all the time to have a notation how to take derivatives with respect to matrices so that you can solve for the minimum of J of theta with like a few lines of algebra rather than writing our pages and pages of matrixes into verses so then we're go ahead and define this new notation first and then and then we'll go ahead and work on the minimization um given a function J um and J is a function of a vector parameters data right I'm going to define the derivative of the gradient of J with respect to three later as self a vector okay and so this is going to be you know an N plus 1 dimensional vector Rho theta as an n plus 1 dimensional vector with indices ranging from 0 to n and so I'm going to find this derivative to be equal to that okay and so on we can actually rewrite the gradient descent algorithm as follows because this is a batch gradient descent and in rewrite gradient descent as updating the parameter vector theta notice there is no subscript I now updating parameter vector theta as the previous parameter minus alpha times the gradient okay and so in this equation all of these quantities you know theta and this gradient vector all of these are n plus 1 dimensional vectors um oh I see it right as using balls out of order wasn't I so more generally um if you have a function f on that maps from the space of matrices a loops excuse me um then maps from say the space of M by n matrices to the space of real numbers so if you have a function you know F of a where a is an M by n matrix so this function in maps from matrices to real numbers the function that takes this input in matrix let me define the derivative with respect to F of the matrix a right now just taking the gradient of F with respect to its input which is which is a matrix I'm going to define this itself to be a matrix okay so the derivative of F with respect to a is itself a matrix and matrix contains all the partial derivatives of F we respect to the elements of a um one more definition is um if a is a square matrix so if a is an N by n matrix number of rows equals number of columns let me define the trace of a to be equal to the sum of a diagonal elements so this is your sum over I of a I I um for those of you that haven't seen this live operator notation before you can think of trace of a as you know the trace operator applied to the square matrix a but it's more commonly written without the parentheses so I usually write the trace of a like this this is this just means the sum of diagonal elements so um here are some facts about the trace operator and about derivatives and notice I'm going to write these without proof you can also teach to prove some of them at the description section um or you can actually go home and so verify the proofs of all of these yourself turns out that um given two matrices a and B the trace of the matrix a times B is equal to the trace of B a okay I'm not going to prove this but you should be able to go home and prove this yourself from without too much difficulty um and similarly the trace of a product of three matrices so if you can take the matrix at the end and you know cyclically permute it to the front since trace of a times B times C just to the trace of C a B so take the matrix C at the back and move it to the front and this is also equal to the trace of BCA we take the matrix B and move it to the front okay um also suppose you have a function f of a which is defined as a trace of a B ok so this is right the trace is a real number so the trace of a B is of a function that takes us in for the matrix a and outputs a real number so then the derivative with respect to the matrix a of this function of Trey's a B um is going to be B transpose this is just another fact that you can prove by yourself by going back and referring to the definitions of traces and matrix derivative I'm not going to prove it it's real work though I lost Lee a couple of easy ones um the trace of a is equal to the trace of a transpose because the case is just the sum of diagonal elements and so if you transpose the matrix the diagonal elements don't change and if no case a is a real number then you know the trace of a real number is just itself so think of a real number as a one by one matrix so the trace of a one by one matrix is just whatever you know whatever that real number is um and lastly this is somewhat tricky one um the derivative with respect to the matrix a of the trace of a be a transpose C is equal to C a B plus C transpose a B transpose and and I won't prove that either this is just algebra and work about yourself okay and so the um I guess key equations the key facts I'm going to use a game about traces and matrix derivatives of all these five ten minutes okay so armed with these things I'm going to UM figure out let's let's try to come up a quick derivation for how to minimize J of theta in as a function of theta in closed form and without needing to use an iterative algorithm to work this out let me define the matrix X this is called the design matrix um to be a matrix containing all the inputs from my training set so you know x1 was was was the vector of inputs of the vector of features my first training example so I'm going to set x1 to be the first row of of this matrix X set my second training examples inputs to be the second row and so on and have M training examples and so that's going to be my arm design matrix X okay just define this matrix capital X as follows and so now let me take this matrix actually multiplied by my parameter vector theta this is stair vation will just take two or three steps so x times theta remember how matrix vector multiplication skills right you take this vector and you multiply by each of the rows of the matrix so X times theta is going to be just you know x1 transpose theta dot down to X M transpose theta and this is of course just two predictions of your hypothesis on each of your M training examples let me also define the Y vector to be the vector of all the target values y1 through yn in my training set so Y vector is an M dimensional vector so X theta minus y containing the map from the previous board is going to be like that right and now X theta minus y this is a vector this is an M dimensional vector if I have M training examples and so I'm actually going to take this vector and take us inner product works with with itself ok so recall that um you know if Z is a vector then Z transpose Z is just some of my Z I squared right that's how you take the inner product of a vector with worth it with itself so I'm going to take this vector at state 2 – y and take the inner product of this vector with itself and so that gives me some from I equals 1 to M F H of X on minus y squared ok since it's just the sum of you know the sum of squares of the elements of this vector and for the 1/2 there then this is this is our previous definition of J of theta ok vise oh yeah I know I feel a long notation at you today so M is the number of training examples and um the number of training examples runs from 1 through m and then is the feature vector that runs from 0 through n that make sense so um so this is this is a sum from 1 through m this um it sort of theta transpose X that's equal to sum from J equals 0 to n of theta I theta J extreme sense okay so so the feature vectors that in that index from 0 through n where X 0 is equal to 1 whereas the training examples actually index from 1 through end so let me clean a few more boards in and take a look take another look at Disney make sure it all makes sense okay okay yeah oh yes thank you SWAT over that whoosh man yes thank you mister feet great eye training example anything else cool so we're actually nearly done with this derivation um would like to minimize J of theta with respect to theta and we've written you know J of theta fairly compactly using this matrix vector notation so in order to minimize J of theta of respect to theta what we're going to do is take the derivative with respect to theta of J of theta and set this to zero and solve for theta okay so we have derivative with respect to theta of that is equal to UM as you mention there will be some steps here that I'm just going to do fairly quickly without proof so is it really true that the derivative of half of that is half of the derivative and I really exchange you know the derivative and then one half it turns out the answer is yes but later on you should go home and look for the lecture notes and make sure that you know you understand and believe why every step is correct I'm going to do things relatively quickly here and you can work for every step yourself more slowly by referring to lecture notes okay so um that's equal to I'm going to expand out this quadratic function so this is going to be okay um and this is just taking a quadratic function and expanding it out by multiplying yard roots and again work for this step day to yourself if you're not quite sure how I did that um so this thing this vector vector product right x2 you know this quantity here this is just J of theta and so it's just a real number and the trace of a real number is just it so done oh thanks that good right um so this this quantity in parentheses this is J of theta and it's just a real number and so the traits of a real number is just the same roll number in second so I'll take a trace operator without changing anything um and this is equal to 1/2 derivative with respect to theta of the trace of armed by the cyclic permutation property of tracer you can take the state or at the end and move it to the front so this is going to be trace of theta times theta transpose X transpose X minus derivative respect to theta of the trace of and again I'm going to take that and bring it to the UM oh sorry you know what I'm actually this thing here is also a real number and the transpose of a real number is just itself right so and take the transpose a real number without changing anything so let me go ahead and just take the transpose of this so there's a real number transpose itself is just the same real number so this is minus the trace of taking the transpose of that gives 1 transpose X theta then minus theta ok and this last quantity Y transpose Y it doesn't actually depend on theta so when I take the derivative of this last term with respect to theta is zero so just drop that term um and I'll see um well the derivative respect to theta of the trace of you know theta theta transpose on X transpose X I'm going to use um I'm going to use one of the facts I wrote down earlier without proof and I'm going to let this be a remote instead of identity matrix there so this is a be a transpose C and using a rule that written down previously that fine lecture notes um I guess I saw one of the balls bit but you have previously um this is just equal to X transpose X theta um so this is C a B which is suggest the identity matrix which we're going to ignore plus X transpose X theta where this is now C transpose a you know again times the identity which you can ignore times B transpose okay and the matrix X transpose X is symmetric so C transpose is equal to C um similarly the derivative respect to theta of the trace of Y transpose theta X um you know this is the arm derivative respect to a matrix a of the trace of B a and this is just X transpose Y this is just B transpose Phi by again but one of the rules that I wrote down earlier and so if you plug this back in we find therefore that the derivative wow this was really bad so we plug this back into our formula for the derivative okay you find that the derivative with respect to theta of J of theta is equal to you know one half x controls etc plus X transpose X theta minus X transpose Y minus X transpose 1 which is just X transpose X theta minus X is equal to 1 okay so we set this as 0 and we get that um which is called the normal equations and um we can now solve this equation for theta in closed form as X transpose X theta inverse times X transpose Y and so this gives us a way for solving for the least squares fit to the parameters in closed form without needing to use an iterative our library innocent okay and using this matrix vector notation I think it I know it so far few I think we did this whole thing in about ten minutes which we couldn't have if it was writing our reams of algebra okay some of you look a little bit dazed but guys this is our first learning algorithm aren't you excited this any quick questions about this before we close to today inverse of X okay don't care what you've arrived you wasn't that just respect us what had what invited pseudo interest in rivers um yeah I it turns out that in cases if x transpose x is non-invertible then you use the pseudo inverse to minimize to solve this but um intensive x transpose x is not invertible that usually means your features were dependent usually means you did something like repeat the same feature twice in your training set um so this is not invertible it turns out the minimum is obtained by the pseudo inverse and so the universe if you don't know what i just said don't worry about it usually won't be a problem yes don't take that off my yeah like we're running over let's closer today and that there are other questions I'll take them offline okay guys

How Hard Can You Hit a Golf Ball? (at 100,000 FPS) – Smarter Every Day 216



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Why Use Binary? – Computerphile



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Surely decimal numbers are easier to understand than binary? So why don’t computers use them? Professor Brailsford explains the relationships between binary, power and simplicity.

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Q&A – How Science is Taking the Luck out of Gambling



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Is stock market investing to do with luck or skill? Why do people counting cards tend to get found out? Are there any games that humans are still better at? Adam Kurcharski answers questions from the audience after his talk.
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From the statisticians forecasting sports scores to the intelligent bots beating human poker players, Adam Kucharski traces the scientific origins of the world’s best gambling strategies. Spanning mathematics, psychology, economics and physics, he reveals the long and tangled history between betting and science, and explains how gambling shaped everything from probability to game theory, and chaos theory to artificial intelligence.

Adam Kucharski is a Lecturer at London School of Hygiene and Tropical Medicine where his research focusses on the dynamics of infectious diseases, particularly emerging infections. Prior to this, he got a degree in mathematics from the University of Warwick, received a PhD in applied mathematics from the University of Cambridge and had a post-doc position at Imperial College London.

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Integration and the fundamental theorem of calculus | Essence of calculus, chapter 8



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What is an integral? How do you think about it?

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What We Cannot Know – with Marcus du Sautoy



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Is it possible that we will one day know everything? Or are there fields of research that will always lie beyond the bounds of human comprehension? Marcus du Sautoy investigates.
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Is it possible that we will one day know everything? Or are there fields of research that will always lie beyond the bounds of human comprehension? Former Christmas Lecturer Marcus du Sautoy will lead us on a thought-provoking expedition to the furthest reaches of modern science.

Marcus du Sautoy is a mathematician and popular science writer and speaker. He delivered the 2006 CHRISTMAS LECTURES on mathematics, titled THE NUM8ER MY5TERIES. He is currently the Charles Simonyi Professor for the Public Understanding of Science at the Oxford University.

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How Science is Taking the Luck out of Gambling – with Adam Kucharski



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From the statisticians forecasting sports scores to the intelligent bots beating human poker players, Adam Kucharski traces the scientific origins of the world’s best gambling strategies.
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Spanning mathematics, psychology, economics and physics, Adam Kucharski reveals the long and tangled history between betting and science, and explains how gambling shaped everything from probability to game theory, and chaos theory to artificial intelligence.

Adam Kucharski is a Lecturer at London School of Hygiene and Tropical Medicine where his research focusses on the dynamics of infectious diseases, particularly emerging infections. Prior to this, he got a degree in mathematics from the University of Warwick, received a PhD in applied mathematics from the University of Cambridge and had a post-doc position at Imperial College London.

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Origins of the Laws of Nature – Peter Atkins



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Thermodynamics. Speed of light. Conservation of energy. Where do the fundamental laws of nature come from?
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Peter Atkins explores the pieces that build up the complexity of the universe and argues that it all came from very little, or arguably from nothing at all.

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Peter Atkins began his academic life as an undergraduate at the University of Leicester, and remained there for his PhD. He then went to the University of California, Los Angeles as a Harkness Fellow and returned to Oxford as lecturer in physical chemistry and fellow of Lincoln College in 1965, where he remained as professor of chemistry until his retirement in 2007. He has received honorary doctorates from universities in the United Kingdom (Leicester), the Netherlands (Utrecht), and Russia (Mendeleev University, Moscow, and Kazan State Technological University) and has been a visiting professor at universities in France, Japan, China, New Zealand, and Israel.

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Greek Philosophers ("Can't Get You Out of My Head" by Kylie Minogue)



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Floating heads, flying Greek letters, and lots of naked statues. Oh, and the Socratic method and Plato’s Cave are in there too. And Aristotle’s Golden Mean.
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Richard Feynman Computer Heuristics Lecture



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Introduction Article to Heuristics and Metaheuristics –
Richard Feynman, Winner of the 1965 Nobel Prize in Physics, gives us an insightful lecture about computer heuristics: how computers work, how they file information, how they handle data, how they use their information in allocated processing in a finite amount of time to solve problems and how they actually compute values of interest to human beings. These topics are essential in the study of what processes reduce the amount of work done in solving a particular problem in computers, giving them speeds of solving problems that can outmatch humans in certain fields but which have not yet reached the complexity of human driven intelligence. The question if human thought is a series of fixed processes that could be, in principle, imitated by a computer is a major theme of this lecture and, in Feynman’s trademark style of teaching, gives us clear and yet very powerful answers for this field which has gone on to consume so much of our lives today.

No doubt this lecture will be of crucial interest to anyone who has ever wondered about the process of human or machine thinking and if a synthesis between the two can be made without violating logic.

x Math The secret to being a successful student



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Learn the secrets to becoming a successful math student. In this video I address the number one quality that all math students need to have to become successful in their math class

How simple ideas lead to scientific discoveries



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Adam Savage walks through two spectacular examples of profound scientific discoveries that came from simple, creative methods anyone could have followed — Eratosthenes’ calculation of the Earth’s circumference around 200 BC and Hippolyte Fizeau’s measurement of the speed of light in 1849.

المترجم: Mohamed Gouraizim
المدقّق: khalid marbou (موسيقى) أحد الأمور المضحكة بخصوص امتلاك دماغ هو عدم قدرتنا على التحكم بالأشياء
التي يلتقطها ويحتفظ بها، الحقائق والقصص. حينما تكبر تصبح الأمور أسوأ في بعض الأحيان هناك أمور
تبقى في الذاكرة لسنوات قبل أن تفهم لماذا تهمك ولماذا هي مهمة بالنسبة لك سأخبركم بـثلاثة أمثلة خاصة بي حينما كان ريتشارد فاينمان
فتى يافعا في مدينة كوينز ذهب ليتنزه رفقة والده وعربته وكرة، فلاحظ أنه حينما يجر العربة تتجه الكرة صوب مؤخرة العربة فسأل والده:
"لماذا تتجه الكرة صوب مؤخرة العربة؟" فأجابه والده:
"إنه القصور الذاتي" فسأله"ما هو القصور الذاتي؟"
فأجابة والده "إنه الاسم الذي أطلقه العلماء على ظاهرة توجه الكرة صوب مؤخرة العربة" لكن الحقيقة، أن لا أحد كان يعلم حصل فايمن فيما بعد على درجات علمية من جامعات MIT وبرينستون،
وحل لغز تشالنجر، وانتهى به المطاف بالفوز بجائزة نوبل في الفيزياء عن مخططات فايمن التي تصف حركة الجسيمات تحت الذرية. ويرجع الفضل في ذالك إلى المحادثة التي كانت مع والده والتي مكنته من فهم أن أبسط الأسئلة قد تدفعك إلى حدود المعرفة الإنسانية، وهناك كان يود أن يلعب. وقد فعل. كان إراتوستينس الأمين الثالث للمكتبة الكبرى في الإسكندرية ولقد قام بالعديد من الإسهامات العظيمة للعلوم. لكن أكثر ما يتذكره الناس قد بدأ بـرسالة تلقاها كأمين مكتبة، من مدينة أسوان، التي تقع في جنوب الإسكندرية. تضمنت الرسالة حقيقة ظلت راسخة في ذهن إراتوستينس وكانت الحقيقة أن الكاتب قال عند الظهيرة يوم الانقلاب الشمسي، عندما نظر إلى أسفل بئر عميق استطاع أن يرى انعكاس صورته في الأسفل ولاحظ أن رأسه هو الذي كان يحجب الشمس دعوني أخبركم أن فكرة كريستوف كلومبس حول اكتشاف كروية الأرض كانت ساذجة تماما، وليست صحيحة بتاتا. في الحقيقة، كل من تلقى تعليمه قد يفهم أن الأرض كروية منذ عصر أرسطو، حيث أثبت أرسطو ذالك بـملاحظة بسيطة. حيث لاحظ أن كل مرة يرى فيها ظل الأرض على القمر يكون دائرياً، و الشكل الوحيد الذي قد يخلق ظلاً دائرياً باستمرار هي الكرة، وبالتالي فالأرض كروية لكن لا أحد كان يعلم كم كان حجمها حتى استقبل إراتوستينس هذه الرسالة والتي بها تلك الحقيقة. فعلم أن الشمس متواجدة مباشرة فوق مدينة أسوان، لأنه بالنظر لأسفل البئر، كان خطا مستقيما بدايةً من أسفل البئر لرأسه وصولا للشمس. علم إراتوستينس بحقيقة أخرى. علم أن العصا المغروسة في الأرض في الإسكندرية في نفس الوقت في نفس اليوم، عند الظهيرة، عند وقت ذروة الشمس، وقت الانقلاب، الشمس تلقي بظلالها بدرجة 7.2 خارج المحور. الآن، إن كنتم تعلمون محيط دائرة، وعليها نقطتان، كل ما يتوجب عليكم معرفته هي المسافة بين هاتين النقطتين، لكي تحصلوا على محيط هذه الدائرة. 360 درجة قسمة 7.2 يعادل 50. أعلم أنه رقم تقريبي وهذا ما يجعلني أرتاب من هذة القصة أيضاً، لكنها قصة رائعة، وسنكملها. يحتاج لمعرفة المسافة بين أسوان والإسكندرية وهذا أمر رائع لأن إراتوستينس كان جيدا في الجغرافيا. في الحقيقة، هو من قدم مصطلح الجغرافيا. كان الطريق بين أسوان والإسكندرية طريقاً تجارياً، ويحتاج التجار لمعرفة كم من الوقت يُستغرق للوصول هناك. كان يحتاج لمعرفة المسافة بالضبط، وبالتالي كان يعرف بدقة كبيرة أن المسافة بين المدينتين هي 500 ميل. اضرب ذالك في 50 وستحصل على 25,000، والتي تعتبر قريبة بـ1% من القطر الفعلي للكرة الأرضية لقد قام بذالك قبل 2,200 سنة. الآن نحن نعيش في عصر حيث تستخدم فيه أجهزة بمليارات الدولارات للبحث عن بوزون هيغز. وحالياً نكتشف جزيئات قد تتنقل أسرع من سرعة الضوء، وكل تلك الإكتشافات صارت ممكنة بفضل التكنولوجيا التي طورت في العقود القليلة الماضية لكن بالنسبة لمعظم التاريخ البشري كان علينا اكتشاف هذه الأشياء باستخدام أعيننا وآذاننا وعقولنا كان أرماند فيزو عالم فيزياء يقوم بإجراء تجارب في باريس في الواقع كان تخصصه تحسين وتأكيد نتائج الآخرين، ربما يبدو هذا الأمر غير مميز، لكنه في الحقيقة يعتبر جوهر العلم، لأنه لا يوجد شيء كحقيقة لا يمكن إثباتها بشكل مستقل. وكان على ذراية بتجارب غاليليو في محاولته لإثبات إذا ماكان للضوء سرعة أم لا. فغاليلو قام بتلك التجربة الجميلة حيث كان هو ومساعده يحملان مصابيح كان كل واحدٍ منهم يحمل مصباحاً،
فيقوم بفتح مصباحه ومساعده كذالك. وحصلوا على التوقيت بشكل جيد، تعرفوا على توقيتهم بشكل جيد ووقفوا على تلين، تبعدان ميلين عن بعض، قاما بنفس الشيء على فرضية إنه لو كان للضوء سرعة ملحوظة، سيلاحظ غاليليو تأخرا في سرعة الضوء القادمة من مصباح مساعده. لكن الضوء كان سريعا جداً بالنسبه لغاليليو. كان بعيداً عن القيمة التي افترضها أن سرعة الضوء أكبر من سرعة الصوت بـحوالي 10 مرات. كان فيزو على علم بهذة التجربة. وقد عاش في باريس، وأنشأ محطتين للتجارب، تبعدان عن بعضهما حوالي 5 أميال ونصف، في باريس. وحل اشكالية غاليليو، وقد قام بذالك بـالاستعانة بقطعة من المعدات البسيطة. قام بإجرائها باستخدام واحدة من هذه. سأقوم بوضع جهاز التحكم جانباً لبضعة ثوان. لأني أريد أن أقحم عقولكم في هذا الأمر. هذه عجلة مسننة. لديها مجموعة من الشقوق وكذلك مجموعة من الأسنان. كان هذا حل فيزو لإرسال نبضات منفصلة من الضوء. قام بوضع شعلة خلف واحدة من هذه الشقوق. إذا وجهت شعلة من خلال واحدة من هذه الشقوق على المرآة، على بعد خمسة أميال، سيرتد الضوء من المرآة. وسيعود إلي من خلال هذا الشق. لكن أمر مثير سيحدث حينما يدير العجلة بسرعة. لاحظ أنه يبدو كباب بدأ بالانغلاق. إلى شعاع الضوء الذي يأتي إلى عينه. لماذا هذا؟ لأنه بسبب أن نبضات الضوء، لا تأتي من خلال نفس الشق، إنها في الواقع تصطدم بسن العجلة. وعندما يدير العجلة بسرعة كافية سيقوم بحبس الضوء تماماً. وبعد ذالك، اعتمادا على المسافة بين المحطتين وسرعة عجلته وعدد الشقوق في العجلة، قام بحساب سرعة ضوء مقاربة بـ 2% للقيمة الحقيقية. قام بتلك التجربة عام 1849. وهذا ما يجعلني حقاً مهتما بـالعلم. كلما كان لدي مشكلة في فهم مفهوم،
أذهب وأبحث عمن اكتشف هذا المفهوم. ألقي نظرة على قصة توصلهم لفهم المفهوم. وماذا يحدث حينما تلقي نظرة على ما كان يفكر فيه المكتشفون لحظة توصلهم لاكتشافاتهم، أنك تفهم أنهم ليسوا مختلفين عنا. جميعنا بشر من لحم ودم. وجميعنا بدأ بـنفس الأدوات. تعجبني فكرة اطلاق مصطلح المجالات الدراسية
على مختلف الفروع العلمية. العديد من الناس يعتقدون أن العلم مغلق، صندوق أسود، لكنه في الواقع مجال مفتوح وجميعنا مستكشفون. والناس الذين حققوا تلك الاكتشافات فقط فكروا أكثر قليلاً بما كانوا ينظرون إليه، وكانوا أكثر فضولا. وفضولهم هذا غير طريقة تفكير الناس عن العالم. وبالتالي هذا ما غير العالم. لقد غيروا العالم، وأنتم كذالك باستطاعتكم ذلك. شكراُ لكم. (تصفيق)

What is Logic? (Philosophical Definition)



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A definition of Logic as a field of philosophy, as well as several types of logic studied in philosophy, including second order logic, non-classical Logic, and modal logic.

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Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!

Information for this video gathered from The Stanford Encyclopedia of Philosophy, The Internet Encyclopedia of Philosophy, The Cambridge Dictionary of Philosophy, The Oxford Dictionary of Philosophy and more!
(#Logic #Modal)

welcome back to Carnegie store today we're going to be continuing our series dumbfounding definitions dizzying distinctions and diabolical doctrines a series sorting through some of the jargon of philosophy in this video we're going to be answering one of the questions at the center of philosophy which is what is logic now logic is one of the most basic areas of philosophies what all the rest of philosophy is built upon logic is the codification of language into a deductive system basically we're taking kind of our flowy flowering ounces of an English language and codifying it quantifying it in something that looks like math logic is the bridge between English and math logic has two main parts semantics the meanings of symbols and logics and syntax the rules that those symbols follow the purpose of semantics is to codify words and concepts in a language and put them into those shapes and symbols that make up a watch language the purpose of the syntax and the rules and the grammar is to show which arguments are valid and which are not valid of course meaning the truth of premise implies the truth of the conclusion now there's a number of different kinds of logic the most basic kind is known as first-order classical logic also just first-order logic classical logic or elementary logic it's what you'll start studying whenever you study logic after first-order logic you can kind of move up to what's known a second-order logic or higher logics contrasting with first-order logic you have non classical logic generally people to deny some element of first-order classical logic and somewhere in between first-order logic and non classical logic you have modal logics in this video very quickly we're going to go through each of these categories and give you some resources or a couple of them if you're interested in learning more so first-order logic can be generally divided into two parts you've categorical logic and propositional logic categorical logic looks like this it comes all the way back in ancient Greece all men are mortal Socrates is a man therefore Socrates is mortal he dropped pictures with Venn diagrams and do things like that it talks about the uses of terms in a language that's what it uses it is the application of predicates to objects if you want more on categorical logic I have a whole series it's about a third of the 100 days of logic propositional logic on the other hand runs arguments like this if it is raining then it is cloudy it is raining therefore it is cloudy its arguments are kind of in this form they build up truth tables and it uses full sentences of a language it connects those sentences with operators like and or it implies propositional calculus is basically another name for first-order logic it's the combination of categorical and propositional logic also known as classical or first-order logic all of these concepts are covered in a series I have known as the 100 days of logic if you're interested in learning about any of these check out that series now modal logics does a number of different modal logics we'll cover four here you have a Lepik modal logic which is the logical possibility in necessity this is the most common logic that people think of when they think of modal logic but you also have other logics that follow the same pattern such as Dejan tuka logic which is the logic of ethics and obligation you have temporal logic the logic of time the past in the future and epistemic or docks a stick or justification logic which is the logic of knowledge belief and justification if you're more interested in any of these logics we have another series called the three months of modal logics that cover all of these with a month for each of them now second-order logic is logic we don't have a series on yet but while first order logic quantified over objects or individuals such as all baseball's around second-order logic quantifies over properties or sets of objects so for all sets of for all properties for example all sets with more than three members are sets that have more than two members second-order logic and higher-order logics are the building blocks of set theory and hopefully we'll get a series on them soon we've already started touching a little bit on set theory but we haven't gotten to it properly non classical logics on the other hand are going to be logics which deny some element of classical logic such as perhaps the law of excluded middle this leads to new types of logic known as non classical logic which will have different conclusions from our original logic some examples may include many valued logic intuitionistic logic and para consistent logic once again it's another topic that I hope to have a series on if you have opinions on what my next big logic series should be on non classical logic or a second-order logics please offer them in the comments below check out my series on the 100 days of logic for more on first-order logic or the three months of modal logics for more on modal logic stay tuned for more logic series on second-order logic and non classical logic coming soon watch this video and more here at Renee DS org and stay skeptical buddy

Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14



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A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.

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"Eigenvectors and eigenvalues" هي واحدة من تلك المواضيع التي يجدها الكثير من الطلاب غير واضحة بشكل خاص. أسئلة مثل "لماذا نفعل هذا" و "ماذا يعني هذا في الواقع" غالبًا ما يتم تركها في بحر من الحسابات دون إجابة. ولأنني وضعت مقاطع الفيديو الخاصة بالسلسلة ، لقد علق الكثير منكم على التطلع إلى تصور هذا الموضوع على وجه الخصوص. وأظن أن السبب في ذلك ليس كثيرا أن الأشياء eigen- معقدة أو سيئة بشكل خاص. في الواقع ، إنها بسيطة نسبيا وأعتقد أن معظم الكتب تقوم بعمل جيد لشرحها. القضية هي ذلك من المنطقي حقًا أن يكون لديك فهم مرئي راسخ للعديد من الموضوعات التي تسبقها. الأهم هنا هو أن تعرف كيف تفكر في المصفوفات كتحولات خطية ، ولكن عليك أيضًا أن تكون مرتاحًا مع الأشياء مثل المحددات والنظم الخطية للمعادلات وتغيير الأساس. الارتباك حول المواد eigen عادة ما يكون له علاقة مع أساس هش في واحدة من هذه المواضيع مما هي عليه في eigenvectors و eigenvalues ​​نفسها. للبدء ، ضع في اعتبارك بعض التحولات الخطية في بعدين ، مثل الذي ظهر هنا. وهو يتحرك على أساس i-hat المتجه إلى الإحداثيات (3 ، 0) و j-hat إلى (1 ، 2) ، بحيث يتم تمثيلها بمصفوفة ، تكون أعمدةها (3 ، 0) و (1 ، 2). التركيز على ما يفعله لمتجه واحد معين وفكر في مدى ذلك الناقل ، الخط الذي يمر من خلال أصله وطرفه. معظم المتجهات سوف تخرج عن نطاقها خلال التحول. أعني ، قد يبدو الأمر من قبيل الصدفة إذا كان المكان الذي هبط فيه الناقل في مكان ما على هذا الخط. لكن بعض المتجهات الخاصة تبقى في نطاقها الخاص ، وهذا يعني أن تأثير المصفوفة على مثل هذا المتجه هو فقط لتمديده أو إسحقه ، مثل العدد القياسي. لهذا المثال بالتحديد ، فإن المتجه الأساسي i-hat هو أحد هذه المتجهات الخاصة. امتداد i-hat هو المحور السيني ، ومن العمود الأول للمصفوفة ، يمكننا أن نرى أن i-hat يتحرك إلى 3 أضعاف نفسه ، لا يزال على ذلك المحور x. ما هو أكثر من ذلك ، بسبب الطريقة التي تعمل بها التحولات الخطية ، أي متجه آخر على المحور السيني هو أيضا امتدت من قبل عامل 3 وبالتالي لا يزال على المدى الخاص به. ناقل أضيق قليلاً الذي يبقى على امتداده خلال هذا التحول هو (-1 ، 1) ، ينتهي الأمر الحصول على تمدد بواسطة عامل 2. ومرة أخرى ، فإن الخطية تعني ذلك أي متجه آخر على الخط القطري الذي يمتد إليه هذا الشخص هو مجرد الحصول على امتدت بها عامل 2. ولهذا التحول ، هذه هي جميع المتجهات مع هذه الخاصية الخاصة للبقاء على امتدادها. تلك التي على محور اكس تمد من قبل عامل 3 وتنتشر تلك الموجودة على هذا الخط القطري بواسطة عامل 2. أي ناقل آخر سيتم تدويره نوعًا ما أثناء التحويل ، خرج من الخط الذي يمتد. كما قد تفكر الآن ، تسمى هذه النواقل الخاصة "المتجهات الذاتية" للتحول ، وقد ارتبط كل من eigenvector بها ، ما يسمى "قيمة الذاتية" ، وهو العامل الوحيد الذي امتد من خلاله أو تم سحقه أثناء التحول. بالطبع ، لا يوجد شيء خاص حول التمدد مقابل السحق أو حقيقة أن قيم eigenvalues ​​هذه كانت إيجابية. في مثال آخر ، يمكن أن يكون لديك متجاوب مع eigenvalue -1/2 ، وهذا يعني أن ناقلات يحصل انقلبت ويسحقها عامل 1/2. لكن الجزء المهم هنا هو أنه يبقى على الخط الذي يمتد من دون أن يدور حوله. وللمحافظة على السبب ، قد يكون هذا أمرًا مفيدًا للتفكير فيه ، النظر في بعض دوران ثلاثي الأبعاد. إذا كان بإمكانك العثور على مُخترِع لهذا التناوب ، ناقل يمتد على المدى الخاص به ، ما وجدته هو محور الدوران. ومن الأسهل التفكير في الدوران ثلاثي الأبعاد من حيث بعض محور الدوران وزاوية الدوران ، بدلاً من التفكير في المصفوفة الكاملة 3-بواسطة 3 المرتبطة بهذا التحول. في هذه الحالة ، بالمناسبة ، يجب أن تكون قيمة eigenvalue المقابلة 1 ، بما أن الدورات لا تمتد أو تسحق أي شيء ، لذا فإن طول المتجه سيبقى كما هو. يظهر هذا النمط كثيرًا في الجبر الخطي. مع أي تحول خطي موصوف في المصفوفة ، يمكنك فهم ما تفعله من خلال قراءة أعمدة هذه المصفوفة مثل بقع الهبوط للمتجهات الأساسية. لكن في الغالب طريقة أفضل للحصول على قلب ما يحدثه التحول الخطي ، أقل اعتمادا على نظام الإحداثيات الخاص بك ، هو ايجاد eigenvectors و eigenvalues. لن أغطي التفاصيل الكاملة حول طرق حساب eigenvectors وقيم eigenvalues ​​هنا ، ولكن سأحاول تقديم نظرة عامة على الأفكار الحسابية التي هي الأهم لفهم مفاهيمي. رمزيا ، وهنا ما تبدو فكرة eigenvector. A هي المصفوفة التي تمثل بعض التحول ، مع v كالمخترع ، و λ هو رقم ، أي القيمة الذاتية المقابلة. ما يقوله هذا التعبير هو أن المنتج المتجه المصفوفة – A مرة v يعطي نفس النتيجة بمجرد تحجيم eigenvector v ببعض القيمة λ. حتى إيجاد eigenvectors وقيمها الذاتية من مصفوفة A ينزل لإيجاد قيم v و λ التي تجعل هذا التعبير صحيحًا. من الصعب أن تعمل معه في البداية ، لأن هذا الجانب الأيسر يمثل مضاعفة متجه المصفوفة ، لكن الجانب الأيمن هنا هو مضاعفة العددية. لذلك دعونا نبدأ بإعادة كتابة ذلك الجانب الأيمن كنوع من مضاعفة متجهات المصفوفة ، باستخدام مصفوفة ، والتي لها تأثير تحجيم أي ناقل بواسطة عامل λ. سوف تمثل أعمدة مثل هذه المصفوفة ما يحدث لكل ناقل أساس ، وكل ناقل أساس هو ببساطة λ ، لذا فإن هذا المصفوفة سيكون لها الرقم λ أسفل القطر مع الصفر في كل مكان آخر. الطريقة الشائعة لكتابة هذا الشخص هي أن تدرج ذلك وتكتبه كـ λ الأوقات I ، أين أنا مصفوفة الهوية مع 1 أسفل القطري. مع النظر إلى كلا الجانبين مثل مضاعفة متجهات المصفوفة ، يمكننا طرح هذا الجانب الأيمن وعامل v. إذاً ، ما لدينا الآن هو مصفوفة جديدة – A ناقص the أضعاف الهوية ، ونحن نبحث عن v المتجه ، بحيث تعطي هذه المصفوفة الجديدة v المتجه صفر. الآن سيظل هذا صحيحًا دائمًا إذا كانت v هي المتجه صفر ، لكن هذا ممل. ما نريده هو aigenvector غير الصفر. وإذا كنت تشاهد الفصلين الخامس والسادس ، ستعرف أن الطريقة الوحيدة الممكنة لمنتج مصفوفة ذات ناقل غير صفري لتصبح صفرًا إذا كان التحويل المرتبط بهذه المصفوفة يسحق الفضاء في بُعد أقل. وهذا الاستنزال يقابله محدد صفري للمصفوفة. لكي تكون ملموسًا ، لنفترض أن المصفوفة تحتوي على أعمدة (2 ، 1) و (2 ، 3) ، والتفكير في طرح كمية متغيرة λ من كل إدخال قطري. الآن تخيل التغيير والتبديل ، وتحويل مقبض لتغيير قيمته. حيث أن قيمة التغييرات، المصفوفة نفسها تتغير ، وبالتالي فإن محدد المصفوفة يتغير. الهدف هنا هو إيجاد قيمة λ التي ستجعل هذا العزم صفر ، مما يعني أن التحويل المعدّل يسحق الفضاء في بُعد أقل. في هذه الحالة ، تأتي النقطة الحلوة عندما يساوي λ 1. بالطبع ، إذا اخترنا بعض المصفوفات الأخرى ، قد لا تكون قيمة eigenvalue بالضرورة 1 ، فقد يتم ضرب البقعة الحلوة بعض القيمة الأخرى لـ λ. إذن هذا نوع كثير ، لكن دعنا نفكك ما يقوله هذا. عندما تساوي λ 1 ، فإن المصفوفة A ناقص the تضغط الهوية على الفضاء على خط. هذا يعني أن هناك متجه غير صفري v ، بحيث يكون ناقص the أضعاف مرات الهوية v مساوية للناقل صفر. وتذكر أن سبب اهتمامنا بذلك هو أنه يعني أن الأوقات v تساوي λ المرات v ، والتي يمكنك قراءتها كقول أن المتجه v هو عامل ملاحي من A ، البقاء على المدى الخاص بها أثناء التحول A. في هذا المثال ، تكون قيمة eigenvalue المقابلة هي 1 ، لذا فإن v سيكون في الواقع ثابتًا في مكانه. وقفة والتأمل إذا كنت بحاجة للتأكد من أن خط التفكير يبدو جيدا. هذا هو الشيء الذي ذكرته في المقدمة ، إذا لم يكن لديك فهم قوي للمحددات ولماذا تتعلق بنظم خطية من المعادلات التي لها حلول غير صفرية ، مثل هذا التعبير من شأنه أن يشعر تماما من فراغ. لرؤية هذا في العمل ، دعنا نرجع إلى المثال من البداية مع المصفوفة التي تكون أعمدةها (3 ، 0) و (1 ، 2). لمعرفة ما إذا كانت القيمة λ هي قيمة ذاتية ، طرح من الأقطار من هذه المصفوفة وحساب المحدد. عند القيام بذلك ، نحصل على بعض الحدود المتعددة التربيعية في λ ، (3-λ) (2-λ). بما أن λ لا يمكن أن تكون سوى قيمة ذاتية إذا كان هذا المحدد هو صفر ، يمكنك أن تستنتج أن القيم الذاتية الوحيدة الممكنة هي λ يساوي 2 و λ تساوي 3. لمعرفة ما هي العوامل الذاتية التي لديها في الواقع واحدة من هذه القيم الذاتية ، يقول λ يساوي 2 ، قم بتوصيل تلك القيمة بـ λ إلى المصفوفة ثم حل لأي نواقل ترسل هذه المصفوفة المتغيرة مائلًا إلى 0. إذا قمت بحساب هذه الطريقة التي تريد بها أي نظام خطي آخر ، سترى أن الحلول هي جميع المتجهات على خط قطري يمتد من (-1 ، 1). وهذا يتوافق مع حقيقة أن المصفوفة غير المعدلة [(3 ، 0) ، (1 ، 2)] لديه تأثير لتمديد كل تلك المتجهات بعامل 2. الآن ، لا يجب أن يكون للتحول ثنائي البعد متواجدة. على سبيل المثال ، ضع في الاعتبار دوران بمقدار 90 درجة. هذا لا يملك أي متابعين ، لأنه يدور كل ناقل من نطاقه الخاص. إذا كنت تحاول بالفعل حساب قيم eigenvalues ​​لدوران كهذا ، فلاحظ ما يحدث. تحتوي المصفوفة على أعمدة (0 ، 1) و (-1 ، 0) ، طرح λ من العناصر القطرية والبحث عن عندما يكون المحدد 0. في هذه الحالة ، تحصل على كثير الحدود λ ^ 2 + 1 ، والجذور الوحيدة لهذا متعدد الحدود هي الأرقام الخيالية i و i. تشير حقيقة عدم وجود حلول رقمية حقيقية إلى عدم وجود أي حركات ذاتية. مثال آخر مثير للاهتمام يستحق الإمساك في الجزء الخلفي من عقلك هو القص. يعمل هذا على إصلاح قبعة i-hat في مكانها وتحريك j-hat one over over، لذلك لها المصفوفة أعمدة (1 ، 0) و (1 ، 1). جميع المتجهات على المحور السيني عبارة عن متجهات ذاتية ذات قيمة ذاتية 1 ، حيث تظل ثابتة في مكانها. في الواقع ، هذه هي eigenvectors الوحيدة. عندما تقوم بطرح λ من الأقطار وحساب المحدد ، ما تحصل عليه هو (1-λ) ^ 2 ، والجذر الوحيد لهذا التعبير هو λ يساوي 1. يتطابق هذا مع ما نراه هندسيًا من أن جميع المتجهات الذاتية لها قيمة ذاتية 1. ضع في اعتبارك رغم ذلك ، من الممكن أيضًا أن تكون لديك قيمة ذاتية واحدة فقط ، ولكن مع أكثر من مجرد خط مليء بالمفاهيم الذاتية. مثال بسيط هو مصفوفة تقاس كل شيء بمقدار 2 ، قيمة eigenvalue الوحيدة هي 2 ، لكن كل متجه في المستوي يصبح متجاوراً ذا قيمة eigenvalue. الآن هو وقت آخر جيد للتوقف والتأمل في بعض من هذا قبل الانتقال إلى الموضوع الأخير. أريد أن أنتهي هنا بفكرة وجود eigenbasis ، التي تعتمد بشكل كبير على أفكار من الفيديو الأخير. ألقِ نظرة على ما يحدث إذا حدث أن تمكَّن لنا المتجهات الأساسية من أن تكون مجرد متجهات ذاتية. على سبيل المثال ، ربما يتم تحجيم i-hat بمقدار -1 ويتم تحجيم j-hat بمقدار 2. كتابة إحداثياتهم الجديدة كأعمدة مصفوفة ، لاحظ أن هذه مضاعفات العددية -1 و 2 ، والتي هي القيم الذاتية من i-hat و j-hat ، الجلوس على قطري لدينا المصفوفة وكل إدخال آخر هو 0. في أي وقت ، تحتوي المصفوفة على صفر في كل مكان بخلاف القطر ، يطلق عليه ، بشكل معقول بما فيه الكفاية ، مصفوفة قطرية. والطريقة لتفسير هذا هو أن جميع المتجهات الأساسية هي متجهات ذاتية ، مع الإدخالات القطرية لهذه المصفوفة كونها قيمها الذاتية. هناك الكثير من الأشياء التي تجعل المصفوفات المائلة أكثر جاذبية للعمل بها. واحد كبير هو ذلك من الأسهل حساب ما سيحدث إذا ضربت هذه المصفوفة بذاتها مجموعة من المرات. بما أن كل واحدة من هذه المصفوفات هي مقياس كل ناقل أساس بواسطة بعض القيم الذاتية ، تطبيق ذلك المصفوفة عدة مرات ، ولنقل 100 مرة ، سوف يتطابق فقط مع توسيع نطاق كل ناقل أساس بواسطة القوة 100 في القيمة الذاتية المقابلة. على النقيض من ذلك ، جرب استخدام القوة المئة عشر لمصفوفة غير قطرية. حقا ، جربها للحظة ، إنه كابوس. وبالطبع ، نادرًا ما تكون محظوظًا جدًا لأن تكون متجهاتك الأساسية هي أيضًا متجهات ذاتية ، ولكن إذا كان للتحول لديك الكثير من العوامل الذاتية ، مثل ذلك الذي يحدث في بداية هذا الفيديو ، بما فيه الكفاية بحيث يمكنك اختيار مجموعة تغطي المساحة الكاملة ، ثم يمكنك تغيير نظام الإحداثيات الخاص بك بحيث تكون هذه الموجات الذاتية هي المتجهات الأساسية الخاصة بك. تحدثت عن تغيير الفيديو الأخير أساس ، لكنني سأذهب من خلال تذكير سريع جدًا هنا كيفية التعبير عن تحويل مكتوب حاليًا في نظام الإحداثيات الخاص بنا إلى نظام مختلف. خذ إحداثيات المتجهات التي تريد استخدامها كأساس جديد ، والتي ، في هذه الحالة ، تعني اثنين من المتجهات الذاتية ، التي تجعل هذه الإحداثيات أعمدة مصفوفة ، والمعروفة باسم تغيير مصفوفة الأساس. عند شطيرة التحول الأصلي وضع تغيير مصفوفة الأساس على حق وعكس تغيير مصفوفة الأساس على يساره ، ستكون النتيجة مصفوفة تمثل هذا التحول نفسه ، ولكن من منظور نظام تنسيق ناقلات أساس جديد. بيت القصيد من القيام بذلك مع eigenvectors هو ذلك مضمونة هذه المصفوفة الجديدة لتكون قطري مع قيم eigen المقابلة لها أسفل هذا القطري. هذا لأنه يمثل العمل في نظام الإحداثيات حيث يحدث ما يحدث للمتجهات الأساسية التي يتم تحجيمها أثناء التحويل. مجموعة من المتجهات الأساسية ، والتي هي أيضا ناقلات ، يسمى ، مرة أخرى ، بشكل معقول بما فيه الكفاية ، "eigenbasis". إذا ، على سبيل المثال ، إذا كنت بحاجة إلى حساب القوة المائة عشر لهذه المصفوفة ، سيكون من الأسهل بكثير أن تتغير إلى أيغبار ، حساب القوة ال 100 في هذا النظام ، ثم تحويل إلى نظامنا القياسي. لا يمكنك فعل هذا مع كل التحولات. على سبيل المثال ، لا يحتوي القص على ما يكفي من العناصر الذاتية لإمتداد المساحة الكاملة. ولكن إذا كنت تستطيع إيجاد eigenbasis ، فإنه يجعل عمليات المصفوفة جميلة حقا. لأولئك منكم على استعداد للعمل من خلال لغز أنيق جدا لنرى كيف يبدو هذا في العمل وكيف يمكن استخدامه لإنتاج بعض النتائج المفاجئة ، سأترك موجه هنا على الشاشة. يتطلب الأمر بعضًا من العمل ، ولكنني أعتقد أنك ستستمتع به. سيكون الفيديو التالي والأخير من هذه السلسلة على مساحات ناقلات مجردة. اراك لاحقا!

Learning about Area | #aumsum



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Learning about Area.
The space occupied by a shape is called its area.

التعلم الذكي للجميع. التعلم عن المنطقة. المساحة التي يشغلها شكل يسمى منطقته. دعونا نجد مساحة هذا المستطيل. للعثور على منطقتها، ونحن بحاجة إلى العثور على عدد الساحات وحدة يمكن أن يصلح في هذا المستطيل. وحدة مربع هو مربع حيث كل جانب يساوي 1 وحدة. يبدو أن لا أحد في المنزل. الذهاب اللعب في مكان آخر. دعونا نجد مساحة المستطيل. في هذا المستطيل، يمكننا أن يصلح 4 صفوف من 2 مربعات لكل منهما. وبالتالي، فإن منطقتها تساوي 4 إلى 2. يساوي 8 وحدات الساحات. هنا، 4 هو طول، 2 هو اتساع. وبالتالي، فإن المنطقة تساوي 4 إلى 2. يساوي طول إلى اتساع. دعونا نجد مساحة مربع. في هذه الساحة، يمكننا أن تناسب 2 صفوف من 2 المربعات لكل منهما. وهكذا، منطقة يساوي 2 إلى 2. يساوي 4 وحدات الساحات. هنا، 2 هو طول كل جانب. وهكذا، منطقة يساوي 2 إلى 2. يساوي الجانب إلى الجانب. 3، 4، 5، 6، 7، 8، 9. النهاية. اشترك في قناتنا على يوتوب: سمارت ليارنينغ فور آل.

Should I Get Further Education (Master's, PhD, MBA, and More)?



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In this video I discuss whether you should get further education beyond a bachelor’s degree. There are many types of further schooling you can go into including a master’s and phd, medical school, nursing school, pharmacy school, law school, MBA, etc. For many careers like becoming a doctor, dentist, or vet you definitely need that further education, but for many other careers it’s not so obvious.

People get an MBA when they want to go into a business related field or management of some kind. An MBA is something that can also help you ease your way into the business field from another field like engineering. However, you should wait a few years and get real world experience before pursuing an MBA because it doesn’t always help and isn’t always needed.

Engineers really don’t NEED to get anything past a bachelor’s in order to start their career. A master’s does help for those who want to go more into research though. The people doing research and design work at companies are typically those with higher degrees.

Students who are considering chemistry, biochemistry, or biology should really consider getting a master’s (assuming they are not going into healthcare like becoming a doctor, pharmacist, etc). It’s been said by a lot of graduates that they felt it was tough to move up in their career with just a bachelor’s in one of those sciences. If you want to do more of the design work in medicine as an example, you’ll need to go beyond a bachelor’s.

For math and physics you also don’t NEED to get a master’s or PhD in order to start your career. Pure math students usually will get a master’s and PhD because they want to work in academia doing research/become a professor. If you want to do physics research as well then you should get a master’s or PhD.

Overall getting further education, specifically a master’s or PhD, is a great way to increase your salary, learn more about a field, qualify for certain jobs, or have a change in your career. Some of the downsides include how much it costs and the time it will take. Make sure to do your own research on the field you want to go into so you know whether it’s worth the time and money.

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this video will be focused on whether you should think about getting further education beyond just a bachelor's that will be a little more broad to capture as many careers as possible but there will be some focus on the audience of this channel which is mostly engineering math and science majors now there are many types of further education you can go into path the bachelors including a master's and PhD medical school nursing school pharmacy school law school veterinary school dental school or an NBA or Masters of Business Administration and so on so obviously jobs like being a doctor pharmacist lawyer vet etc require further education which will not be the focus of this video because plain and simple you need further education for those careers you should pick a major that fulfills all the prereqs you need and really focus on making yourself as qualified of a candidate as possible including a high GPA relevant experience good letters of rec or whatever it is the school you're attending needs many are competitive so you really want to do well even in undergrad but the rest of this video will be focused on the STEM fields as well as business so first some statistics on money the average version with a bachelor's degree in the u.s. is a salary of 56,000 and so although the Masters has an average of 68,000 which is a difference of $12,000 per year this will differ in different occupations which you can see in a Bureau of Labor Statistics article linked in the description the average cost of a master's tuition has a large range it said between 30,000 to 120,000 but let's just say it costs you 40,000 all in loans so yes a master's degree is a big financial decision but if you take all these averages and assume the tuition was all in loans then a person with a master's degree over 40 years would make about $500,000 more in their lifetime then account for the $40,000 in loans plus interest and let's say – two years worth of salary that you don't have because you are in a master's program you're still net around $300,000 more in your lifetime so yes the master's degree definitely pays off in the long run if your pay will definitely increase I like playing around with these numbers and showing them to you guys but remember this is a huge gender there's no way to predict your salary and how you'll advance throughout your career regardless of your degree many articles even said that a master's degree might not even boost your salary that much in some occupations so just be aware and do your own research on the career you want to go into now first I'm going to briefly discuss getting an MBA before I go into the engineering science and math graduate degrees an MBA is for those who are looking to go into a business-related field or management or something similar otherwise it probably won't be financially worth it people in various careers like engineering might want to switch the business side of things as well so they go back and get an MBA so they are more qualified for the job but after asking and looking around there were some things to note one is that most people should not get their MBA right after getting their bachelor's nearly everyone recommended that you go into the workforce for a few years like maybe three or five and figure out whether you actually want to get your MBA figure out whether it's actually required for what you want to do will it be financially worth it and then you can also talk to people at your job to learn more and make a more informed decision there are situations where it might help to get one right after a bachelor's like if you've been working for a business and they offer to pay for it or there's a joint program where you get a bachelor's then an MBA otherwise get your bachelor's and then try to find work in fact I read a Forbes article from an MBA graduate who said that he went to get his MBA right after undergrad only because it would be paid for but if you could go back he said you would probably not do this again it was stated a lot that an MBA is not some guarantee that you will be successful or get a huge pay increase so it's important to make an informed decision before spending tens of thousands of dollars for the degree it definitely does open doors especially at large fortune 500 companies but it's not any sort of guarantee and I noticed a lot of debate around it as well if an MBA is not for you there are other options such as a masters in finance economic statistics or accounting and these aren't the same but they do help with careers in Business and Finance and so on so now let's talk about engineering or whether you should pursue further education now I'm assuming that you've got a bachelor's in engineering and we're going to master's in the same type of engineering or be something a little different like computer engineering than software engineering as an example so do you need a master's in engineering no with just a bachelor's you will be qualified for a large majority of jobs and are able to start your career but there were two majors I consistently heard of people saying you should really get a master's and those were biomedical engineering and engineering physics for biomedical engineers you should either consider getting a master's in it or get a bachelor's in it then specialize in something in a master's program like electrical or mechanical engineering this is one of the fastest growing engineering fields at the moment but even the biomedical engineers that I talked to we're all getting their master's degree and when I asked them why the reason was pretty much the same and that it's much more important for their field engineering physics is a much less common major but I read that more than half of those students go on and pursue either a master's in physics or some type of engineering you get a good foundation on both engineering and physics but people pursue further education so they can become more of an expert on one and this is not to say that you cannot get a job with just a bachelor's obviously that would be too much of a generalization but going through lots of forums and talking to people with these degrees it came up for these two majors so much that you should really consider a masters that I really needed to address it now for most people watching again you don't need a masters but a big reason besides money or learning more about the subject to consider a masters and even a PhD is if you want to go into research and do the true design work with just a bachelor's it's less likely you'll be doing the initial design work on the newest technologies and innovations for example aerospace engineers work on the propulsion of a spacecraft they analyze commonly used propulsion methods and determine how to make everything just right for the spacecraft that's being worked on so that it can provide the right thrust to the spacecraft there's enough fuel for the mission and that's a tanker size correctly and so on but with a masters or PhD you could design the next breakthrough and propulsion methods that are being worked on that could propel the spacecraft to greater speeds as an example so yell in one case you're working on something entirely new and doing the design and research well in the other case you are doing the engineering work that takes the design and figures out how to implement it with the spacecraft again this is not to say that with a masters or PhD you will be guaranteed a job like this or that with a bachelor's you could never get a job like this there's a lot of stuff in this video that's not so black and white in fact I've heard of companies in which people with bachelor's degree we're doing the design work because they worked up to that position but generally speaking the people who work in these areas have more advanced degrees also a common thing to see on an engineering job page is something with the requirements like at least three to four years experience and a bachelor's degree or one to two years experience and a master's degree so you see how you can qualify with both degrees but they kind of count your master's as experience when you enter the workforce you don't need as much extra experience before being qualified for some of those slightly higher up positions now moving onto the sciences first when it comes to chemistry biochemistry and biology these are the most common degrees to get before going into healthcare like becoming a nurse doctor dentist pharmacist etc and actually as a side tangent psychology was listed as one of the most popular undergrad degrees to get from medical school but with these majors if you don't go into healthcare or something related you should really consider getting a masters with just a bachelor's in these Sciences it's likely you'll start out as a lab tech where you mostly will follow procedures given you by a superior it'll be much harder to further your career this isn't the only possible job but a common one but if you want to do the more academically challenging design work of creating new drugs and medicine or doing forensics work or whatever it is that you want to do you want a masters or even a PhD this will allow you to do research group up faster and do more exciting work at a company or even become a professor now moving on to physics for this major someone with the PhD so that if you want to be a quote physicist as in someone who actually does research in physics you usually need to go beyond just a bachelor's now people with just a bachelor's in physics do have job opportunities they go into a variety of fields often outside of physics like engineering jobs programming or software developer jobs finance jobs etc so it really just depends on you but for the physics research jobs like studying biotechnology astrophysics plasma physics laser physics nanotechnology etc a masters and / a PhD at least will help you get into more of the research you can get jobs in those sectors with just a bachelor's but for long-term careers and research you likely need further education now moving on to math for pure math you will definitely want to get a masters and most likely a PhD as well many of these students will work in academia and do research let's become a professor which will of course require a PhD for applied math it's not as necessary to get a master's because just like physics you can get jobs in software development computer science finance and more but by getting a master's at all for one just make your application look better but to can open you up to mathematical research jobs in industry like doing algorithm design at a company like Google or of uber lanes to optimize its dynamic pricing and shortest route algorithms I knew of a math professor whose previous job was doing cryptography for the government you can do research at big defense companies and so on something also to consider is getting a masters in something that uses applied math like maybe statistics or computer science this will make you more marketable because of your broad background as well as a graduate degree in an applied field and note when I talk about research there's academic research like at a university and various labs and there's Industrial Research which is often done at companies that is aimed at making products so academic research has much more range you can work on so much as long as someone is willing to fund it you could study black holes in space theorems and pure mathematics lasers renewable energy methods and so on and obviously these are for different degrees but there's a lot you could do it could be product related or not but Industrial Research would be like working at Lockheed Martin Intel Apple Toyota and doing research on new types of technology for the of making a product to be sold so things like nanotechnology harvesting energy from vehicles high speed circuits prosthetic body parts and so on are important to those companies so research is more specific but for something that will be put out into the market now I've said before that there are companies that will pay for further education for you which can make the decision way easier you shouldn't rely on this but do know it's an option because that would save you tens of thousands of dollars but note that a lot of people say they will start working after school then go back to get their masters later and end up not doing it the freedom of not having homework and exams can be hard to walk away from so just be aware of everything obviously people do it but just might not be as easy as you think so reasons to get a further education is so you're more qualified for a certain job that you want or you want to learn more about the field you want to increase your salary or you want to change in career some reasons to not get further education include the cost sometimes it's just not right for people depending on their situation it will take time an average of two years for a master's and it might not even guarantee advancement or a salary increase so make sure you do your own research on specifically what you want to go into and note this video does not cover how competitive certain professional schools are or how big the job market is for some of those degrees there's a lot of controversy and debate even from people with these degrees as to whether it was worth it so definitely do your own research but hopefully this gave you a starting point and I hope you guys comment below any specific questions you have because I know this was mostly a general video but if you like this information don't forget to Like and subscribe and I'll see you all next time

Basic Division for Kids | #aumsum



Views:981273|Rating:3.96|View Time:6:2Minutes|Likes:2306|Dislikes:607
Basic Division.
Do you know why we divide? We divide to distribute things equally.
There are 8 carrots, they need to be divided equally among 4 rabbits. How many carrots will each rabbit get?
First take away 4 carrots so that each rabbit gets 1 carrot each. 4 carrots are still left.
Take away these also one by one. No more carrots are left. All the carrots are distributed and each rabbit has 2 carrots.
We have taken away 4 carrots 2 times. This can be written as 8 minus 4 is equal to 4. 4 minus 4 is equal to 0.
Thus, 8 divided by 4 is equal to 2.
This means for equal distribution, that is, for division, we do repeated subtraction and the answer is the number of times we subtract the same number.

شعبة الأساسية. الساحر في فلوب مشاهدة. هل تعرف لماذا نقسم؟ نحن نقسم لتوزيع الأمور على قدم المساواة. علامة الانقسام هو. هناك 8 الجزر، فإنها تحتاج إلى تقسيم بالتساوي بين 4 الأرانب. كم عدد الجزر سوف تحصل على كل أرنب؟ أولا يسلب 4 الجزر بحيث يحصل كل أرنب 1 الجزرة لكل منهما. لا تزال 4 الجزر. يسلب هذه أيضا واحدا تلو الآخر. لا يترك المزيد من الجزر. وتوزع كل الجزر ولكل أرنب 2 الجزر. لقد أخذنا بعيدا 4 الجزر 2 مرات. هذا يمكن أن تكون مكتوبة 8 ناقص 4 يساوي 4. 4 ناقص 4 يساوي 0. وهكذا، 8 مقسوما على 4 يساوي 2. وهذا يعني للتوزيع المتساوي، وهذا هو، للتقسيم، ونحن نكرر الطرح والجواب هو عدد المرات التي طرح نفس العدد. دعونا نفهم الانقسام بمساعدة مثال آخر. تقسيم هذه الكرات 12 بين 3 صناديق. باستخدام نفس المفهوم من الطرح المتكرر، طرح 3 مرارا وتكرارا من 12. 12 ناقص 3 يساوي 9. 9 ناقص 3 يساوي 6. 6 ناقص 3 يساوي 3. 3 ناقص 3 يساوي 0. 1 مرة. 2 مرات. ثلاث مرات. 4 مرات. كما فعلنا الطرح 4 مرات، 12 مقسوما على 3 يساوي 4. النهاية. اشترك في قناتنا على يوتوب: سمارت ليارنينغ فور آل.

Thinking outside the 10-dimensional box



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Visualizing high-dimensional spheres to understand a surprising puzzle.
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Podcast!
Check out Ben Eater’s channel:

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Animations largely made using manim, a scrappy open source python library.

If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
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数学在二维/三维空间中有许多巧妙的几何论证思维方法, 十分迷人 在这两个维度下, 我们只在二维/三维数对中转换, 且可以借助许多空间上的手段去具象化它们以帮助理解 例如,想象一个用原点为中心半径为一的圆 你相当于在寻找每个可能的数对使其满足该特征 即 x² + y² = 1 且 这里的用处是,许多在代数环境下看起来难以理解的东西 在几何环境下变得清晰起来, 且反之亦然 实话说我们频道一直都在靠这 在两个环境中互相转化来获取好处 因为它为我们提供了一个非常多的智慧来让我们连接两个 看似独立的想法,而且我说的不仅是一对或多个数字或空间上来来回回的大体思想。

Marathi – Syllogism tricks – MPSC / IBPS PO / SBI / BANK PO CLERK / Maharashtra exams



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