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

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إذا طلبت منك أن تفكر في شيء صعب ما هو أول شيء يتبادر إلى ذهنك؟ حسناً، إذا كنت مثلي ربما فكرت في بعض معادلات الرياضيات الطويلة و المعقدة، أليس كذلك؟ بالنسبة لمعظم الناس، تعتبر الرياضيات واحدة من أكثر المواد الصعبة. فهي عويصة ومعقدة، ولسوء الحظ لهذه الأسباب، اعتقد الكثير من الناس أنهم لن يحبوا الرياضيات أبداً. وهذا غير صحيح لأن الرياضيات مهارة يمكن تعلمها تماماً مثل أي مهارة أخرى. ولأنك تشاهد هذا الفيديو نأمل أنك لست من هؤلاء الأشخاص. وآمل أنك تؤمن حتى ولو لدرجة بسيطة بأنك ستصبح أفضل في الرياضيات و لديك الدافع للقيام بذلك. وإذا كنت كذلك، السؤال الذي يطرح نفسه هو كيف تصبح أفضل في الرياضيات؟ حسناً، لحسن الحظ، هذا واحد من الأسئلة التي إجابتها بسيطة. إذا كنت ترغب في أن تصبح أفضل في الرياضيات عليك القيام بحل الكثير والكثير من المسائل الرياضية. وكلما كانت أصعب كان ذلك أفضل. لأن المسائل الصعبة تزيد من فهمك وتقودك الى حلول جديدة. ولكن، في سياق دراسة الرياضيات والعمل من خلال هذه المسائل الصعبة ستصادف في نهاية المطاف مسائل محيرة، وستصبح عالقاً تماماً وعندما تصل إلى هذه النقطة، من المهم أن تعرف كيفية حل هذه المشاكل، لأن هذه المسائل ستزيد من مستوى فهمك وتبني مهاراتك. وهذا ما أريد التركيز عليه في هذا الفيديو. أريد أن أقدم لكم التقنيات العملية لحل هذه المسائل والتي تبدو للوهلة الأولى أنها لا يمكن لأحد أن يحلها في البداية، أود أن أركز على نصيحة ذكرها عالم الرياضيات الهنغاري (جورج بوليا) في كتابة "كيف تحلها" عام 1945. وهي كالتالي: " إذا لم تستطع حل مسألة ما، إذن فإن هناك مسألة أسهل يمكنك حلها، اعثر عليها." وهذا في رأي، أهم تقنية لفهمها وتطبيقها عملياً عند محاولة حل المسائل الرياضية الصعبة. لأن الرياضيات مبنية على رياضيات. بُنيت المفاهيم المعقدة من مفاهيم بسيطة وإذا لم يكن لديك فهم جيد للمبادئ الأساسية، إذن ستصبح المسائل المعقدة محيرة بالنسبة لك. إذن، إذا مررت بمسألة ما ولم تستطع حلها، أولاً، تعرف على مكوناتها أو العمليات التي يتُطلب القيام بها. غالباً، ستكون المسائل المعقدة ذات خطوات عديدة. الآن، ما يمكنك فعله في هذه الحالة هو تقسيم المسألة إلى مسائل متعددة وذلك من شأنه أن يعزل إما المكونات أو العمليات. أريد أن أبين ذلك عملياً لذلك دعونا نطبق ذلك من خلال مثال سريع الآن، لدي مسألة سهلة جداً لذلك دعونا نقوم بشىء أكثر تعقيداً قليلاً إذن، هذه مسألة جمع و تستخدم الرمز اليوناني، سيغما. وتعني أننا سنقوم بإضافة سلسلة من التعبيرات والتي تستخدم المتغير الذي يبدأ عند 1 وينتهي عند 4 ولكن، إذا لاحظت أن مسألة الجمع هذه بها (أس كسري) الآن، ربما البعض منكم أذكياء في الرياضيات ويمكنه حل مسائل كهذه بسهولة لكن، من الممكن أنك لا تملك فهم جيد لقواعد الجمع أو للأسس الكسرية. لذا، عند حلك لمسألة تحتوي على الأثنين معاً، فإنك تحتار إذن، على افتراض أن هذا ما يحدث، دعونا نقسم هذه المسألة إلى مسألتين أبسط كلاً منهما يركز على مفهوم واحد فقط من المفاهيم الأساسية. أولا، دعونا نصنع مسألة جمع أسهل والتي تتخلص من الأس الكسري تماماً الآن، كل ما عليك القيام به هو تكرار هذه الجملة أربع مرات ثم تضيف الإجابات والتي تعطينا الإجابة النهائية (66). والآن دعونا ننتقل إلى الأس الكسري. الآن، سأحله بطريقة سريعة جداً لأن هذا ليس درساً على الأسس الكسرية يمكنك إعادة كتابته كأربعة إلى قوى ثلاثة مضروبة في قوى النصف وبعد ذلك، يمكنك إعادة كتابة ذلك مرة أخرى مع الجذر التربيعي لأربعة مرفوعة إلى قوى الثلاثة. وبمجرد أن تفعل ذلك، تحصل على الجواب (8) الآن، الفكرة الأساسية من حل هذه المسائل البسيطة ذات المفهوم الواحد هو السيطرة على المفهوم الأساسي أو العملية التي تعمل عليها. لذا، إذا حللت بعض المسائل ولاتزال تشعر أنك غير واثق من استيعابك للمفهوم استمر في الحل حتى تصبح واثقاُ. تذكر، يعني الإتقان أنك لن تكون قادراً على ارتكاب الأخطاء ليس فقط الحصول على إجابة صحيحة مرة واحدة فقط على أي حال، بمجرد اتقانك للمكونات الأساسية عن طريقة العزل والتفكيك الآن يمكنك العودة للمسألة الأكثر تعقيداً التي تجمع المفاهيم كلها. في هذه المرحلة، يجب أن تكون قادرة على حل المفاهيم المفككة بسهولة وهو ما يعني أنه يمكنك من خلال كل قوتك العقلية حل مسائل جديدة ومبتكرة إذا تعلق الأمر بالترادف. الآن، هناك طريقة واحدة إضافية لتبسيط المسائل الصعبة التي أريد أن أتحدث عنها وقد تكون خمنتها مسبقاً إذا كنت قد أعرت انتباهاً للأمثلة. لم أستخدم أرقاماً معقدة جداً. لم أستخدم ارقاماً طويلة. لم أستخدم النقاط العشرية. لم أستخدم كسوراً كبيرة. والتزمت بالحد المنخفض من مسائل الجمع. يمكن للأرقام الكبيرة والمعقدة مع الكثير من النقاط العشرية أن تصرف انتباهك بعيداً عن المفاهيم والعمليات المفترض عليك القيام بها. لذا، إذا لم تستطع حل مسألة بها هذا النوع من الأرقام قم بحل مسألة مماثلة بأرقام صغيرة التي من السهل اضافتها وحسابها في رأسك، بهذه الطريقة من الممكن أن لا يكون لديك خبرة في المفاهيم . وبطبيعة الحال، أحياناً لديك فهم ضعيف للمفاهيم والعمليات نفسها لتتمكن من التعامل معها وحلها. وفي هذه الحالة، فقد حان الوقت للتعلم. انظر جيداً في كتابك، والق نظرة على ملاحظاتك. أو ابحث على أمثلة على الأنترنت والتي يمكنك اتباعها خطوة بخطوة ليمكنك أن ترى كيف وصل الناس إلى الحلول، وذلك باستخدام هذه المفاهيم. وإذا كنت بحاجة إلى ذلك، يمكنك الحصول على الحل خطوة بخطوة للمسألة التي تعمل عليها أيضاً. هناك العديد من الأدوات التي يمكنك استخدامها للقيام بذلك. وهناك أداتين أود التركيز عليهما في هذا الفيديو و التي هي أفضل الأدوات التي عثرت عليها هما (WolframAlpha) و (Symbolab) كل من هذه المواقع تسمح لك بكتابة معادلة والحصول على حلها وأيضاً ( gypha) و (Symbolabs) والتي تمكنك من متابعة الحلول. الفرق بين الاثنين: أن (WolframAlpha) أكثر قوة وقدرة، ولكنه يتطلب منك أن تدفع جزء مقدماً إذا كنت تريد للحصول على تلك الحلول خطوة بخطوة. على النقيض من ذلك، في حين أن كتابة المعادلات في (Symbolab) أبطأ قليلاً وأقل سهولة من (WolframAlpha) فإن حلول المسائل مجانية. بغض النظر عن الأداة التي اخترتها النقطة الأساسية هي أنه في بعض الأحيان قد يكون من المفيد أن نرى الحل خطوة بخطوة لمسألة لم تستطع حلها. ولكن، هناك تنبيهان مهمان جداً اولاً في المقام الاول وقبل كل شي، قبل أن تنطلق سريعاً إلى تلك المواقع لإيجاد الحل، اسأل نفسك "بصراحة، هل استخدمت ذهني لحل هذه المسألة أولاً ؟ " أن تجهد نفسك ذهنياً لحل المسألة بنفسك أمر مطلوب حيث أنه يزيد من قدراتك. و سوف يجعلك أفضل في الرياضيات أما مجرد معرفة الحلول لن يفيدك الآن، إذا كنت بحاجة إلى حل، فلا بأس. ابحث عنه، اتبع الخطوات وتأكد من أن تفهم كيف صار هذا الجواب ولكن، بمجرد فعل ذلك، تحدى نفسك للعودة وإعادة صياغة المسألة دون النظر إلى المرجع. من المهم أن تنتبه لهذا الأمر لأنه إذا كنت تود أن يتحسن مستواك في الرياضيات فإن الفكرة الأساسية هي إتقان المفاهيم التي تتعامل معها. الخطرالذي يكمن في النظر إلى الحلول هو أنه في مجال الرياضيات فإنه من السهل متابعة الحل خطوة بخطوة وفهم ما يجري. ولكن هذا يختلف تماماً عن أن تكون قادراً على القيام بذلك بنفسك. وهذا يقودني إلى نصيحتي النهائية لك. و هذا مهم بشكل خاص لمن يدرسون الرياضيات ومكلفين بحل الواجبات المنزلية. لا تتعجل في حل مسائل الرياضيات. وأعلم أنه من المغري محاولة حل الواجبات المنزلية بأسرع ما يمكن وقد قدمت فيديو يتكلم عن ذلك ولكن، مع الرياضيات والعلوم أو أي نوع من المواضيع المعقدة التسرع سيسبب مشكلة لك. لأنه مع العجلة، لن تتمكن من إتقان المفاهيم. فأنت وجهت قدراتك إلى الحل فقط وكان جل همك هو أن تنتهي من حل الواجب وفي وقت لاحق، عندما ستجلس في غرفة الاختبار، أو سيتوجب عليك تطبيق ما تعلمته في العالم الحقيقي ستحصل حينها على درس قاسي حول الاشياء التي تجهلها لذا، دعونا نلخص ما تعلمناه إذا كنت ترغب في التحسن في الرياضيات وترغب في تحسين قدراتك في حل هذه المسائل الصعبة أولاً، حدد المفاهيم أو العمليات المستخدمة في المسألة ومن ثم قم بعزلها. قم بحل المسائل التي تتضمن مفهموم واحد ومن ثم أتقن كل مهارة على حدة يمكنك أيضاً تبسيط المسألة من خلال ترك جميع المفاهيم كما هي لكن بتبديلها بأرقام سهلة التعامل. إذا كنت بحاجة إلى مساعدة في فهم المفاهيم تصفح كتابك أو ابحث في الأنترنت ابحث عن مسائل مشابهة، أو استخدام أداة مثل ( WolframAlpha) أو (Symbolab)
للحصول على الحلول خطوة بخطوة للمسألة التي تعمل عليها وأخيراً، لا تتعجل في حل الواجبات المنزلية. تأكد من أنك تركز باهتمام على اتقان المفاهيم، ليس فقط أن تنتهي نأمل أن هذه النصائح أعطتك الثقة للتعامل مع بعض مسائل الرياضيات الصعبة وتطوير مهاراتك الحسابية. وعلى تلك المذكرة، أريد أن أذكر لكم أحد اقتباسات الفيزيائي العظيم، ريتشارد فاينمان، الذي قال: " لا أؤمن بفكرة أن هناك عدد قليل من الأشخاص المميزين قادرين على فهم الرياضيات بينما يعتبر البقية أشخاصاً عاديين" ، الرياضيات هي من اختراع البشر وليست خارج حدود فهم البشر" في نهاية المطاف، مستواك في الرياضيات وأي شىء يتعلق بهذا الموضوع يبدأ في التحسن مع وجود الثقة وأثناء حلك للمسائل و تشغيل دماغك سوف تزيد ثقتك بطبيعة الحال وتصبح شيئاً لازماً فيك إذا كنت مهتماً بأن تبدأ التعلم الآن، المكان المثالي لكي تبدأ رحلتك من خلاله هو (Brilliant). منصة تعليم تستخدم التدريب العملي على حل المسائل كأساسيات لمساعدتك على تعلم الرياضيات والعلوم وعلم الحاسوب، وبفعالية كبيرة. أنا أدرس دورة في أساسيات علم الحاسوب في الموقع وبينما كنت في المرحلة الأول في الخوارزميات خضت اختبار قصير أجبرني على أن أتعلم مهارة جديدة في الرياضيات لم أتعلمها قبل فاضطررت الى البحث في موقع الويكي وفي بعض المسائل المشابهة وفي النهاية اضطررت لأن أخرج ورقة من دفتر الملاحظات ورسمت الخوارزميات خطوة بخطوة حتى أتمكن من فهم ما يجري. هذه الطريقة منحتني تجربة مميزة وفعالة في التعلم لم أتعلمها في محاضراتي الجامعية. وهذه الأنواع من التحديات هي التي تجبرك على البحث تعتبر أساساً في جميع دوراتهم. وتشمل الاحتمالات والمنطق وحساب التفاضل والتكامل علم الفلك، ذاكرة الكمبيوتر وأكثر من ذلك. بالإضافة إلى الدورات المنظمة لدى Brilliant أيضاً تحديات أسبوعية والتي يمكن أن تستفيد منها في تحسين مهاراتك ويمكنك في الموقع أن تتحدث مع زملائك الطلاب ولديهم موقع ويكي مفصل ومفيد مع الكثير من التفسيرات والأمثلة. لذا، إذا كنت تريد تجربة Brilliant اضغط على الرابط في صندوق الوصف واشترك مجاناً لتبدأ التعلم. وكمكافأة، إذا كنت من بين أول 200 مشترك، ستحصل أيضاً على تخفيض 20٪ من اشتراكك السنوي. أريد أن أتقدم بالشكر الجزيل لـ Brilliant لرعايتها هذا الفيديو ومساعدتها في دعم هذه القناة وكما هو الحال دائماً، شكراً جزيلاً للمشاهدة. إذا وجدت هذا الفيديو مفيداً قم بضغط زر الإعجاب لدعم هذه القناة وربما يمكنك أن تشاركه مع صديق يمكن أن يستفيد منه أيضاً يمكنك الاشتراك هناك وقم بضغط زر الجرس لتكون أول من يعرف عند نزول فيديو جديد أو انقر هنا للحصول على نسخة مجانية من كتابي حول كيفية الحصول على أفضل الدرجات. يمكنك أيضاً النقر هنا للاستماع لأحدث حلقة بودكاست عن كيف تسحق مقابلات العمل ويمكن النقر هنا لمشاهدة فيديو آخر تابع لهذه القناة شكراً لمشاهدة وأراكم في الفيديو القادم
ترجمة: فريق أُترجِم @autrjim

Lecture 1 | The Fourier Transforms and its Applications



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Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood provides an overview of the course, then begins lecturing on Fourier series.

The Fourier transform is a tool for solving physical problems. In this course the emphasis is on relating the theoretical principles to solving practical engineering and science problems.

Complete Playlist for the Course:

EE 261 at Stanford University:

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this presentation is delivered by the Stanford center for professional development we are on the air okay welcome one at all and as I said on the TV when you were walking in but just to make sure everybody knows this is e 261 the Fourier transform and it's applications Fourier transforms at all Fourier and my name is Brad Osgood circulating around are two documents that give you information about the class there is a general description of the class course information how we're going to proceed some basic bookkeeping items I'll tell you a bit more about that in just a second and also a syllabus in a schedule and I'll also say a bit more about that in just a second let me introduce our partners in crime in this course we have three courses fins thomas john thomas one stand up where's Thomas there we go Rajiv Agarwal I smell that right very good Reggie going to stand up is Rajiv and Michael medias okay so far am gonna correct that okay that's like a Metis everybody thank you all right so now and we will be setting up times for the review sessions and so on all right you know so that'll be that will be forthcoming we have a web page for the course some of you may have already visited that but let me give you the and it's the addresses on the one the sheet one of the sheets that's being passed around but let me write that up now so you can be sure to visit it and register for the class because it is on the web page that you will find course handouts course information I will email people via the web page alright so you have to be registered I five to send an announcement to the class post an announcement and send out an email then that'll be done through the web page and you have to be registered on the Paige in order to get those emails I won't be doing it through access all right so it is at like many of the other classes HTTP slash slash however you do those where the colons go or is it here EE class Sanford et you you can find it very easily edu slash e 261 okay go there if you have not already and and register yourself for the class all right now let me say a little bit about the information that you have I want to say a little bit more about the mechanics I'll talk more about the content in just a second the let me say a bit about the syllabus and schedule and the course reader the syllabus is as I said on the on the top an outline of what we're going to be doing I hope a fairly accurate outline of what we're going to be doing but it's not a contract alright so there will be a natural ebb and flow of the course as things go along and when we get to particular material or what we cover in what order this is more or less I say accurate but it is not written in stone what you should use it for however is to plan your reading so I things will be much better for all of us if you read along with with the material as the syllabus has the schedule basically outlines all right because there's there times what I'm going to want to want to skip around a little bit there are times and I'm going to derive things there are times when I'm not going to derive things and you'll get much more out of the lectures our time together if you've read the material thoroughly before you come to class so that's one thing I ask you to do we have two exams scheduled we have a midterm exam and a final exam I'm going to schedule the midterm exam midterm exam is is already actually on here at least tentatively sort of toward the end of October we'll have it outside of class that is it'll be a sit down regular exam but I want to do it for 90 minutes rather than 15 min 15 minutes is just too short at a time for a class like for material like this so it'll be a 90 minute exam and we'll schedule it several sessions outside of class this is the way I've usually done it hasn't been any problem it's worked out alright for everybody so we'll have you know alternate times and so on and the final exam is scheduled by the char's office do not come to me right before the final exam saying oh I scheduled a trip out of town I hope that's not a problem right you know what the dates are ahead of time we'll also have regular problem sets none of these things that I'm saying should be new to you you've been through the drill many times the first problem the problem sets are going to be I had a starling innovation last time I taught the course where I handed out the problem sets on Monday and had them do the following Wednesday so you actually had you know like a week and a half to do the problem set so there was overlap between the two and people thought that was just a brilliant idea so we're going to do that again this year except for the first problem set and I decided it was not such good policy to hand at the very first problem set on the very first day of class so I'll hand that out on Wednesday and I'll post that also or at least I'll post it on that sure I'll hand it out it'll be available on Wednesday and it'll be due the following Wednesday and again these sorts of things are pretty routine for you I'm sure even through many times it will be practice although again not necessarily every time without fail to have MATLAB problems on the homework one or two MATLAB problems on the homework so I'm going into the assumption that people have some experience with using MATLAB you don't have to be terribly advanced and also access to using MATLAB so if you do not have experience using MATLAB and you do not have access to MATLAB get some experience and get some access won't be hard okay now let me say us a little bit about some of the things this is the course reader for the course it's available at the bookstore and also available on the course website all right doesn't have the problems in it but it has the material that were going to be covering in class now this is a basically a stitched together set of lecture notes that I've been using for a number of years in the class and I sort of tinker with it every time I teach the class but because it is a stitched together set of lecture notes they're the organization is sometimes a little bit odd like you have an appendix in the middle of the chapter and what that means is it was used to be an appendix to a set to a particular lecture that went on that particular day and it never got moved to anywhere else all right so the organization can be a little bit funny you can help on this all right that is if you find typos if you find errors if you find things that are less than clear in their in their wording if you want if you if you have some other ideas for you apples or other explanations please tell me I am working on this I have to say that the I'm hope because these these were written as a set of lecture notes these are meant to be a good and I hope helpful companion for the class that is they're meant to be read and they're meant to be used so you can help as generations of students in the past have helped try to refine these and turn them into something that's really a good accompaniment to to the class as we go on okay one other thing that's special this quarter is the class is as always taped and the lecture notes the lectures are going to be available to everybody but this time for the first time the lecture is going to be available to the world all right Stanford is decide on an experimental basis we're sort of competing with MIT here I think to try to make some classes some of the materials for some classes available to the world all right so the Elektra notes are going to be everything's going to be done through the website but instead of needing a Stanford ID to view the tape lectures I think anybody in the world can view these lectures was a little bit daunting I have to watch my language to try to dress well all right so we'll see what we'll see what goes with that I will however issue a warning I will not answer the world's email all right I will answer email from the class but I will not answer and I think I speak for the TAS here the TAS neither will answer the world's email on this all right how we're going to keep the world out of our inboxes I'm not sure exactly whether this is going to be a problem or not but at any rate that's what's happening okay all right any questions about that any questions about the mechanics of the course or what your expectations should be what my expectations of you are okay all right now I always like to take an informal poll actually when we start this class that's what it's a number of times now and it's always been a mixed crowd and I think that's one of the things that's attractive about this class so let me ask who are the E's in this class who are an electrical engineering your undergraduate or graduate all right so that's a pretty strong show of hands but let me also ask who are the non E's in this class all right that's also a pretty strong show of hands the EES are as is typical the majority of the students in the class but there's also a pretty strong group of students in this class who are not elect engineers by training by desire by anything all right and they usually come from all over the place I was looking at the web I was looking at the classes before I got the class and I think there's some people from chemistry somebody from chemistry anybody from chemists I thought there were somebody up see back there all right and other some people from Earth Sciences uh somebody from somebody is talking actually from Earth Sciences this morning somebody from Earth Science okay where else I think there was an Emmy couple of Emmys maybe yeah all right now that's important to know I think the course is very rich in material all right rich in applications rich in content and it appeals to many people for many different reasons okay for the ease and who are taking the class you have probably seen a certain amount of this material I don't want to say most of the material but you probably seen a fair amount of this material scattered over many different classes but it's been my experience that one of the advantages of this class for electrical engineering students either undergraduate or graduate students is to see it all in one piece all right to put it all in your head at one time at least once all right because the subject does have a great amount of coherence it really does hang together beautifully for all the different and varied applications there are core ideas and core methods of the class that it is very helpful to see all at once alright so if you have seen the material before that's fine I mean that is I mean that you can you can draw on that and draw on your experience but don't deny yourself the pleasure of trying to synthesize the ideas as we go along I mean there's nothing so pleasurable as thinking about something you already know trying to think about it from a new perspective try think about it from a new point of view trying to try to fold it into some of the newer things you'll be learning so I have I've heard this from electrical engineering students many times in the past that it's a it's a pleasure for them to see the material all together at once it may seem like a fair amount of review and in some cases it will be but not in all cases and even if it is a review they're often slightly different twists or slightly new takes on things that you may not have seen before I may not have thought of quite in quite that way so so so that is my advice to the electoral engineering students for the students who have not seen this material before they're coming out of from a field and maybe only heard you know secret tales of the Fourier transform and its uses well I hope you enjoy the ride because it's going to be a hell of a ride a heck of a ride as we go along alright now for everyone I sort of feel like I have to issue I don't know if I call this a warning or just sort of a statement a principal or whatever this is a very mathematical class this is one of the sort of Holy Trinity of classes in the Information Systems lab in electrical engineering the electril engineering is a very broad department and split up into a number of laboratories along research lines I am in the Information Systems lab which is sort of the mathematical part of the subject there's a lot of signal processing coding Theory imaging and so on and this course has been for a number of years taught by faculty sort of thought of as a cornerstone in the signal processing although it has a lot of different applications to a lot of different areas the other courses in that Holy Trinity are 263 dynamically near dynamical systems and 270 a statistical signal processing who's taken to say whose likes let me ask you so because this is also very common who's taken to 63 in the class also a strong majority and who's taken to 78 yeah ok so there's a fair a little little bit less but still number of people we will actually see not so much with 270 oh well actually with both classes with 263 in 278 you'll actually see some overlap that I also hope you find interesting the language will be slightly different the perspective will be slightly different but you see this material in this class melding over into the other classes and vice-versa and again I think it's something that you can really draw on and I hope you enjoy all right so it is those those classes and the perspective that we take the faculty your teaching those classes is a pretty mathematical one but it's not a class in theorems and proofs you can breathe a heavy sigh of relief now all right I can do that but I won't all right I will derive things I'll derive a lot of a number of formulas I'll derive it and I'll go through those derivations or I'll hope that you go through the derivations in the book when I hope and I think that they will be helpful all right and when in some case that is there's an important technique or there's an important idea that you'll see not only in the tick Euler instance but over all that you'll see the same sort of derivation the same sort of ideas be applied not only for one formula but for other sorts of formulas and also in some cases to my mind as twisted as that may be I sometimes think of the derivation of a formula almost as identical with a formula I mean to use the formula effectively almost as to know the derivation because it's to know where it applies and to know how it applies and where to expect to use it all right so that's why I will go through those things for the purpose of teaching a certain amount of technique and for the purposes of sort of having those techniques really at your fingertips so that you can apply them again in a situation that may not be quite identical with with what we did but will be similar enough so that the simp so that the ideas may apply in this situation that's that's very important we will also do plenty of different sorts of applications but again because the field the subject is so varied and because the clientele because the students in the class are also varied will try to take applications from different areas will have applications from electrical engineering but will also have applications from physics and from other areas i i've also done in the past and will see if i get to this some applications from Earth Sciences for example and we'll just see how they go so we all have to cut each other a little bit of slack and if an application or particular area is not exactly to your liking well chances are it might be to somebody's liking to your right or left so you say cut everybody should cut each other a little slack and just enjoy the ride I should also say that many of the more specialized applications are found in more specialized courses all right so we will touch on a lot of things and I will use the words that are used in a lot of different courses and a lot of different subjects but we won't always do see an application to its bitter end so to speak or we won't do every pot we certainly won't do every possible application because there are just so many of them so you will find you will not run out of ways of using the Fourier transform and Fourier analysis techniques in any classes here they go it goes on and on and on but we'll only be able to see a certain amount of a certain amount of that all right and actually that leads to a very important point release of the start of the class that is where do we start all right that is this subject which is so rich and so diverse forces you forces me forces all of us to make hard choices in some ways about where what we're going to cover where we're going to start what direction we're going to go and all the different choices are defensible you will find books out there that take very different taps toward the subject they take different starting points they have different emphases they go off in different directions and you can make a good argument for any one of those choices but you have to make a choice so for us we are going to choose I have chosen not we me I have chosen to start the class with a brief discussion of Fourier series and go from there to the Fourier transform all right whereas it is also very common choice to forget about Fourier series and maybe pick them up a little bit along Angier or pick them pick them up a little bit on the edges or assuming that everybody seen Fourier series then go right into the fray transform I don't want to do that because I think that the subject of Fourier series is interesting enough in it we're not going to do very much with it but it's interesting enough in itself again it's something you may have seen in different context but it provides a natural transition to the study of the Fourier transform and it is historically actually the way the subject developed okay so that's how we're going to that's how we're going to do things will start with Fourier series and use them as a transition to Fourier transform now first of all what is this concerned with overall I it may be a little bit too strong a statement but for our purposes I want to identify the idea of Fourier series as almost identified with the study of periodic phenomena alright so for us it's identified most strongly with a mathematical analysis of periodic phenomena now it certainly shouldn't be necessary for me to justify periodic phenomena as an important class of phenomena you have been studying these things for your entire life pretty much ever since the first physics course you ever took where they do the harmonic oscillator and then the second physics course you took where they did the harmonic oscillator and then the third physics course you took rhythm they did the harmonic oscillator you have been studying periodic phenomena alright so that shouldn't be a controversial choice Fourier series goes much beyond that but it is first and foremost for us associated with a study of periodic phenomena the Fourier transform in although again it doesn't maybe doesn't do it's just justice completely is can be viewed as a limiting case of Fourier series it has to do with a study of the mathematical analysis on phenomena so if you want to contrast Fourier series and Fourier transforms then that's not a bad rough-and-ready way of doing it doesn't it say it doesn't capture everything but it captures something so Fourier transform as a limiting case and in a meeting that I'll make more precise later is limiting case of Fourier series Fourier series of free series techniques is identified with or has to do with is concerned with how about that for weaseling way out of it is concerned with the analysis of non periodic phenomena so again it doesn't say everything but it says something and one of the things that I hope you get out of this course especially for those of you who have had some of this material before are these sort of broad categorizations that help you sort organize your knowledge all right it's a very rich subject you've got to organize it somehow otherwise you'll get lost in the details all right you want to have certain markers along the way that tell you how to think about it how to organize it what what what a particular formula what cat it what general category it fits under okay now it's interesting is that the ideas are sometimes similar and sometimes quite different and sometimes it's the situation is simpler for periodic phenomena sometimes the situation is more complicated for periodic phenomena so it's not as though there's sort of a one-to-one correspondence of ideas but that's one of the things that we'll see and one of the reasons why I'm starting with Fourier series is to see how the ideas carry over from one to the other see where they work and see where they don't work alright some ideas carry easily back and forth between the two some phenomena some ideas some techniques some don't and it's interesting to know when they do and when they don't sometimes the things are similar and sometimes they're not now in both cases there are really to kind of inverse problems there's a question of analysis and there's the question of synthesis two words that you've used before but it's worthwhile reminding what they mean in this context the analysis part of Fourier analysis is has to do with breaking a signal or a function I'll use the term signal and function pretty much interchangeably alright I'm a mathematician by training so I tend to think in terms of functions but electrical engineers tend to think in terms of signals and they mean the same thing all right so analysis has to do with taking a signal or a function and breaking it up into its constituent parts and you hope the constituent parts are simpler somehow then the complicated signal that as it comes to you so you want to break up a signal into simpler constituent parts I mean if you don't talk in just in terms of signals here or you don't use exactly that language that's the meaning of the word analysis I think close enough whereas synthesis has to do with reassembling a signal or reassembling a function from its constituent parts a signal from its constituent parts kind of stitch one alright and the two things go together all right you don't want one without the other you don't want to you don't want to break something up into its constituent parts and then just let it sit there all these little parts sitting on the table with nothing to do you want to be able to take those parts maybe modify those parts maybe see which parts are more important than other parts and then you want to put them back together to get that to get either the original signal or a new signal and the process of doing those things are the two aspects of Fourier analysis I use I use the word analysis they're sort of in a more generic sense now the other thing to realize about both of these procedures analysis and synthesis is that they are accomplished by linear operations series and integrals are always involved here both analysis and synthesis free analysis analysis and synthesis are accomplished by linear operations this is one of the reasons why the subject is so I don't know powerful because there is such a body of knowledge on and such a deep and advanced understanding of linear operations linearity will make this a little bit more explicit as I go as we go on further but I wanted to point it out now because I won't always point it out all right because when I say linear operations when I'm thinking of here integrals in series all right eg ie integrals and series both of which are linear operations the integral of a sum is the sum of the integrals the integral of a of a constant times a function is a constant interval the function and so and similarly with sums alright because of this one often says or one often thinks that Fourier analysis is part of the study of linear systems alright in engineering there's there's a there's their courses called linear systems and so on and sometimes Fourier analysis is thought to be a part of that because the operations involved in it are linear I don't think of it that way I mean I think it's somehow important enough on its own not to think of it necessarily as subsumed in a larger subject but nevertheless the fact that the operations are linear does put it in a certain context in some in some ways in some cases more general context that turns out to be important for many ideas alright so often so you see you often hear that Fourier analysis Fourier analysis is a part of the subject of linear systems the study of linear systems so I don't think that really does complete justice to Fourier analysis because of because of the particular special things that are involved in it but nevertheless you will you'll hear that okay now let's get launched alright let's start with with the actual subject of Fourier series and the analysis of periodic phenomenon a periodic phenomena and Fourier series as I said it certainly shouldn't be necessary for me to sell the importance of periodic phenomena as something worth studying you see it everywhere all right the study of periodic phenomena is for us the mathematics and engineering or mathematics and science and engineering of regularly repeating phenomena that's what's always involved there's some pattern that repeats and it repeats regularly right so it's the mathematics and engineering so this is an engineering course I'll put that before science or maybe I won't even mention science mathematics and engineering of regularly repeating patterns I'm relieving a couple of terms here I'm leave all these terms somewhat vague what does it mean to be regular what does it mean to repeating what is a pattern in the first place but you know what you know what I mean you know it when you see it and the fact you can mathematically analyze it is what makes the subject so useful now I think although again it's not ironclad trouble is this subject is so rich that every time I make a statement I feel like I have to qualify it well it's often true but it's not completely true and sometimes it's not really true at all but most of the time it's true that it's helpful but not always helpful but most of the time helpful occasionally helpful to classify periodicity as either periodicity in time or periodicity in space all right you often see periodic phenomena as one type or the other type although they can overlap so you often periodic phenomena often are either periodicity in time a pattern repeats in time over and over again you wait long enough and happens again so for example harmonic motion so eg harmonic motion a pendulum I think bobbing on a string G harmonic motion or periodicity in base periodicity in space the city in space alright now what I mean he is there is often a physical quantity that you are measuring that is living on some object in space one dimension two dimensions whatever that has a certain amount of symmetry alright and the periodicity of the phone on is a consequence of the symmetry of the object so it's often the cow giving example just a second so here you have say some some physical quantity physical not always but often you know physical quantity distributed over a region with symmetry the region itself repeats all right the region itself as a repeating pattern all right so the periodicity of the phenomenon the periodicity of the physical quantity that you're measuring is a consequence of the fact that it's distributed on on over some region that itself has some symmetry so the periodicity arises from the symmetry for periodicity here of the object of the of the physical quantity that you're measuring arises because the periodicity of the are the symmetry of the object where tributed where it lives I'll give you an example there from the symmetry matter of fact I'll give you the example the example that really started the subject and we'll study this is the distribution of heat on a circular ring so eg the distribution of heat on a circular ring alright so the object the the physical quantity that you're interested in is the temperature but it's a temperature associated with a certain region and the region is a ring all right the ring has circular symmetry it's around okay so you're measuring the temperature at points on the ring and that's periodic because if you go once around you're at the same place so the temperature is periodic as a function of the spatial variable that describes where you are on the ring time is not involved here position is involved all right it's periodic in space not periodic in time periodic in a spatial variable that gives you the position and the periodicity arises because the object itself is symmetric because the object repeats that's why this sort of example is why one often sees and this actually turns out to be very far-reaching and quite deep that free analysis is often associated with questions of symmetry in a sort of most mathematical form you often find for a series developed in and in this context and Fourier transform is developed in the context of symmetry so you often see so you see Fourier analysis let me just say free analysis analysis is often associated with problems or just not off with with analysis of questions that have to do with that have some sort of relying symmetry so let me say often associated with problems with symmetry just leave it very general this is the very first of all that for the problem of distribution of heat on a ring we're going to solve that problem that was the problem that Fourier himself considered alright they introduced some of the methods into the into the whole subject let's launch everything all right so again it's not periodicity in time its periodicity in space and for those of you who have had or may have courses in this that the mathematical framework for this very general way of looking for a analysis is group theory because the theory of groups in mathematics is a way of mathematize the ADEA of symmetry and then one extends the ideas for elseís into to take into account of groups that is to say to take into account the symmetry of certain problems that you're saying and it really stays very quite it's quite far-reaching we're not going to do it we'll actually have a few occasions to to go to go into this but but with a light touch all right I'm just telling you I'm just giving you some indication of where the subject goes all right now what are the mathematical descriptors of periodicity well nothing I've said so far I'm sure it is new to you at all you just have to trust me that at some point before you know it some things I say to you will be new I hope but one of the mathematical descriptions of periodicity again that in the two different categories say the numbers the quantities that you associate with either either a phenomena that's periodic and timer function or a phenomenon that's periodic in space for periodic and time for periodicity in time you often use the frequency all right frequency is the word that you hear most often associated with a phenomena that is periodic in time you use frequency the number of repetitions the number of cycles in a second say if a pattern is repeating whatever the pattern is again if I leave that term sort of undefined or sort of vague it's the number of repetitions of the pattern in one second or over time all right that's the most common descriptor mathematical descriptor of a phenomenon is periodic that's periodic in time for a function for a phenomenon is periodic in space you actually use the period that's the only word that's really in use in general for the particulate well one thing a time so for periodicity in space you use the period all right that is sort of the physical measurement of how long the long the pattern is before repeats somehow all right the measurement of how whether its length or some other quantity measurement of how let me just say how big the pattern is that repeats they're not the same all right they have a different feel they rise off from from different sorts of problems that's probably too strong a statement but I think I think it's fair to say that mathematicians tend to think in terms of mostly in periodic they tend to think in terms of the period of a function or the period is the description of periodic behavior whereas engineers and scientists tend to think of systems evolving in time so they tend to think in terms of frequency they tend to think of how often a pattern repeats over a certain period of time all right that's like everything else is that statement has to be qualified but I get tired of qualifying every statement so I'll just leave it at that now of course the two phenomena are not completely separate or not always completely separate they come together periodicity and time and periodicity in space come together in for example wave motion all right that is traveling disturbance a travelling periodic disturbance so the two notions of periodicity come together two notions here periodicity and time periodicity in space come together in EEG wave motion understood very generally here as a periodic as a regularly repeating pattern that changes in time that moves because more jumps up a little bit I think of their skipping so a regular a moving a subset regularly moving disturbance you know a group of freshmen through the quad you know just they're everywhere mostly regular mostly moving all right now there again the two descriptors come in the frequency and the wavelength so again you have frequency and wavelength you have frequency nu and wavelength usually associated usually denoted by this is for periodicity in space and for periodicity and time frequency nu for periodicity in time that's the number of times and repeats in one second this is cycles per second the number of times that the pattern repeats in one second so for example you fix yourself at a fix your position in spate both time and space are involved so you fix yourself at a point in space and the phenomenon washes over you like a water wave all right and you count the number of times you're hit by the wave in a second and that's the frequency that's the number of times that the phenomenon comes to you for periodic for periodicity and time the function the phenomenon comes to you for periodicity in space you come to the phenomenon so to speak all right so I fixed myself at a point in time the wave washes over me at a certain characteristic frequency over and over again regularly repeating it comes to me new times per second the wavelength you fix the time and allow the platen and see what the phenomena looks like to distribute it over space so for periodicity in space fix the time and see how the phenomena is distribute to see the pattern distributed over space distributed my writing is getting worse distributed then the length of one of those a complete to speak is the period or the wavelength length is a term that's associated with the periodicity in space for a traveling traveling phenomena for a wavelet wave for wave motion so the length the length of the disturbance I say one complete disturbance if I can say that one complete pattern is the wavelength now like I say ever since you were a kid you've studied these things and especially don't know the number by lambda but I bring it up here because of the one important relationship between frequency and wavelength which we are going to see in a myriad of forms throughout the quarter that is there's a relate in the case of wave motion there is a relationship between the frequency in the wave length determined by the velocity and there could be two different phenomena all right periodicity in time and periodicity in space may not have anything to do with each other but if you have a wave traveling if you have a regularly repeating pattern over time then they do have something with to do with each other and they're governed by the formula distance equals rate times time which is the only formula that governs motion all right so there's a relationship between frequency and wavelength that is distance equals rate times time I love writing this in a graduate course because it's the up the equation in calculus actually in all of calculus I think this is pretty much the only equation used in very clever ways but the only equation and in our case if the rate is the velocity of the wave then this translate V is the velocity the rate of the wave of the motion and the equation becomes as I'm sure you know many times lambda that's the distance that this this the the wave travels in one cycle it traveling it's traveling at a speed V if it goes nu cycles in one second then it goes one cycle in 1 over nu seconds let me say that it going to make sure I got that right if it goes nu cycles in one second if it just passed you nu times in one second then in 1 over nu seconds it rushes past you once rushing past you once means you've gone through one wavelength so distance equals rate times time the time it takes to go one wave length is 1 over nu seconds so I have lambda equals V times 1 over nu or lambda nu equals V again a formula européenne many times now why did I say this if you've seen it many times because I never have the confidence that I can talk my way through that formula for one thing so I always have to do it secondly it exhibits a reciprocal relationship to quantities all right there's a reciprocal relationship you can see it more clearly over here where the constant of proportionality or inverse proportionality is the velocity all right lambda is proportional to the reciprocal of the frequency or the restore the frequency is proportional to the reciprocal of the wavelength at any rate or the or expressed this way lambda times nu is equal to V so there's a reciprocal relationship between the frequency and the wavelength all right this is the first instance when you talk about periodicity of such reciprocal relationships we are going to see this everywhere all right it's one of the characteristics of the subject hard to state as a general principle but but they're plain to see that in the prop in in in the analysis and the synthesis of signals using methods from Fourier series or Fourier analysis there will be a reciprocal relationship between the two between the quantities involved all right I'm sorry for being so general and but you'll see this play out in case after case after case and it is something you should be attuned to all right all right so you may never have thought about this in these types of simple enough formula you've used millions of times all right you may not have thought about it somehow in those terms but I'm asking you to think about stuff use you once saw in very simple context and how those simple ideas sort of cast shadow into much more involved situations all right the reciprocal relationship between as well as we'll learn to call it the reciprocal relationship between the two domains of Fourier analysis the time domain in the frequency domain or the tie or the store the time domain and the spatial domain or the spatial domain in the frequency domain and so on is something that we will see constantly alright and I will point that out but if I don't point it out you should point out to yourself you should be attuned to it because you will see it and it's one of those things that helps you organize your understanding of the material because sometimes when you're called upon to apply these ideas in some context that you haven't quite seen you have to ask yourself it's at least the good starting place is to ask yourself questions like well should I expect a reciprocal relationship here you might lead you to guess what the formulas should be or guess what the relationship should be so you say well somehow I want to use for a analysis to do this problem so I'm sure I should be looking for some sort of reciprocal relationship the quantities that I'm interested in somehow should be related in some kind of reciprocal way and what that might mean might be more or less involved depending on the particular kind of problem but you'll see it trust me you'll see it okay right now we're almost done for today why does mathematics come into this in the first place I mean periodicity is evidently sort of a very physical type property why is it allow any kind of mathematical description well it does because there are very simple maybe not so simple mathematical functions that exhibit periodic behavior and so can be used to model periodic phenomena so math comes in because there are simple mathematical functions that model that are periodic that repeat and so can be used to model periodic phenomena I am speaking of course of our friends the sine and cosine now you may think again we've only talked about elementary things in very elementary contexts but you know I have a PhD in this subject and I get excited talking about sines and cosines I mean you know and it's not just creeping old age I mean I think there you know there's a lot there's a lot to reflect on here and sometimes the miraculous nature of these things cosine of I'll use I'll use T is the variable cosine of T and sine of T our periodic of period two pi that is cosine of T plus two pi is equal to cosine of T for all values of T and sine of two pi + t + 2 pi is equal to sine of T why dead silence because the sine and cosine are item don't tell me I want to do it because this I'll do it over here because the sine and the cosine are associated with periodicity in space because the sine of the cosine are associated with an object that regular repeats the simplest object that the regularly repeats does circle you didn't meet sine and cosine that way first you met sine and cosine in terms of ratios of psiy lengths of sides in triangles that's fine but that's an incomplete definition the real way of understorey way but the but them but the more sophisticated way the ultimately more far-reaching way of understanding sine and cosine is as associated with the unit circle where the cosine of t is the x-coordinate and the sine of t is the y-coordinate and T is Radian measure I'm not going to go through this in too much detail but the point is that the sine of the cosine are each associated with the phenomenon of periodicity in space they are periodic because if you go once around the circle that is to say T goes from T to T plus 2 pi you're back where you started from all right that's why it's periodicity in space all right that's the definition of sine and cosine that exhibits their their periodic phenomena not the definition in terms of right triangles it's not the definition it's not that the definition in terms of right triangles is wrong it just doesn't go far enough it's incomplete all right it doesn't reveal that fundamental link between the trigonometric functions and periodicity and it is fundamental if not for that mathematics could not be brought to bear on the study of periodic phenomena and furthermore this clear and will quit in just a second that is not just 2pi but any multiple of 2pi positive or negative I can go clockwise or I can go counterclockwise I can say the cosine of t plus 2pi n is the same thing as cosine of 2t and the sine of 2pi t plus 2pi n is the sine of T for n any integer n 0 plus or minus 1 plus or minus 2 and so on and so on the interpretation is that when n is positive I'm going count and it is just an interpretation is just a convention when n is positive I'm going counter clockwise around the circle when n is negative I'm going clockwise around the circle but it's only when you make the connection between periodicity and space and the sign of the cosine that you see this fundamental property all right now all right I think we made it out of junior high today that's that was my goal all right what is what is most amazing and what and what was what we'll see you next time is that such simple functions can be used to model the most complex periodic behavior all right the simple from such simple things some simple acorns mighty oaks grow or whatever you excuse me whatever whatever stuff you learn out there that the simple these simple functions that associated with such a simple phenomena can be used to model the most complex really the most complex periodic phenomena and that is the fundamental discovery of Fourier series all right and is the basis of Fourier analysis and we will pick that up next time thank you very much see you then

MATH & GEOMETRY Vocabulary and Terminology in English



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Do you need to speak about or understand mathematics or geometry in English? This lesson teaches you all the terminology you need to translate your mathematics knowledge into English. This video will be especially important for students who are studying in an English-speaking country, and for professionals who need to work with English speakers. I’ll also explain the correct sentence structures we use to talk about common mathematical operations in English. For example: “One plus one equals two”, “one and one is two”, “if you add one and one, you get two”, and many more. This lesson covers terminology about: operations (+ – * /), fractions, decimals, exponents, roots, shapes, measurements, angles, triangles, and much more. Don’t let English stand in the way of your mathematics!

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Hi. Welcome to www.engvid.com. I’m Adam. In today’s video I’m going to look at some math. Now, I know this is an English site, don’t worry, I’m not actually going to do any math. Philosophy and English major, so math not my favourite, but I will give you some math terminology, words that you need if you’re going to do math. Now, a lot of you might be engineers or you might be students who came from another country to an English-speaking country, and you go to math class and you know the math, but you’re not sure of the wording. Okay? So this is what we’re looking at, terminology, only the words that you need to go into a math class or to do some math on your own. Okay?

We’re going to start with the very basics. You know all these functions already. I’m just going to give you some ways to talk about them, and then we’ll move on to some other functions and other parts. So, you know the four basic functions: “addition”, “subtraction”, “multiplication”, and “division”. What you need to know is ways to say an equation. Right? You know an equation. “1 + 1 = 2”, that’s an equation. “x2 + y3 = znth”, that’s also an equation which I’m not even going to get into.

So, let’s start with addition. The way to talk about addition. You can say: “1 plus 1”, “plus”, of course is “+” symbol, that’s the plus symbol. “1 plus 1 equals 2.” 2 means the total, is also called the “sum”. Now, you can also say: “The sum of 1 and 1 is 2.” You can also just say, without this part: “1 and 1 is 2.” So you don’t need the plus, you don’t need the equal; you can use “and” and “is”, but it means the same thing. Everybody will understand you’re making… You’re doing addition. Sorry. Doing addition, not making. If you add 1 and 1, you get 2. Okay? So: “add” and “get”, other words you can use to express the equation. Now, if you’re doing math problems, math problems are word problems. I know a lot of you have a hard time understanding the question because of the words, so different ways to look at these functions using different words, different verbs especially.

If we look at subtraction: “10 minus 5 equals 5”. “5”, the answer is also called the “difference”. For addition it’s the “sum”, for subtraction it’s “difference”. “10, subtract 5 gives you 5.” Or: “10 deduct”-means take away-“5”, we can also say: “Take 5 away”… Oh, I forgot a word here. Sorry. “Take 5 away from 10, you get”, okay? “10 subtract 5”, you can say: “gives you 5”, sorry, I had to think about that. Math, not my specialty. So: “Take 5 away from 5, you get 5”, “Take 5 away from 5, you’re left with”, “left with” means what remains. Okay, so again, different ways to say the exact same thing. So if you see different math problems in different language you can understand what they’re saying. Okay?

Multiplication. “5 times 5”, that’s: “5 times 5 equals 25”. “25” is the “product”, the answer to the multiplication, the product. “5 multiplied by 5”, don’t forget the “by”. “5 multiplied by 5 is 25”, “is”, “gives you”, “gets”, etc.

Then we go to division. “9 divided by 3 equals 3”, “3”, the answer is called the “quotient”. This is a “q”. I don’t have a very pretty “q”, but it’s a “q”. “Quotient”. Okay? “3 goes into… 3 goes into 9 three times”, so you can reverse the order of the equation. Here, when… In addition, subtraction, multiplication… Well, actually addition and multiplication you can reverse the order and it says the same thing. Here you have to reverse the order: “goes into” as opposed to “divided by”, so pay attention to the prepositions as well. Gives you… Sorry. “3 goes into 9 three times”, there’s your answer. “10 divided by 4”, now, sometimes you get an uneven number. So: “10 divided by 4” gives you 2 with a remainder of 2, so: “2 remainder 2”. Sometimes it’ll be “2R2”, you might see it like that. Okay? So these are the basic functions you have to look at. Now we’re going to get into a little bit more complicated math things. We’re going to look at fractions, exponents, we’re going to look at some geometry issues, things like that.

Fractals: The Geometry of Chaos – Christmas Lectures with Ian Stewart



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Using a simple set of mathematical rules, a very intricate geometry can be created in the form of fractals. The Sierpinski triangle is a famous example.
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Ian Stewart gave the 1997 Christmas Lectures “The Magical Maze” about hows how maths governs almost every aspect of our lives, ranging from our birthdays to American game shows, calling in at panthers, petals, and the logic of chaos.

In his fourth lecture, Ian Stewart sets out to bring order into chaos by using mathematics.

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one of the best-known features of Jupiter is the Great Red Spot it's been there for hundreds of years it's a big storm a vast hurricane on Jupiter and when the Voyager spacecraft got to look at that it noticed you get complicated turbulent vortices spinning off what we see on Jupiter is turbulence big vortices smaller vortices very small vortices and if we could look closely it's smaller and smaller and smaller ones that's the example of chaos but it's chaotic geometry any geometrical object with that structure is called a fractal so fractals are the geometry of chaos they're unlike all the usual things you get in geometry you take something like a circle I've got a circle and I make it bigger and bigger and bigger look more closely at it it just gets flatter and flatter and flatter nothing interestingly for start with the sphere and make it bigger and bigger it just seems flatter and flatter like a plane this is why a lot of people thought the earth was flat in ancient times fractals are different when you magnify them you see more and more and more structure and yet that very intricate geometry can be created by very simple mathematical rules and the next demonstration is going to show us an example of a fractal pattern created by simple mathematical rules okay now I need some assistance for doing this as well young man there okay and your name is okay Richard now what we've got here for you that's C you might as well stand there here's a whole pot of stuff there's a triangle mark one two three pile of red dots a three-sided die it looks like an ordinary die but it's got a one on that side and it's also got a one on the opposite side and the same for the two and the three so you hold that for a minute whatever you throw with that is either a one two or a three and you've also got a ruler so what we do is we start by putting a dot somewhere in the Triangle I'll put it in the middle now roll that it's a two I want you to find the point halfway between that dot and corner – I'll pop it in there if you tell me where to put it that's it let's have a measure okay what's that okay so about there and now do it again three okay that's over there so halfway between your new dot and three yeah okay and that's right we put that one back and what's more one halfway between your new dot and one right and that would be okay that's great okay thank you very much so there's some simple mathematical rules roll the die start from wherever you are and go halfway towards the corner that the die has given you and just repeat the questions what shape would you expect to see well the most likely thing you'd expect to see is just a sort of smeary miss in the middle of the triangle let me show you what happens if you do it thousands and thousands of times again we'll use the computer and we'll see exactly what pattern builds up okay we start with random looking dots nothing very interesting just sitting in the Triangle but now we'll just speed it up a little bit keep going and you'll start to see very very beautiful surprising mathematical pattern appearing here this is called the sierpinski gasket gasket all of your car mechanics will know a gasket is the thing you take out of your car and it's full of holes of different sizes sierpinski vas lab sierpinski was a Polish mathematician who invented this shape for totally different reasons it's got big holes in the middle and then in the pieces around there smaller holes and smaller holes and smaller holes and smaller holes and mathematically those holes go on forever and that's typical of fractals simple rules create structure on all scales even the finest you

Studying Mathematics in Göttingen



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Well prepared for research and business: The research-oriented Master’s Degree programme in Mathematics (M.Sc.) offers numerous options for individual specialisation. With excellent capacity for analytical thought, graduates of Göttingen are in high demand in various areas of research as well as in industry and business. This degree also allows students to pursue doctoral studies for example in one of the interdisciplinary research training groups at the University of Göttingen. This graduate programme takes place in an international and interdisciplinary atmosphere which characterises the reputation of Mathematics at Göttingen.

in a t-33 there was a copper wire which they used for communicating via binary code when Carl Friedrich Gauss and will Henry the physicist developed the first electromagnetic Telegraph of the world and this application laid the cornerstone for modern telecommunication at that time carl friedrich gauss was the director of the observatory here he was working in living getting is a very modern university which at the same time has a long history in the beginning of the 20th century many important mathematicians did their research here in getting these days i'm working on my master's thesis here at the Institute for mathematical statistics and getting I'm working on blind separation using multi scale techniques it has an important application in cancer genetics I hope that one day geneticists will use my results to detect copy number variations and cancer cells one real nice thing about cutting in ASEA close collaboration between the different Institute's that we have so we are working for example on apply topology and applied geometry but we are closely collaborating with the Institute of mathematical statistics because there are many questions which were interesting also from the statistical perspective so students learn both aspects topological aspects and aspects of Statistics topology geometry also play on a significant role especially geometry dance and physics and what you can see here are so called elastic rods that if you twist one end while all the other and fixed and they form these flecked amines and some of you may still remember these structures from the times when telephones had ports and then you would have this annoying in our research group we're developing a theory for good public transportation systems as a partner from practice we're working with the Danish railways as a student during my master's degree I've already been working with scientists from this research group the atmosphere is very familiar you can approach anyone everyone here is very helpful if you've got any problems this is the place where I learned to work in the team the level of the courses is very high but in my opinion it pays off to invest yourself the courses are organized in cycles including lectures seminars several cycles start every year and continue for the whole programme with these cycles you are perfectly prepared for your masterpieces in getting in we've got big departments for pure and applied mathematics for example we consider minimal surfaces what do complex numbers have to do with soap bubbles I myself would like to do PhD in getting them it's possible to choose the topic between math and physics that connects both fields after the master's thesis it's possible to pursue one of our PhD programs such as mathematical structures in modern quantum physics and this is a program at the interface of pure mathematics and theoretical physics we have a number of international students from all over the world who are currently working within this program and all of these programs are part of the Graduate School Gauss here in getting in girls to prince of mathematics he made several important contributions in mathematics such as normal or Gauss distribution the historical importance of göttingen was a good reason for me to choose this university other important reasons to come here for the low cost of living and the spoken standard German as a hostage you can live directly in the city center I for example live in the house where cows used to live while he was in getting in that's something really special there are short ways you can use your student ID as a ticket for the city bus but you can also take the train with it for free and go to Hamburg or cousin Gooding is in the center of Germany during my Master's I've been studying in Seville going abroad is not a problem I've been in Birmingham UK for three months in order to write my master thesis I wanted to go to Edinburgh and it worked out it's a student advisory service helps you a lot with the organization of your semester abroad the mathematicians are quite connected also in getting the city itself is a real student city with a broad variety of ideas for living such as those common gardens where everyone can basically participate this is the graph of cows and even from here you can see the cows via valets after sunset you

MIT AI: Revolutionary Ideas in Science, Math, and Society (Eric Weinstein)



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Eric Weinstein is a mathematician, economist, physicist, and managing director of Thiel Capital. He formed the “intellectual dark web” which is a loosely assembled group of public intellectuals including Sam Harris, Jordan Peterson, Steven Pinker, Joe Rogan, Michael Shermer, and a few others. Follow Eric on Twitter: and look out for a podcast that he may be starting soon.

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the following is a conversation with Eric Weinstein he's a mathematician economist physicist and a managing director of teal Capital he coined the term and you could say is the founder of the intellectual dark web which is a loosely assemble group of public intellectuals that includes Sam Harris Jordan Peterson Steven Pinker Joe Rogan Michael Shermer and a few others this conversation is part of the artificial intelligence podcast at MIT and beyond if you enjoy it subscribe on youtube itunes or simply connect with me on twitter at Lex Friedman spelled Fri D and now here's my conversation with Eric Weinstein are you nervous about this specialist okay the bus policia you mentioned Kung Fu Panda is one of your favorite movies it has the usual profile master student dynamic going on so who was who has been a teacher that significantly influenced the direction of your thinking and life's work so if you're the Kung Fu Panda who was your Shifu oh well it's interesting because I didn't see Shifu as being the teacher who was the teacher who way Master Oogway the turtle oh the turtle right they only meet twice in the entire film and the first conversation sort of doesn't count so the magic of the film in fact it's point yeah is that the teaching that really matters is transferred during a single conversation and it's very brief and so who played that role in my life I would say either my grandfather Harry Rubin and his wife Sophie Rubin my grandmother or Tom Lehrer Tom Lehrer yeah in which way if you give a child Tom Lehrer records what you do is you destroy their ability to be taken over by later malware and it's so irreverent so witty so clever so obscene that it destroys the ability to lead a normal life for many people so if I meet somebody who is usually really shifted from any kind of neurotypical presentation I'll often ask them are you a Tom Lehrer fan and the odds that they will respond are quite high Tom layer is poisoning pigeons in the park Tom layer that's very interesting there's small number of Tom Lehrer songs that broke into the general population poisoning pigeons in the park the element song and perhaps the Vatican rag so when you meet somebody who knows those songs but doesn't know are you judging me right now aren't you harshly no but you're Russian so I dad as the you known Nikolai Ivanovich Lobachevsky that's us yeah yeah that was a song about plagiarism that was in fact plagiarized which most people don't know from Danny Kaye where Danny Kaye did a song called Stanislavski of the musky arts and so Tom Lehrer did this brilliant job of plagiarizing a song about and making it about plagiarism and then making it about this mathematician who worked in non Euclidean geometry that was like giving heroin to a child it was extremely addictive and eventually led me to a lot of different places one of which may have been a PhD in mathematics and he was also at least a lecturer in mathematics I believe at Harvard something like that I just had dinner with him in fact when my son turned 13 we didn't tell him but his Bar Mitzvah present was dinner with his hero Tom Lehrer and Tom Lehrer was 88 years old sharp as a tack irreverent and funny as hell and just you know there very few people in this world that you have to meet while they're still here and that was definitely one for our family so that wit is a reflection of intelligence in some kind of deep way like where that would be a good test of intelligence whether you're Tom Lehrer fan so what do you think that is about wit about that kind of humor ability to see the absurdity in existence well do you think that's connected to intelligence or we just two Jews on a mic that appreciate that kind of humor no I think that it's up connected to intelligence so you can see it there's a place where Tom Lehrer decides that he's going to Lampoon Gilbert of Gilbert & Sullivan and he's going to outdo Gilbert with clever meaningless wordplay and he has forget the policy he's doing Clementine as if Gilbert and Sullivan wrote and he says that I missed her depressed her young sister named mr. this mr. de pester she tried pestering sisters a festering blister you best her resistor say aye the sister persisted the mister resisted I kissed her all loyalty slipped when he said when she said I could have her her sister's cadaver must surely have turned in its crypt that's so dense it's so insane yeah that that's clearly intelligence because it's hard to construct something like that if I look at my favorite Tom Lehrer Tom Lehrer lyric you know there's a perfectly absurd one which is once all the Germans were warlike and mean but that couldn't happen again we taught them a lesson in 1918 and they've hardly bothered us since then right that is a different kind of intelligence you know you're taking something that is so horrific and you're you're sort of making it palatable and funny and demonstrating also just your humanity I mean I think the thing that came through as as Tom Lehrer wrote all of these terrible horrible lines was just what a sensitive and beautiful soul he was who was channeling pain through humor and through grace I've seen throughout Europe throughout Russia that same kind of humor emerge from the generation of world war two it seemed like that humor is required to somehow deal with the pain and the suffering of that that war created well you do need the environment to create the broad Slavic soul I don't think that many Americans really appreciate Russian humor how you had to joke during the time of let's say article 58 under Stalin you had to be very very careful you know that the concept of a Russian satirical magazine like crocodile doesn't make sense so you have this cross-cultural problem that there are certain areas of human experience that it would be better to know nothing about and quite unfortunately Eastern Europe knows a great deal about them which makes the you know the songs of Vladimir Vysotsky so potent the you know the prose of Pushkin whatever it is you have to appreciate the depth of the Eastern European experience and I would think that perhaps Americans knew something like this around the time of the Civil War or maybe you know under slavery and Jim Crow or even the harsh tyranny of the coal and steel employers during the labor Wars but in general I would say it's hard for us to understand and imagine the collective culture unless we have the system of selective pressures that for example Russians were subjected to yeah so if there's one good thing that comes out of war its literature art and humor music oh I don't think so I think almost everything is good about war except for death and destruction right without the death they would bring and the romance of it the whole thing is nice well this is why we're always caught up in war we have this very ambiguous relationship to it is that it makes life real and pressing and meaningful and at an unacceptable price and the price has never been higher so just jump in a into AI a little bit you are in one of the conversation you had or one of the videos you described that one of the things AI systems can't do and biological systems can itself replicate in the physical world oh no in the physical world well yeah the physical robots can self-replicate but but you this is a very tricky point which is that the only thing that we've been able to create that's really complex that has an analog of our reproductive system is software but nevertheless software replicates itself if we're speaking strictly for the replication in this kind of digital space so I mean just to begin and you ask a question do you see a protective barrier or a gap between the physical world and the digital world let's not call it digital let's call it the logical world versus the physical world why illogical well because even though we had let's say Einstein's brain preserved it was meaningless to us as a physical object because we couldn't do anything with what was stored in it at a logical level and so the idea that something may be stored logically and that it may be stored physically are not necessarily we don't always benefit from synonymous I'm not suggesting that there isn't a material basis to the logical world but that it does warrant identification with a separate layer that need not invoke logic gates and zeros and ones and so connecting those two worlds the logical world in the physical world or maybe just connecting to the logical world inside our brains is brain you mentioned the idea of out out telogen s– artificial app telogen artificial intelligence yes this is the only essay that Jon Brockman ever invited me to write that he refused to publish an edge why well maybe it wasn't it wasn't well written but I don't know the idea is quite compelling is quite unique and new and at least from my view a stance point maybe you can explain it sure what I was thinking about is why it is that we're waiting to be terrified by artificial general intelligence when in fact artificial life is terrifying in and of itself and it's already here so in order to have a system of selective pressures you need three distinct elements you need variation within a population you need heritability and you need differential success so what's really unique and I've made this point I think elsewhere about software is that if you think about what humans know how to build that's impressive so I always take a car and I say does it have an analogue of each of the physical physiological systems does it have a skeletal structure that's its frame does it have a neurological structure has an on-board computer as a digestive system the one thing it doesn't have is a reproductive system but if you can call spawn on a process effectively you do have a reproductive system and that means that you can create something with variation heritability and differential success now the next step in the chain of thinking was where do we see inanimate non intelligent life outwitting intelligent life and I have two favorite systems I try to stay on them so that we don't get distracted one of which is the Oh freeze orchid subspecies or subclade I don't know what to call it a type of flower yeah it's a type of flower that mimics the female of a pollinator species in order to dupe the males into engaging it was called pseudo copulation with the fake female which is usually represented by the lowest petal and there's also a pheromone component to fool the males into thinking they have a mating opportunity but the flower doesn't have to give up energy energy in the form of nectar as a lure because it's tricking the males the other system is a particular species of muscle lamp bacillus in the clear streams of Missouri and it fools bass into biting a fleshy lip that contain its young and when the bass see this fleshy lip which looks exactly like a species of fish that the baths like to eat the the young explode and clamp on to the gills and parasitize the bass and also lose the bass to redistribute them as they eventually release both of these systems you have a highly intelligent dupe being fooled by a lower life-form and what is sculpting these convincing lures it's the intelligence of previously duped targets for these strategies so when the target is smart enough to avoid the strategy those weaker mimics fall off they have terminal lines and only the better ones survive so it's an arms race between the target species that is being parasitized getting smarter and this other less intelligent or non intelligent object getting as if smarter and so what you see is is that artificial intelligence artificial general intelligence is not needed to parasitize us it's simply sufficient for us to outwit ourselves so you could have a program let's say you know one of these Nigerian scams that writes letters and uses whoever sends it Bitcoin to figure out which aspects of the program should be kept which should be varied and thrown away and you don't need it to be in any way intelligent in order to have a really nightmarish scenario being parasitized by something that has no idea what it's doing so you you you phrase a few cots have really eloquently so let me try to uh as a few directions this goes so one first on the way we write software today it's not common that we allow it to self modify hope we do have that ability now we have the ability it's just not common it's not just common so so your your thought is that that is a serious worry if there becomes it's all Spotify encode is available now so there are different types of self modification right there's a personalization you know your email app your gmail is self-modifying to you after you log in or whatever you can think of it that way but ultimately it's central all the information is centralized but you're thinking of ideas where you're completely so this is an unique entity operating under selective pressures and it changes well you just if you think about the fact that our immune systems don't know what's coming at them next but they have a small set of spanning components and if it's if it's a sufficiently expressive system in that any shape or binding region can be approximated with with the Lego that is present then you can have confidence that you don't need to know what's coming at you because the combinatorics are sufficient to reach any configuration needed so that's a beautiful thing well terrifying thing to worry about because it's so within our reach whatever I suggest these things I do always have a concern as to whether or not I will bring them into being by talking about them so uh there's this thing from open e I said next next week to talk to the founder of open AI this idea that their text generation the new the new stuff they have for generating text is they didn't want to bring it they didn't want to release it because they're worried about the I'm kind of lighted to hear that but they're going to end up really yes so that's the thing is I think talking about it I'm well at least from my end I'm more a proponent of technology preventing techni so further innovation preventing the detrimental effects of innovation well we're a we're sort of tumbling down a hill at accelerating speed so whether or not we're proponents or it doesn't mean it may not matter but I do not well I do feel that there are people who have held things back and you know died poorer than they might have otherwise been and we don't even know their names I don't think that we should discount the idea that having the smartest people showing off how smart they are by what they've developed maybe a terminal process I'm very mindful in particular of a beautiful letter that Edward Teller of all people wrote to Leo Szilard where Ziller was trying to free how to control the use of atomic weaponry at the end of World War two and tell her rather strangely because many of us view him as a monster showed some a very advanced moral thinking talking about the slim chance we have for survival and that the only hope is to make Warren thinkable I do think that not enough of us feel in our gut what it is we are playing with when we are working on technical problems and I would recommend to anyone who hasn't seen it a movie called the bridge over the bridge on the river kwai about I believe captured British POWs who just in a desire to do a bridge well end up over collaborating with their Japanese captors well now you're making me question the unrestricted open discussion of ideas and AI I'm not saying I know the answer I'm just saying that I could make a decent case for either our need to talk about this and to become technologically focused on containing it or need to stop talking about this and try to hope that the relatively small number of highly adept individuals who are looking at these problems is small enough that we should in fact be talking about how to contain them well the way ideas the way innovation happens what new ideas develop Newton with calculus whether if he was silent the idea would be would emerge elsewhere well in the case of Newton of course but you know it was in case of AI how small is the set of individuals out of which such ideas would arise well the idea is that the researchers we know and those that we don't know who may live in countries that don't wish us to know what what level they're currently at are very disciplined and keeping these things to themselves out of course I will point out that there's a religious school in Kerala that developed something very close to the calculus certainly in terms of infinite series in in I guess religious prayer and and in Ryman prose so you know it's not that Newton had any ability to hold that back and I don't really believe that we have an ability to hold it back I do think that we could change the proportion of the time we spend worrying about the effects what if we are successful rather than simply trying to succeed note that we'll be able to contain things later beautifully put so on the idea of all telogen s– what form treading cautiously is we've agreed as we tumbled down the hill what can top ourselves can we can we cannot well form do you do you see it taking so one example Facebook Google what do want to I don't know a better word you want to influence users to behave a certain way and so that's one kind of example of all telogen s– is systems perhaps modifying the behavior of their these intelligent human beings in order to sell more product of different kind but do you see other examples of this actually emerging in just take any parasitic system you know make sure that there's some way in which that there's differential success heritability and in variation and those are the magic ingredients and if you really wanted to build a nightmare machine make sure that the system that expresses the variability has a spanning set so that it can learn to arbitrary levels by making it sufficiently expressive that's your nightmare so it's your nightmare but it could also be as it's a really powerful mechanism by which to create well powerful systems so are you more worried about the the negative direction that might go versus the positive so you said parasitic but that doesn't necessarily need to be what the system converges towards it could be what is it not hirsutism the dividing line between parasitism and symbiosis is not so clear that's what they tell me about marriage I'm still so I know well yeah I did we could go into that too but no I think we have to appreciate you know are you infected by your own mitochondria right right yeah so you know in marriage you fear the loss of Independence but even though the American therapeutic community may be very concerned about codependence what's to say the codependence isn't what's necessary to have a stable relationship in which to raise children who are maximally k-selected and require incredible amounts of care because you have to wait 13 years before there's any reproductive payout and most of us don't want our 13 year olds having kids as a very tricky situation to analyze and I would say that predators and parasites Drive much of our evolution and I don't know whether to be angry at them or thank them well ultimately they I mean nobody knows the meaning of life or what even happiness is but there is some metrics did you tell you again they didn't that's why all the poetry books are about they you know there's some metrics under which you can kind of measure how good it is that these ACI systems are roaming about so your mores you're more nervous about software than you are optimistic about ideas of yeah self-replicating Lars I don't think we've really felt where we are you know occasionally get a wake-up 9/11 was so anomalous compared to everything we've out everything else we've experienced on American soil that it came to us as a complete shock that that was even a possibility what it really was was a highly creative and determined R&D team deep in the bowels of Afghanistan showing us that we had certain exploits that we were open to that nobody had chosen to express I can think of several of these things that I don't talk about publicly that just seemed to have to do with how relatively unimaginative those who wish to cause havoc and destruction have been up until now the great mystery of our time of this particular little era is how remarkably stable we've been since 1945 when we demonstrated the ability to use nuclear weapons and anger and we don't know why things like that haven't happened since then we've had several close calls we had mistakes we've had brinksmanship and what's now happened is that we've settled into a sense that oh it's it'll always be nothing it's been so long since something was at that level of danger that we've got a wrong idea in our head and that's why when I went on the Ben Shapiro show I talked about the need to resume above-ground testing of nuclear devices because we have people whose developmental experience suggests that when let's say Donald Trump and North Korea engage on Twitter oh it's nothing it's just posturing everybody's just in it for money there's that there's an a sense that people are in a video game mode which has been the right call since 1945 we've been mostly in video game mode it's amazing so you're worried about a generation which has not seen any existential but we've lived under it see you're younger I don't know if any again you came from from Moscow there was a TV show called the day after it had a huge effect on a generation and growing up in the US and he talked about what life would be like after a nuclear exchange we have not gone through an embodied experience collectively where we've thought about this and I think it's one of the most irresponsible things that the elders among us have done which is to provide this beautiful garden in which the thorns are cut off of the of the rose bushes and all of the edges are rounded and sanded and so people have developed this totally unreal idea which is everything's going to be just fine and do I think that my leading concern is AGI or my leading concern is thermonuclear exchange or gene drives or any one of these things I don't know but I know that our time here in this very long experiment here is finite because the toys that we've built are so impressive and the wisdom to accompany them has not materialized and I think it's we actually got a wisdom uptick since 1945 we had a lot of dangerous skilled players on the world stage nevertheless no matter how bad they were managed to not embroil us in something that we couldn't come back from the Cold War yeah and the distance from the Cold War you know I'm very mindful of there was a Russian tradition actually of on your wedding day going to visit a memorial to those who gave their lives can you imagine this or you your happiest day of your life you go and you pay homage to the people who fought and died in the Battle of Stalingrad I'm not a huge fan of communism I gotta say but there were a couple of things that the Russians did that were really positive in the Soviet era and I think trying to let people know how serious life actually is is the Russian model of seriousness is better than the American model and maybe like you mentioned there was a small echo of that after 9/11 but that we wouldn't let it form we talked about 9/11 but it's 912 that really moved the needle when we were all just there and nobody wanted to speak we suddenly we witness something super serious and we didn't want to run to our computers and blast out our deep thoughts and our feelings and it it was profound because we woke up briefly there you know I talked about the gated institutional narrative and that sort of programs our lives that I've seen it break three times in my life one of which was the election of Donald Trump well another time was the fall of Lehman Brothers when everybody who knew that Bear Stearns wasn't that important knew that Lehman Brothers met AIG was next and the other one was 9/11 and so if I'm 53 years old and I only remember three times that the global narrative was really interrupted that tells you how much we've been on top of developing events you know I mean we had the murrah Federal Building explosion but it didn't cause the narrative to break wasn't profound enough around nine twelve we started to wake up out of our slumber and the powers that be did not want to coming together they you know the the admonition was go shopping and the powers would be was what is that force as opposed to blaming individual we don't know so whatever that whatever that forces there's a sound holdin of it that's emergent and there's a component of it that's deliberate so give yourself a portfolio with two components some amount of it is emergent but some amount of it is also an understanding if people come together they become an incredible force and what you're seeing right now I think is there are forces that are trying to come together and their forces that are trying to push things apart and you know one of them is the globalist narrative versus the national narrative where to the global a globalist perspective the National Bank's in essence that they're temporary they're nationalistic they're jingoistic it's all negative to people in the national more in the national idiom they're saying look this is where I pay my taxes this is where I do my army service this is where I have a vote this is where I have a passport who the hell are you to tell me that because you've moved into some place that you can make money globally that you've chosen to abandon other people to whom you have a special and elevated duty and I think that these competing narratives have been pushing towards the global perspective from the elite and a larger and larger number of disenfranchised people are saying hey I actually live in a in a place and I have laws and I speak a language I have a culture and who are you to tell me that because you can profit in some faraway land that my obligations to my fellow countrymen are so so much diminished so these tensions between nations and so on ultimately you see being proud of your country and so on which creates potentially the kind of things that led to Wars and so on they ultimately it is human nature and it is good for us for wake-up calls of different guys well I think that these are tensions and my point isn't I mean nationalism run amok is a nightmare an internationalism run amok is a nightmare and the problem is we're trying to push these pendulums to someplace where they're somewhat balanced where we we have a higher duty of care to those who share our log our laws and our citizenship but we don't forget our duties of care to the global system I would think this is elementary but the problem that we're facing concerns the ability for some to profit at the abandoned by abandoning their obligations to others within their system and that's what we've had for decades he mentioned nuclear weapons I was hoping to get answers from you since one of the many things you've done as a economics and maybe you can understand human behavior why the heck we haven't blown each other up yet but okay so well good I know the answer yes it's a it's a fast it's it's really important to say that we really don't know and a mild uptick in wisdom a mild uptick in wisdom that's well Steven big pink it wasn't who I've talked with his a lot of really good ideas about why but no I I don't trust his optimism listen I'm Russian so I never trusting I was that optimist no no it's just that you're talking about a guy who's looking at a system in which more and more of the kinetic energy like war has been turned into potential energy like unused nuclear weapon Beata Filipa and you know now I'm looking at that system and I'm saying okay well if you don't have a potential energy term then everything's just getting better and better yeah wow this has beautifully put only in physicists good okay not a physicist is that a dirty word no no I wish I were a physicist me too my dad's a physicist I'm trying to live up that probably for the rest of my life he's probably gonna listen to this too so you did yeah so your friend Sam Harris worries a lot about the existential threat of AI not in the way that you've described but in the more well he hangs out with Elon I don't know so are you worried about that kind of you know about the about either robotic systems or you know traditionally defined AI systems essentially becoming super intelligent much more intelligent in human beings and getting well they already are and they're not when seen as a collective you mean well I mean I can mean all sorts of things but certainly many of the things that we thought were peculiar to general intelligence or do not require general intelligence so that's been one of the big awakenings that you can write a pretty convincing sports story from stats alone without needing to have watched the game so you know is it possible to write lively prose about politics yeah no not yet so we were sort of all over the map one of the one of the things about chess that you'll there's a question I once asked on Quora that didn't get a lot of response which was what is the greatest brilliancy ever produced by a computer in a chess game which was different than the question of what is the greatest chimera played so if you think about brilliance ease is what really animates many of us to think of chess as an art form those are those moves and combinations that just show such Flair panache and and and insole computers weren't really great at that they were great positional monsters and you know recently we've started seeing brilliance ease yeah and so if you're grandmasters have identified with that without Fazil that things work quite brilliant yeah so that's it that's it you know that's an example of something we don't think that that's a GI but in a very restricted set set of rules like chess you're starting to see poetry of a high order and and so I'm not I don't like the idea that we're waiting for Asia a GI is sort of slowly infiltrating our lives in the same way that I don't think a worm should be you know that C elegans shouldn't be treated as non conscious because it only has 300 neurons and maybe just has a very low level of consciousness because we don't understand what these things mean as they scale up so am I worried about this general phenomena sure but I think that one of the things that's happening is that a lot of us are fretting about this in part because of human needs we've always been worried about the Golem right well the gums the artificially created life you know it's like Frankenstein to ash or characters it's a Jewish version and Frankenberg frankerz yeah that's make sense that's right so the but we've always been worried about creating something like this and it's getting closer and closer and there are ways in which we have to realize that the whole thing is kind of with the whole thing that we've experienced are the context of our lives is almost certainly coming to an end and I don't mean to suggest that we won't survive I don't know and I don't mean to suggest that it's coming tomorrow it could be three hundred five hundred years but there's no plan that I'm aware of if we have three rocks that we could possibly inhabit that are sensible within current technological dreams the earth and the Moon and Mars and we have a very competitive civilization that is still forced into violence to sort out disputes that cannot be arbitrated it is not clear to me that we have a long-term future until we get to the next stage which is to figure out whether or not the Einsteinian speed limit can be broken and that requires our source code our source code the stuff in our brains to figure out what we mean by our source code the source code of the context whatever it is that produces the quarks the electrons the neutrino our source code I got it so this is your idea best stuff that's written in a higher-level language yeah yeah if that's right you're talking about the low-level bits so that's what is currently keeping us here we can't even imagine you know we have Harebrained Schemes for staying within the Einsteinian speed limit you know maybe if we could just drug ourselves and go into a suspended State or we could have multiple generations I think all that stuff is pretty silly but I think it's also pretty silly to imagine that our wisdom is going to increase to the point that we can have the toys we have and we're not going to use them for 500 years speaking of Einstein I had a profound break that when I realized you're just one letter away from the guy yeah but I'm also one letter away from Feinstein it's well you get to pick okay so unified theory you know you've worked you you enjoy the beauty of geometry well I don't actually know if you enjoy it you certainly are quite good at its trouble before trembled before it that if you're a religious that is one of the can I have to be religious it's just so beautiful you will tremble anyway I just read I sign his biography and one of the ways one of the things you've done is tried to explore a unified theory talking about a 14 dimensional observers that has the 4G space-time continuum embedded in in it i I just curious how you think and how philosophically at a high level about something more than four dimensions how do you try to what doesn't make you feel talking in the mathematical world about dimensions that are greater than the ones we can perceive is is there something that you take away that's more than just the math well first of all stick out your tongue at me okay now on the front of that yeah there was a sweet receptor and next to that were salt receptors in two different sides a little bit farther back there were sour receptors and you wouldn't show me the back of your tongue where your bitter receptor with I'm sure the good side always okay that was four dimensions of taste receptors but you also had pain receptors on that tongue and probably heat receptors on that tongue so let's simply get one of each that would be six dimensions so when you eat something you eat a slice of pizza and it's got some some some hot pepper on it maybe some jalapeno you're having six dimensional experience dude do you think we overemphasize the value of time as one of the dimensions or space well we certainly overemphasize the value of time because we like things to start and end or we really don't like things to end but they seem to but what if you flipped one of the spatial dimensions into being a temporal dimension and you and I were to meet in New York City and say well where where and when should we meet say how about I'll meet you on 36th in Lexington at 2:00 in the afternoon and eleven o'clock in the morning that would be very confusing well so it's a convenient for us to think about time you mean all right we happen to be in a delicious situation in which we have three dimensions of space and one of time and they're woven together in this sort of strange fabric where we can trade off a little space for a little time but we still only have one dimension that has picked out relative to the other three it's very much Gladys Knight and the pips so which one developed four who did we develop for these dimensions or did the dimensions or were they always there and it doesn't well do you imagine that there isn't a place where there are four temporal dimensions two and two of space and time or three of time in one of space and then would time not be playing the role of space why do you imagine that the sector that you're in is all that there is I certainly do not but I can't imagine otherwise I mean I I haven't done ayahuasca or any any of those drugs that hope to one day but I said up doing ayahuasca you could just head over to building two that's where the mathematicians are that's where they hang just to look at some geometry we'll just ask about pseudo Romani and geometry that's what your interest is okay or you could talk to a shaman and end up in Peru and then it's an extra money for I won't be able to do any calculations if that's how you choose to go about it well a different kind of calculation so decide yeah one of my favorite people Edward Frenkel Berkeley professor author of love and math great title for a book said that you were quite a remarkable intellect to come up with such beautiful original ideas in terms of unified theory and so on but you are working outside academia so one question in developing idea as a truly original truly interesting what's the difference between inside academia and outside academia when it comes to developing such you know it's a terrible choice terrible choice so if you do it inside of academics you are forced to constantly show great loyalty to the Consensus and you distinguish yourself with small almost microscopic heresies to make your reputation in general and you have very competent people and brilliant people who are working together who are informed very deep social networks and have a very high level of behavior at least within mathematics and at least technically within physics theoretical physics when you go outside you meet lunatics and crazy people mad men and these are people who do not usually subscribe to the consensus position and almost always lose their way and the key question is will progress likely come from someone who is miraculously managed to stay within the system and is able to take on a larger amount of heresy that is sort of unthinkable in which case that will be fascinating or is it more likely that somebody will maintain a level of discipline from outside of academics and be able to make use of the freedom that comes from not having to constantly affirm your loyalty to the consensus of your field so you've characterized in ways that I could academia in this particular sense is declining you are posted to plot the older population of the faculty is getting larger the younger is getting smaller and so on so what's which direction of the – are you more hopeful about well the baby boomers can't hang on forever what's it first of all in general true and second of all in academia but that's really what what this time is about is the baby we didn't we're used to like financial bubbles that last a few years in length and then pop yeah the baby boomer bubble is this really long-lived thing and all of the ideology all of the behavior patterns the norms now for example string theory is an almost entirely baby-boomer phenomena it was something that baby boomers were able to do because it required a very high level of mathematical ability you know you don't think of string theory as an original idea oh I mean it was original to Veneziano it probably is older than the baby boomers and there are people who are younger than the baby boomers who are still doing string theory and I'm not saying that nothing discovered within the large strength theoretical X is wrong quite the contrary a lot of brilliant mathematics and a lot of the structure of physics was elucidated by string theorists what do I think of the deliverable nature of this product that will not ship called string theory I think that it is largely an affirmative action program for highly mathematically and geometrically talented baby boomer physics physicists so that they can say that they're working on something within the constraints of what they will say is quantum gravity now there are other schemes you know there's like asymptotic safety there are other things that you could imagine doing I don't think much of any of the major programs but to have inflicted this level of loyalty through a Shibboleth well surely you don't question XY question almost everything in the string program and that's why I got out of physics when you called me a physicist it was a great honor but the reason I didn't become a physicist wasn't that I fell in love with mathematics as I said Wow in 1984 1983 I saw the field going mad and I saw that mathematics which has all sorts of problems was not going insane and so instead of studying things within physics I thought it was much safer to study the same objects within mathematics there's a huge price to pay for that you lose physical intuition but the point is is that it wasn't a North Korean re-education camp either are you hopeful about cracking open Einstein five theory in a way that has been really really understanding whether it's the Uniting everything together with quantum theory and so on I mean I'm trying to play this role myself to do it well the extent of handing it over to the more responsible more professional more competent community so I think that they're wrong about a great number of their belief structures but I do believe I mean I have a really profound love-hate relationship with this group of people I think the physics side oh yeah because the mathematicians actually seem to be much more open minded and well they are in there aren't they're open minded about anything that looks like great math right right they'll study something that isn't very important physics but if it's beautiful mathematics then they'll have they have great intuition about these things as good as the mathematicians are and I might even intellectually at some horsepower level give them the edge the theoretically reticle physics community is bar none the most profound intellectual community that we have ever created it is the number one there is nobody in second place as far as I'm certain look in their spare time in the spare time they invented molecular biology well what was the original molecular biology you're saying for something like Francis Crick I mean a lot of a lot of the early molecular biologists well physicists yeah I mean you know the Schrodinger wrote what is life and that was highly inspirational I mean you have to appreciate that there is no community like the basic research community in theoretical physics and it's not something I'm highly critical of these guys I think that they were just wasted that you know decades of time with and your religious devotion to their Mis conceptualization of where the problems were in physics but this has been the greatest intellectual collapse ever witnessed within academics you see it as a collapse or just a lull oh I'm terrified that we're about to lose the vitality we can't afford to pay these people we can't afford to give them an accelerator just to play with in case they find something at the next energy level these people created our economy they gave us the rad lab and radar they gave us two atomic devices to end World War two that created the semiconductor and the transistor to power our economy through Moore's law as a positive externality of particle accelerators that created the world wide web and we have the insolence to say why should we fund you with our taxpayer dollars no the question is are you enjoying your physics dollars right these guys sign the world's worst licensing agreement and if if they simply charged for every time you used a transistor or a URL or enjoyed the piece that they have provided during this period of time through the terrible weapons that they developed or your communications devices all of the things that power our economy I really think came out of physics even to the extent the chemistry came out of physics and molecular biology came out of physics so first of all you have to know that I'm very critical of this community second of all it is our most important community we have neglected it we've abused it we don't take it seriously we don't even care to get them to rehab after a couple of generations of failure all right no one I think the youngest person to have really contributed to the standard model of theater article-level was born in 1951 all right Frank will check and almost nothing has happened that in theoretical physics after 1973-74 that sent somebody to Stockholm for a theoretical development that predicted experiment so we have to understand that we are doing this to ourselves now with that said these guys have behaved abysmally in my opinion because they haven't owned up to where they actually are what problems they're really facing how definite they can actually be they haven't shared some of their most brilliant discoveries which are desperately needed in other fields like gauge theory which at least the mathematicians can can share which is an upgrade of the differential calculus of newton and leibniz and they haven't shared the importance of renormalization theory even though this should be standard operating procedure for people across the sciences dealing with different layers and different levels of phenomena and so shared you mean communicated in such a way that this it disseminates throughout the different signs these guys are sitting both theoretical physicists and mathematicians are sitting on top of a giant stock pile of intellectual gold all right they have so many things that have not been manifested anywhere I was just one Twitter I think I mentioned the harbor man switch pitch that shows the self duality of the tetrahedron realized as a linkage mechanism now this is like a triviality and it makes an amazing toy that's you know built a market hopefully a fortune for Chuck Hoberman well you have no idea how much great stuff that these priests have in their monastery so it's truly a love and hate relationship for you yeah well it sounds like it's more on the love this building that we're in right here yes is the building in which I really put together the conspiracy between the National Academy of Sciences the National Science Foundation through the government university industry research roundtable to destroy the bargaining power of American academics using foreign labor with on microfiche not in the basement oh yeah that was done here in this building is that weird and I'm truly speaking with a revolutionary and a radical no no no no no no no no no no no at an intellectual level I am absolutely garden-variety I'm just straight down the middle the system that we are in this this university is functionally insane Harvard is functionally insane and we don't understand that when we get these things wrong the financial crisis made this very clear there was a long period where every grown-up everybody with a tie who spoke in a you know in baritone with the right degree in at the end of their name which talking about how we banished volunteer volatility we were in the Great Moderation okay they were all crazy and who was who was right it was like Nassim Taleb right Nouriel Roubini now what happens is is that they claimed the market went went crazy but the market didn't go crazy the market had been crazy and what happened is is that it suddenly went sane well that's where we are with academics academics right now is mad as a hatter and it's it's absolutely evident I can show you a graph after graph I can show you the internal discussions I can show you the conspiracies Harvard's dealing with one right now over its admissions policies for people of color who happened to come from Asia all of this madness is necessary to keep the game going what we're talking about just on where around the topic of revolutionaries is we're talking about the danger of an outbreak of sanity yeah you're the guy pointing out the elephant in the room here and the elephant has no clothes see how that goes I was gonna talk a little bit to uh Joe Rogan about this man at a time well I think you're you have some you just listen to you you could probably speak really eloquently to academia on the difference between the different fields so you think there's a difference between science engineering and then the humanities in academia in terms of tolerance that they're willing to tolerate so from my perspective I thought computer science and maybe engineering is more tolerant to radical ideas but that's perhaps innocent of me is that I always you know all the battles going on now are a little bit more in the humanity side and Gender Studies and so on have you seen the American Mathematical Society publication of an essay called get out the way and not what's what's the idea is that white men who hold positions within universities and mathematics should vacate their positions so that young black women can take over or something like this that's in terms of diversity which I also want to ask you about but in terms of diversity of strictly ideas sure do you think because you're basically saying physics as a community has become a little bit intolerant to some degree to new radical ideas or at least you you say that's changed a little bit recently which is that even string theory is now admitting okay we don't this doesn't look very promising in the short term right so the question is what compiles if you want to take the computer science metaphor what will get you into a journal will you spend your life trying to push some paper into a journal or will it be accepted easily what do we know about the characteristics of the submitter and what gets taken up and what does not all of these fields are experiencing pressure because no field is performing so brilliantly well that it's revolutionizing our way of speaking and thinking in the ways in which we've become accustomed but don't you think even in theoretical physics a lot of times even with theories X string theory you could speak to this it does eventually – what are the ways that this theory would be testable and so ultimately although look there's this thing about popper and the scientific method that's a cancer in a disease and the minds of very smart people that's not really how most of the stuff gets worked out it's how it gets checked all right so there is a dialogue between theory and experiment but everybody should read Paul Dirac's 1963 American Scientific American article where he you know it's very interesting he talks about it as if it was about the Schrodinger equation and Schrodinger's failure to advance his own work because of his failure to account for some phenomenon the key point is that if your theory is a slight bit off it won't agree with experiment but it doesn't mean that the theory is actually wrong but Dirac could as easily have been talking about his own equation in which he predicted that the electrons should have an antiparticle and since the only positively charged particle that was known at the time was the proton Heisenberg pointed out well shouldn't your antiparticle the proton have the same mass as the electron and doesn't that invalidate your theory so I think that Dirac was actually being quite potentially quite sneaky and talking about the fact that he had been pushed off of his own theory to some extent by Heisenberg but look we've fetishized the scientific method and popper and falsification because it protects us from crazy ideas entering the field so you know it's a question of balancing type 1 and type 2 error and we're pretty we were pretty maxed out in one direction the opposite of that let me say what comforts me sort of biology or engineering at the end of the day does the thing work yeah you can test the crazies away and the crazy eight well see now you're saying but some ideas are truly crazy and some are are actually correct so well there's pre correct currently crazy yeah right and so you don't want to get rid of everybody who's pre correct and currently crazy the problem is is that we don't have standards in general for trying to determine who has to be put to the sword in terms of their career and who has to be protected as some sort of giant time-suck pain in the ass who may change everything do you think that's possible creating a mechanism of those select well you're not gonna like the answer but here it comes song boy it has to do with very human elements we're trying to do this at the level of like rules and fairness it's not going to work because the only thing that really understands this yeah read that read the double-helix it's a book oh-ho-ho-ho-ho you have like to read this book not only did Jim Watson half discover this three-dimensional structure of DNA he's also one hell of a writer before he became an ass that no he's tried to destroy his own reputation I knew about the ass I didn't know about the good writer Jim Watson is one of the most important people now living and as I've said before Jim Watson is too important a legacy to be left to Jim Watson and that book tells you more about what actually moves the dial and there's another story about him which I do don't agree with which is that he stole everything from rosalind Franklin I mean the the problems that he had with rosalind Franklin are real but we should actually honor that tension in our history by delving into it rather than having a simple solution Jim Watson talks about Francis Crick being a pain in the ass that everybody secretly knew was super brilliant and there's an encounter between chargaff came up with the the equimolar relations between the nucleotides who should have gotten the structure of DNA and Watson and Crick and you know he talks about missing a shiver in the heartbeat of biology and stuff is so gorgeous it just makes you tremble even thinking about it look we know very often who is to be feared and we need to fund the people that we fear the people who are wasting our time need to be excluded from the conversation you see and you know maybe we'll make some errors in both directions but we have known our own people we know the pains and the asses that might work out and we know the people who are really just blowhards who really have very little to contribute most of the time it's not 100% but you're not going to get there with rules right it's using some kind of instinct I mean I to be honest I'm gonna make you roll your eyes for a second but and the first time I heard that there is a large community of people who believe the earth is flat actually made me pause and ask myself the question why would there be such a community yeah is it possible the earth is flat so I had to like wait a minute I mean then you go through a thinking process that I think is really healthy it ultimately ends up being a geometry thing I think it's an interesting it's an interesting thought experiment at the very least well is I don't I do a different version I say why is this community stable yeah that's a good way to analyze it what interesting that whatever we've done has not erased the community so you know they're taking a longshot bet that won't pan out you know maybe we just haven't thought enough about the rationality of the square root of two and somebody brilliant we'll figure it out maybe we will eventually land one day on the surface of Jupiter and explore it right these are crazy things that will never happen so much as social media operates by AI algorithms you talked about this a little bit recommending the content you see so on this idea of radical thought how much should a I show you things you disagree with on Twitter and so on in Twitter or at verse in it about these nice clothes yeah yeah cuz you don't know the answer no no no look we've been that they've pushed out this cognitive Lego to us that will just lead to madness it's good to be challenged with things that you disagree with the answer is no it's good to be challenged with interesting things with which you currently disagree but that might be true so I don't really care about whether or not I disagree with something or don't disagree I need to know why that particular disagreeable thing is being pushed out is it because it's likely to be true is it because is there some reason because I can write I can write a computer generator to come up with an infinite number of disagreeable statements that nobody needs to look at so please before you push things at me that or disagreeable tell me why there is an aspect in which that question is quite dumb especially because it's being used to almost very generically by these different networks to say well we're trying to work this out but you know basically how much do you see the value of seeing things you don't like not you disagree with because it's very difficult to know exactly what you articulated which is the stuff that's important for you to consider that you disagree with that's really hard to figure out the bottom line is the stuff you don't like if you are a Hillary Clinton supporter you may not want to you it might not make you feel good to see anything about Donald Trump that's the only thing algorithms can really optimize for currently everything no they can do better this is weird think so now we're engaged in some moronic back-and-forth where I have no idea why people who are capable of building Google Facebook Twitter are having us in these incredibly low level discussions do they not know any smart people do they not have the phone numbers of people who can elevate these discussions they do but this then optimizing for a different thing and they are pushing those people out of those rooms they're they're optimizing for things we can't see and yes profit is there nobody nobody's questioning that but they're also optimizing for things like political control or the fact that they're doing business in Pakistan and so they don't want to talk about all the things that they're going to be bending to in Pakistan so that we're involved in a fake discussion you think so you think these conversations at that depth are happening inside Google you don't think they have some basic metrics under user engagements you're having a fake conversation with us guys we know you're having a fake conversation I do not wish to be part of your fake conversation you know how to cool you know these units you know high availability like nobody's business my Gmail never goes down almost see you think just because they can do incredible work on the software side with infrastructure they can also deal with some of these difficult questions about human behavior human understanding human you're not you thinking I mean I've seen that I've seen the developers screens that people take shots of inside of Google yeah and I've heard stories inside of Facebook and Apple we're not we're engaged they're engaging us in the wrong conversations we are not at this low level here's one of my favorite questions why is every piece of hardware that I purchase and in in tech space equipped as a listening device where's my physical shudder to cover my lens we had this in the 1970s a cameras that had lens caps you know how much would it cost to have a security model pay five extra bucks why is my indicator light software controlled why when my camera is on do I not see that the light is on by putting it as a something that cannot be bypassed why have you set up my all my devices it's some difficulty to yourselves as listening devices and we don't even talk about this this is this thing is total yeah well I hope these discussions are happening about privacy this is their different more difficult thing you're giving it's not just privacy yeah it's about social control we're talking about social control why do I not have controls over my own levers just have a really cute UI where I can switch I can dial things or I can at least see what the algorithms are you think that there is some deliberate choices being made here is emergence and there is intention there are two dimensions and the vector does not collapse onto either axis but the idea that anybody who suggests that intention is completely absent is a child that's really beautifully put and like many things you've said is gonna make me connections can I turn this around slightly look yeah I sit down with you and you say that you're obsessed with my feet uh-huh I don't even know what my feet is what are you seeing that I'm not I was obsessively looking through your feed on Twitter because it was really enjoyable because there's the Tom layer element is the humor in it by the way that feed is Eric or once yeah i'm twitter edgar ik are weinstein answers it why why did i find any enjoyable or what it was I seeing what are you looking for why are we doing this what is this podcast about I know you've got all these interesting people I'm just some guy is sort of a podcast gift sort of vodcast you know you're wearing a tie I mean not even we're not even a serious interview searching for meaning for happiness for a dopamine rush so short term and long term and how are you finding your way to me what it what it what is I don't honestly know what I'm doing to reach you the representing ideas which feel common sense to me and not many people are speaking so it's kind of like the dog the intellectual dark web folks right they these folks from Sam Harris to Jordan Peterson to yourself are saying things where it's like you're like saying look there's an elephant he's not wearing any clothes and I say yeah yeah let's have more of that conversation that's how I'm finding you I'm desperate to try to change the conversation we're having I'm very worried we've got an election in 2020 I don't think we can afford four more years of a misinterpreted message which is what Donald Trump was and I don't want the destruction of our institutions they all seem hell-bent on destroying themselves so I'm trying to save theoretical physics trying to save the New York Times trying to save our various processes and I think it feels delusional to me that this is falling to a tiny group of people who are willing to speak out without getting so freaked out that everything they say will be misinterpreted and that their lives will be ruined through the process I mean I think we're in an absolutely bananas period of time and I don't believe it should fall to such a tight number of shoulders to shoulder this way so I have to ask you on the capitalism side you mentioned that technology is killing capitalism or it has effects that are unintended but not what economists would predict or speak of capitalism creating I just want to talk to you about in general the effect of even an artificial intelligence or technology automation taking away jobs in these kinds of things and what you think is the way to alleviate that whether the and rank presidential candidate with universal basic income ubi whether your thoughts there how do we fight off the negative effects of technology that aren't your software guy right yeah a human being is a worker is an old idea yes a human being has a worker is a different object all right yeah so if you think about object-oriented programming as a paradigm a human being has a worker and a human being has a soul we're talking about the fact that for a period of time the worker that a human being has was in a position to feed the soul that a human being has however we have two separate claims on the value in society one is as a worker and the other is as a soul and the soul needs sustenance it needs dignity it needs meaning it needs purpose as long as your means of support is not highly repetitive I think you have a while to go before you need to start worrying but if what you do is highly repetitive and it's not terribly generative you weren't in the cross hairs of four four loops and while loops and that's what computers excel at repetitive behavior and when I say repetitive I mean meat I mean things that have never happened be through combinatorial possibilities but as long as it has a looped characteristic to it you're in trouble we are seeing a massive push towards socialism because capitalists are slow to address the fact that a worker may not be able to make claims a relatively on languished median member of our society still has needs to reproduce needs to head to dignity and when capitalism abandons the median individual or you know the bottom tenth or whatever it's going to do it's flirting with revolution and what concerns me is that the capitalists aren't sufficiently capitalistic to understand this you really want to court authoritarian control in our society because you can't see that people may not be able to defend themselves in the marketplace because the marginal product of their labor is too low to feed their dignity as a soul so it my great concern is that our free society has to do with the fact that we are self organized I remember looking down from my office in Manhattan when Lehman Brothers collapsed in thinking who's going to tell all these people that they need to show up at work when they don't have a financial system to incentivize them to show up at work so my complaint is first of all not with the Socialists but with the capitalists which is you guys are being idiots you're courting revolution by continuing to harp on the same old ideas that well you know try and try harder bootstrap yourself yeah to an extent that works to an extent but we are clearly headed in place that there's nothing that ties together our need to contribute and our need to consume and that may not be provided by capitalism because it may have been a temporary phenomena so check out my article on anthropic capitalism and the new gimmick economy I think people are late getting the wake-up call and we would be doing a better job saving capitalism from itself because I don't want this done under authoritarian control and the more we insist that everybody who's not thriving in our society during their reproductive years in order to have a family is failing at a personal level I mean what a disgusting thing that we're saying what would horrible message who who the hell have we become that we've so bought into the chicago model that we can't see the humanity that we're destroying in that process and it's I hate I hate the thought of communism I really do my family has flirted with it decades past it's a wrong bad idea but we are going to need to figure out how to make sure that those souls are nirn nourished and respected and capitalism better have an answer and I'm betting on capitalism but I got to tell you I'm pretty disappointed with my team so you're still on the capitalism team you just uh there's a theme here graphical reticle capital right right on capitalism yeah I want I think hyper capitalism is gonna have to be coupled to hyper socialism you need to allow the most productive people to create wonders and you've got to stop bogging them down with all of these extra nice requirements you know nice is dead good has a future nice doesn't have a future because nice ends up with with goo legs damn that's a good line okay last question you tweeted today a simple quite insightful equation saying imagine that every unit F of Fame you picked up s stalkers and H haters so I imagine s and H or dependent on your path to fame perhaps a little bit but it's not as simple people always take these things literally when you have like 280 characters to explain yourself [Laughter] Soumya that's not a mathematical no there's no law okay okay I just said why I put the word imagine because I have loved a mathematician desire for precision you imagine that this were true but it was a beautiful way to imagine that there is a law that has those variables in it and you've become quite famous these days so how do you yourself optimize that equation with the peculiar kind of Fame that you have gathered along the way I want to be kinder I want to be kinder to myself I want to be kinder to others I want to be able to have heart compassion or these things are really important and I have a pretty spectrum II kind of approach to analysis I'm quite literal I can go full Rainman on you at any given moment no I can yeah its faculties of autism if you like and people are gonna get angry because they want autism to be respected but when you see me coding or you see me doing mathematics I'm you know I speak with speech apnea uh me right Debra dinner you know yeah we have to try to integrate ourselves and those tensions between you know it's sort of back to us as a worker and us as a soul many of us are optimizing one to thee at the expense of the other and I struggle with social media and I struggle with people making threats against our families and I struggle with just how much pain people are in and if there's one message I would like to push out there you're responsible everybody all of us myself included was struggling struggle struggle mightily because you it's nobody else's job to do your struggle for you now with that said if you're struggling and you're trying and you're trying to figure out how to better yourself and where you failed where you've let down your family your friends your worker is all this kind of stuff give yourself a break you know if if if it's not working out I have a life long relationship with failure and success there's been no period of my life where both haven't been present in one form or another and I I do wish to say that a lot of times people think this is glamorous I'm about to go you know do a show with Sam Harris people are gonna listen in on two guys having a conversation on stage it's completely crazy when I'm always trying to figure out how to make sure that those people get maximum value and that's why I'm doing this podcast you know just give yourself a break you owe us you owe us your struggle you don't owe your family or your co-workers or your lovers or your family members success as long as you're in there and you're picking yourself up recognize that this this new situation with the economy that doesn't have juice to sustain our institutions has caused the people who've risen to the top of those institutions to get quite brutal and cruel everybody is lying at the moment nobody's really a truth teller try to keep your humanity about you try to recognize that if you're failing if things aren't where you want them to be and you're struggling and you're trying to figure out what you're doing wrong which you could do it's not necessarily all your fault we are in a global situation I have not met the people who are honest kind good successful nobody that I've met this chick is checking all the boxes nobody's getting all tens so I just think that's an important message that doesn't get pushed out enough either people want to hold society responsible for their failures which is not reasonable you have to struggle you have to try or they want to say you're a hundred percent responsible for your failures which is total nonsense beautifully put Eric thank you so much for talking today thanks for having me buddy you

Math Antics – Circles, What Is PI?



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hi welcome to math antics we've learned a lot about geometry so far but there's one really important geometric shape that we still need to cover and that shape is a surgery since the invention of the wheel circles have been extremely important to all humanity raagh make wheels thanks Grogg in fact you probably see circles almost everywhere you turn but mathematically what is a circle well in geometry a circle is defined as the set of all points that are equidistant or the same distance from another single point and the best way to understand what that means is to see it in action so here's a single point to start with and now let's start drawing points that are equidistant from it this point is a foot away to the right now let's make another point a foot away but in another direction let's say up here now let's make another one also a foot away but in another direction right here now let's make another right here I'm getting tired but do you see what's happening the more equidistant points we add the more the pattern looks like a circle and that's why a circle is defined as the set of points that are equidistant from a center point but of course we usually don't see it as a set of points because there's infinitely many of them so they form a continuous circle okay now let's learn about the parts that make up a circle first of all we have the original point that we started with that's called the center or the origin of the circle next we have the distance that we use to draw all of the equidistant points that form a circle that distance is called the radius the radius is important because it's the distance from the center of a circle to any other point on the perimeter of that circle and even though a circle only has one radius dimension you can draw as many radius lines as you want to usually you'll only see one radius line drawn since it's the same length no matter where you draw it another important circle dimension is called the diameter the diameter is the distance across the circle if you start at one point on the circle and then draw a line straight through the center to the other side that distance is the diameter as you can see the diameter is really just the same as two radius lines drawn in exactly opposite directions so for any circle the diameter is always exactly twice as long as the radius all of the equidistant points we drew combined to form the perimeter of the circle remember that perimeter is just the distance all the way around a shape but because a circle is a special shape the perimeter of a circle gets a special name it's called the circumference the circumference is the distance all the way around a circle we're going to learn how to calculate the circumference of any circle in the next video we'll also learn how to calculate the area of any circle but before we can learn those things we first need to learn about pi grog mate by sorry grog not that kind of pun in math the word Pi which is spelled P I refers to a very special number in fact it's so special that it gets its own symbol this Greek letter here is the symbol for the number pi but if pi is just a number why don't we write it like that why do we need to use a special symbol for it that's a good question and I'll get to that in just a minute but first let's learn what pi really is by seeing how it relates to a circle it turns out that pi is really a ratio now if you're not sure what a ratio is you can watch our video about them but basically a ratio is just a relationship between two numbers that's written like a fraction pi is the ratio of two different distances on a circle it's the ratio of the distance around a circle to the distance across the circle and what do we call those two distances yep the circumference and the diameter so pi is the relationship of the sir cumference to the diameter and as you'll see in a minute because pi is a ratio it's the same number for any circle no matter how big or small okay but what number is it what's the value of pi well to figure that out have a look at these two circles one big and one small we're gonna imagine that our circles diameters are flexible like a piece of string and that we can wrap them around the outside edges circumference –is of the circles so for each circle if we start at the top and wrap the diameter around the circumference we see that one diameter is not enough to go all the way around so let's get another diameter and keep going where the first diameter stopped hmm two diameter still isn't enough to go all the way around it looks like we're gonna need to get a third diameter and keep going oh so close three diameters is almost enough but it looks like we're gonna need just a little bit more to form a full circumference that little bit more turns out to be about 0.14 diameters that means that it takes 3.14 diameters to equal one circumference for any circle big or small so the value of pi is always 3.14 well okay pi is a little more complicated than that 3.14 is really just pi rounded off to two decimal places and we actually have to round pi off because it's a type of number that's called irrational an irrational number has decimal digits that never end and never repeat grog confused yes irrational numbers are confusing but seeing some more of PI's decimal digits will help you understand what I mean to be more precise pi is 3.14159 two six five three five eight nine seven nine three two three eight and the decimal digits just keep on going forever without repeating pretty amazing huh but the good news is that saying pi is 3.14 is usually close enough for most math problems so that's all you really need to memorize and that's why we use a symbol for pi in equations we could write PI with just two decimal places or we could write it with five decimal places to be more accurate or we could write it with hundreds of decimal places to be super accurate or we could just use the symbol to represent the true value which is infinitely accurate okay here on this video we learned what a circle is and we've learned about the important parts of the circle the center the radius the diameter and the circumference we've also learned about a very special number called pi pi is the ratio of a circles circumference to its diameter and its value is about 3.14 no matter what size the circle is in our next video about circles we're going to learn how we can use the number pi to find the circumference and the area of any circle and even though there's not much math you can actually practice in this section don't worry there'll be lots of practice problems in the next section to make up for it thanks for watching math antics and I'll see you next time mmm bra good at math learn more at math antics comm

The complex geometry of Islamic design – Eric Broug



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View full lesson: In Islamic culture, geometric design is everywhere: you can find it …

Circles (Part-1) – Class 9 – Mathematics | Geometry (NCERT) – Sameer Kohli || Study Khazana



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Circles (Part-1) – Class 9 – Mathematics | Geometry (NCERT) – Sameer Kohli || Study Khazana

Find all CBSE Class 9 Mathematics tutorials at

Watch the complete lecture on https:// www.studykhazana.com

About the Lecture: Circles is a part of geometry and Geometry section is of 38 marks which means this is very important section and each and every topic discussed here is very important.

You will learn to prove all the theorem in proper manner. The theorems are proved in step by step manner. Each step as explained by sir values. You will do quick revision of all the theorem done in this chapter. You will even get tips to score full in exams.

About the Topic: In Circles, you will learn to:

(1) Define the terms related to circles

(2) Prove the following theorem:
– Equal chords of the circle subtend equal angle at centre.
– The perpendicular from the centre to the chord bisects the chord.
– There is one and only one point passing through three non-collinear point.
– Equal chords of the circle are equidistant from the centre.
– Congruent arcs of the circle subtend equal angles at the centre.
– The angle subtended by the arc at the centre of the circle is double the angle subtended by it at any remaining part of the circle.
– Angle of the same segment of a circle are equal.
– Angle in a semi-circle is a right angle
– The sum of either pair of opposite angles of a cyclic quadrilateral is supplementary.

About Sameer Kohli: Mr. Sameer Kohli is our the most interactive teacher of Study Khazana. He teaches the students of class IX, X, XI and XII. Further he has 18 years of experience and this experience has made many students score the best result. A topic taught by sir cannot be forgotten. He has a zeal to teach and he always comes up with a new enthusiasm, a new story and a new trick. His lectures are boon for all the students who want to succeed. In his lectures, he has shared some secrets of success. If you wish to be successful, you should watch his lectures. These lectures can be a life changing experience for you.

Complete Principles of Management Lecture is available here:

Principles of Management is the first chapter and accounts for about 6-8 marks of the Business Studies. This chapter is a part of Unit-2 and includes the following topics: Principle of Management- Concept, nature and significance, Fayols Principles of management, Taylors of scientific management- principles and techniques.

Principles of Management are broad and general guidelines for decision making and behavior management. It has certain nature/features/Characters and they are Universal Applicability, General guidelines, Formed by practice and experimentation, Flexibility, Mainly Behavior, Cause and Effect, Relationship and Contingent. All the characteristics have been explained with some practical examples. Thereupon you will come across the significance or importance of management which really matters in decision making in an organisation and run an organisation. Then you will learn about the Fayol’s Principle of Management which is very important in examination point of view. Henri Fayol was a french management theorist. He graduated from the Mining Academy of St. Etienne in 1860 in Mining Engineering. Under his management principles his company reached an excellent position. He developed 14 principles of management which act as guidelines for the manager. You will discuss all the principles.

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4th Dimension Explained By A High-School Student



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There are many theories out there. This is one of those theories.

Inspired by Flatlands.

imagine that this folder is a dimensional plane now assuming that is no height and no depth what would this mean it would mean that it's a one-dimensional world so if hypothetically an organism was living inside of it it would only be able to move in a linear path forward and backwards in a straight line now if we go to the second dimension we have two dimensions we have width and we have length so hypothetically if an organism lived inside of here then it would be able to move up down left right than anywhere else in between and a two dimensional world is comprised of an infinite series of one dimensional worlds stacked upon each other just as our three-dimensional world which has depth and length and height is comprised of an infinite series of two dimensional worlds so this now that I have stamped many folders upon each other we have three dimensions we have depth we have length and we have width now what happens if you keep going on from here on and we would have a four dimensional world what exactly is a fourth dimension in order to understand this we need to understand how two mentions are perceived we live in the three-dimensional world but despite that we actually view things it to be two dimensionally take a perfect sphere for example if you're looking at sphere it looks just like a regular two-dimensional circle the only way that you could tell it's a sexual sphere instead of a circle is because of the hues of light down so just like in a two-dimensional world if a organism in the two-dimensional world was looking upon circle but the light would make it appear to be lighter at one end and darker at the middle also if you if an object is moving farther closer and farther away from you you don't actually perceive that it's getting closer and farther away you see that it's beginning smaller or larger and then you assume that it's getting either farther away or closer but but say that object was to grow and perfectly equilibrium so that it was growing at the same speed that it was shrinking as you move it farther away that me would not be able to tell without any lights or if there were details on the object that it's moving or growing at all it may have assumed that since we perceived things to be in two dimensions that a two-dimensional organism would see in one dimension so as we're watching this rubber brand expand as it moves farther away from this little organism in the two-dimensional world it's does not actually perceive anything that's happening to it because it's growing at the same speed that is moving farther away but since we can actually see it from the three-dimensional world and perceive things to be in two dimensions we can see things for how they actually are the reason that the two dimension organism doesn't see things the way they really are is because the two-dimensional organism sees things in one dimension just how we three-dimensional creatures see things in two dimensions so in a way we don't really see our world the way it truly is a four dimensional creature I've ever seen our three-dimensional world in three dimensions would be able to see through things to be able to see absolutely everything just as we could see if there was several organisms spread along a 2d environment on your floor you'd be able to see inside houses you would be able to see inside of people so if a two-dimensional world a flat surface has just made of an infinite amount of lines then the 3d world is just made out of an infinite amount of planes so the forty world logically is made out of an infinite amount of 3d objects though they're not just put together like um you would like building blocks it's that's not how the 4d world is that just be 3d again so in order to understand this we need to understand the logical progression of mathematics in our world imagine that this connects piece represents the first dimension it's simply a straight line which is basically what the first dimension looks like and if you add three more of these straight lines and connect them to do so that adjacent sides are perpendicular and opposite sides our parent law then you have the basic shape of the second dimension you have a square now if you keep going from here and you add full hand it so that there's a total of four squares and all adjacent sides are perpendicular and all opposite sides are parallel then you end up with obviously a cube so if you're trying to keep going from here and you would have a four dimensional basic shape you would have eight what's called a tesseract now I cannot show you a tesseract but you need to understand that it's basically four cubes that are within each other that have all adjacent sides perpendicular and all pair and all opposite sides parallel yet there are three lines connect four lines connect into each vertex so a tesseract would look somewhat like this picture now that that's not exactly what it looks like because this is a two-dimensional depiction and obviously not all the lines are straight so I cannot show you what assessor Act actually looks like because we cannot perceive things in the third dimension you cannot even imagine what the tesseract looks like you cannot physically it you cannot in your mind picture the fourth of nitrogen or a fourth dimensional shape and you can keep going on from the fourth dimension even you can go to the fifth dimension the sixth dimension the seventy first dimension it doesn't matter theoretically there are an infinite amount of spatial dimensions a common misconception of the fourth dimension is that the fourth dimension is time now while some argue that goodbye going forward and backwards in time if you move forward the same distance and backwards the same then you would end up in the same place you started just like in the fourth dimension and while that may seem logical if you think about it really doesn't make sense if you imply that the fourth dimension is actually time well first of all time is not spatial there's a difference between space and time quite obviously and assuming that all dimensions are according to a pattern then that doesn't really make sense either because saying the 4th dimension is time every dimension has time in it so that would mean that the 4th dimension is special in some way which doesn't really make any sense another reason this doesn't make sense is that we so very very very slightly travel through time whenever we move due to the distance that light takes to get to our body if a group of astronauts were to get in a spaceship and they are to go very very very close to the speed of light then they would and they they went around in this how impossible nearly the speed of light spaceship for a few months and then afterwards they returned earth they would find that earth had actually progressed a few years so they had moved forward in time I move in that quickly another interesting concept involved in the 4th dimension is there many physicists and even mathematicians may say that the dimensions are very very slightly curved because if you really think about it nothing can be truly absolutely infinite so imagine that a the first dimension the line is just very very slightly curved so that after a very long time it will end up creating a circle so as suggested by many physicists if you keep going in the same direction then you will end up where you are where you started after a very very long amount of time obviously and the same thing would happen to the second dimension if it's just a square and then you extend it very very slight leave and a curve and it will eventually make a sphere and the same thing happens in our dimension except it will form a very very slightly curved third dimension which will form a four dimensional universe basically so what this kind of means is that our 3-dimensional world is within a four dimensional world and the four dimensional worlds within the fifth dimensional world and so on now I did say that nothing can be truly infinite but if this is true and a dimension is really within another dimension within another dimension within another dimension then I'm implying that there an infinite amount of dimensions which is the only problem I really have with the seer I'm not sure if it ever stops or if infinite infinity is really even possible we don't know that thanks for watching my video and I hope this gave you a better idea of what the fourth dimension is

The hardest problem on the hardest test



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A difficult geometry puzzle with an elegant solution.
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Solution to the puzzle mentioned at the end:

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A different write-up of this solution:

1992 Putnam with this problem:

A problem with a similar flavor came up on the 2005 Putnam A6. Give it a try! In the solutions for that problem, by the way, the that Calvin Lin is a friend of mine who works at Brilliant.

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.

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你们听说过普特南数学竞赛吗? 这是一个面向美国本科生的数学竞赛 竞赛时长6小时 一共只有12道题目 而且分成上下两场 每场3小时 每道题的得分最低0分 最高10分 所以满分就是120分 然而 尽管每年来参加这个竞赛的 很明显都是那些对数学相当感兴趣的学生 但是分数的中位数也常常只有一两分 所以说 这个竞赛很难 而且对于每场的六个问题来说 从第一题到第六题 难度也在上升 当然了 难不难是因人而异的 但是说到第五题和第六题 虽然它们都是超难竞赛中的压轴难题 但通常来说 它们也有着最为精妙的解法 稍微转换一下视角 极具挑战性的问题就不再无从下手了 我想在这里和大家分享一道题 好早以前某届普特南数学竞赛中的第六题 关注这个频道的朋友们都知道 我并不会直接公布答案 而且这道题的答案出人意料地简短 如果可能的话 我愿意花点时间让你明白 你如何能够独立探索出问题的答案 以及重要的想法是怎么来的 也就是说 视频重点关注的是解答问题的过程 而不是用到这个思路的问题本身 总之 问题是这样的: 在球面上随机选择四个点 然后考虑以它们为顶点的四面体 那么球心落在四面体内部的概率是多少? 花点时间 稍微琢磨一下这个问题 你可能会想 哪些四面体包含了球心 哪些四面体没有 如何系统地区分这两种情况 还有 你要怎样处理这种问题呢? 要从何下手呢? 通常来说 从问题的简单情形入手是一个好办法 所以我们把问题简化到二维来 此时 你要在圆上随机选择三个点 用符号标记更加方便 所以用P_1 P_2 P_3表示这三个点 问题就是 这三个点形成的三角形包含圆心的概率是多少? 我觉得你也会认为这样更容易想象了 但是问题依然很困难 所以你想知道 能不能简化一下问题 找到一个立足点 方便继续思考 或许你会想 先固定P_1和P_2的位置 只允许第三个点移动 你这么做的时候 在脑子里比划比划 你可能会注意到一个特殊区域 一段弧 P_3落在这条弧上时 三角形就包含圆心 否则就不包含 具体来说 如果分别过P_1和P_2以及圆心做直线 这两条直线会把圆分成四段弧 如果P_3恰好落在P_1和P_2对面的弧上 那么三角形就包含圆心 如果P_3落在其他弧上 那就不行 我们假定取到圆上任一点的概率相同 那么P_3落在这条弧上的概率是多少? 答案就是这段弧的长度除以圆的周长 也就是这段弧占整个圆的比例 那这个比例是多少呢? 很明显 这和前两个点的位置有关 如果它们之间相差90° 那么对应的弧就是1/4个圆 如果这两个点离得很远 比例就会更接近1/2 如果这两个点靠得很近 比例就更接近0 花点时间思考一下 P_1和P_2都是随机选出的 而任一点被取到的概率相同 那这段弧的平均长度是多少呢? 你或许会固定P_1 只考虑P_2所有可能的位置 这两条直线形成的所有可能的夹角 也就是从0°到180°之间的所有角度 出现的概率都一样 所以 从0到0.5之间的每个比例出现的概率也一样 这就意味着 平均比例就是0.25 所以 如果这段弧的平均长度是圆周长的1/4 第三个点落在这段弧上的平均概率就是1/4 也就是说 三角形包含圆心的概率是1/4 但是我们能把它推广到三维情况吗? 一共四个点 如果其中三个点固定不动 第四个点要落在球面的哪些位置 才能让形成的四面体包含球心? 和之前一样 我们分别过这三个点以及球心做直线 这些直线两两确定一个平面 画出这些平面也很有用 你可能注意到了 这些平面把球面分成了八个区域 每个区域都有点像球面三角形 要想让形成的四面体包含球心的话 第四个点就必须落在前三个点相对的球面三角形上 和二维情况不同 让最初的三个点变化 然后求这个区域的平均面积 这就很困难了 掌握多元微积分的朋友可能会想: 用曲面积分试试看 不妨拿出纸笔算一算吧 但是这并不简单 当然了 这道题本来就应该很难 这是普特南数学竞赛的第六题啊 你指望它能有多简单? 更何况 算出了三角面积又有什么用呢? 你可以回头看看二维的情况 仔细想想 是否可以通过别的方法解出问题 “1/4”这个答案看上去异常简洁 这也提出了一个问题:“4”代表着什么? 我之所以要特别为这个问题做一期视频 主要的原因之一是 接下来的事情 对求解数学问题有广泛的参考价值 回想我们之前画出的以P_1与P_2为端点 通过圆心的两条线 这两条线让问题更容易思考了 一般而言 只要你往原问题中添加了新东西 使得问题的概念更简洁的话 那就看看 你能否将整个问题只用加进来的新东西来重新叙述 在这里 我们不再考虑随机选择三个点 而是考虑 随机选择两条过圆心的直线 每条直线都对应着圆上的两个点 所以就二选一 确定哪个端点是P_1 类似地 在另一条直线上确定哪个端点是P_2 随机选择一条直线 然后二选一确定端点 这种做法就等于随机在圆上选取一点 乍一看 这让人感觉有点绕弯子 但之所以用这种方式来考虑随机过程 是因为这能让事情变得简单许多 我们依然认为点P_3不过是圆上的任意一点 但我们是在扔两次硬币前就确定好P_3了 因为你看嘛 一旦这两条线和第三个点定死了以后 P_1和P_2落在哪里 依扔硬币的结果而定就只有四种情况了 每种情况都是等可能的 但是 有且仅有一种情况 能使得P_1和P_2在圆上落在P_3的对面 因此三点围成的三角形能包含圆心 所以说 不管这两条线或者P_3点最后落在哪里 扔硬币总是会给我们1/4的机率使得 三角形包含圆心 这就很巧妙了 重新修改一下我们随机选点的顺序 1/4这个答案就以一种很不一样的方式蹦了出来 重要的是 这种推导过程可以毫无痕迹地被推广至三维 再来一次 这次我们从选四个随机点开始 想象一下 任选三条穿过球心的线 再随便来一个点P_4 第一条线与球面相交于两点 扔个硬币 决定一下哪个点是P_1 类似地 扔硬币决定P_2和P_3的落点 扔硬币就有8种等可能的结果 但有且仅有一种结果 能使P_1 P_2和P_3处于与P_4相对的位置上 所以这8个概率一样的结果中 有且仅有一个 能让我们得到包含球心的四面体 这个结果再一次以巧妙的方式出现在了我们的面前 但是 这的确很简洁啊 这就是此问题的一个确实的解了 但是必须承认 到目前为止我的讲解都是基于几何直观的 如果你有点好奇 应该如何不依靠几何直观来写出这个解的话 我在简介里留了个解答的链接 它用线性代数的语言解答了这个问题 这在数学中是挺常见的: 理解问题并知道关键是一回事 但有相关背景知识 能更正式更清晰地阐述这个理解 基本就完全是另一码事了 这种能力也是数学系本科生要花大时间来培养的 你应该从本题中学到的并不是这个解本身 而是如果是你来做这道题的时候 应该怎么找到关键的想法 也就是说 不断去找这个问题的简化版本 直到你能找到个落脚点为止 在这么做的过程中 如果你发现有什么新添的结构能为自己所用 就试试看能不能根据这些新构造来重述整个问题 视频结束之前 我还有一个关于概率的问题 这个问题取自这期视频的赞助商Brilliant.org 假设八个参加普特南数学竞赛的学生环坐在一起 竞赛很难 所以每个学生都试图抄邻座的答案 而且是随机选择一个邻座来抄 现在圈出所有没被抄到的学生 被圈出的学生数目的期望值是多少? 这个问题很有趣 对吧~ Brilliant.org就是个能让你用许许多多类似的问题 来锻炼自己解题能力的地方 这也的确是最好的学习方式 你能找到数不胜数的有趣问题 而它们都被组织得很有深度 所以你一定会在解题方面有所提高的 如果你还想探索更多的可能性 他们也有相当棒的概率课程 当然 他们也有各种其他的数理课程 所以你怎么着都能找到点你感兴趣的东西的 我嘛 我是已经用这个好久了 如果你访问brilliant.org/3b1b 他们就知道你是从这个频道来的了 前256位通过此链接访问的同学可以得到他们高级会员的八折优惠 如果你考虑升级一下的话 我就是用的这个 还有 如果你心痒难耐想看这个问题的解答的话 插一句 这个解使用了概率论中的一个策略 该策略在其他情况下也很有用 我也在简介中留了链接 点一下你就能直接看到答案了

How Earth Moves



Views:14379068|Rating:4.87|View Time:21:37Minutes|Likes:368451|Dislikes:9928
It’s here! Science stuff, mind-blowing stuff, Vsauce stuff, oh my!! THE CURIOSITY BOX:

Jake’s video about The Curiosity Box:

Minute Physics on why December days are the longest:

StandUpMaths on calendars and leap days:

Tom Scott on the Equation of Time:

My video on what would happen if the Earth stopped spinning:

GREAT visuals showing how Earth moves around the sun:

George Washington’s birthday:

real-time sub solar point location:

Lahaina noon images from the Oahu Astrophotography club:

analemma:

great solargraph and analemma images:

interactive seasons and ecliptic simulator:

Nasa video of seasonal movement of Earth:

Tropical year:

Earth rotation specifics:

How Earth moves through the universe:

minute physics on cab:

PBS spacetime on the cosmic microwave background:

CMB rest frame:

Wikipedia:

wikicommons images:

To explore space, I highly recommend these:

music by
and

Awesome 3D graphics by Eric Langlay:
Lame 2D stuff by me.

you hey Vsauce Michael here do you have a best friend who is there for you 24/7 365 sorry that's not really good enough if you're a friend truly had your back they would be there for you 24 point zero zero zero zero zero zero six seven three sixty five point two four two one eight nine one also George Washington was born on February 22nd 1732 at least that's what we're told however his family Bible says he was born on February 11th 1730 1/2 so which is it mister I cannot tell a lie oh and don't even ask about 1752 in Russia 17:52 looked pretty normal but check out what the British Empire was up to that year nothing really out of the ordinary except ember the second was followed by the 14th were eleven days just deleted where'd they go what happened then it is time to question time and how the earth moves reconciling both of these things has led to some pretty strange things we all love time-lapse videos of the stars moving across the sky but really we are the ones who are tumbling through the universe on a giant wet rock vehicle called earth with a windshield called the sky as viewed from above the North Pole we spin counterclockwise west chases east I always remember this by thinking of the US as a weird Mane headed animal with Texas and Florida legs running forward but we don't just spin we also revolve around the Sun on a plane tilted 23.4 degrees relative to our spin it's kind of nauseating at this scale but from this perspective you can see that the Sun rising and setting is just the earth pointing you towards and then away from the Sun this motion causes your sunrise your noon the moment when the Sun is highest in your sky before your Sun set to more closely investigate this movement let's talk about meridians you are on one at this very moment your meridian is just aligned from where you are right now straight towards the North and South Poles it's a line of longitude as opposed to the horizontal lines that lay flat when north or south is up that we call flat attune or actually latitude the Sun is highest in the sky to you your noon when your Meridian is pointed right at the Sun a cool thing happens at this moment all shadows around you point directly towards one of Earth's poles unless you're on the sub solar point the subsolar point is the point on Earth's surface directly below the Sun it's always somewhere you can check its current location online links as always in the description on the subsolar point shadows fall straight down so they can easily disappear twice a year the subsolar point crosses over Hawaii the only place in the US where it hits land and when it does it is called Lahaina noon meaning cool Sun straight vertical objects look unnatural during this brief time like they don't belong as if they were photoshopped in without regard for reality in Honolulu a sculpture by Isamu Noguchi called sky gate casts a twisted shadow all day every day except during Lahaina noon when its shadow is a perfect circle you may not live in a place where the Sun ever appears directly overhead but once every Earth rotation the subsolar point falls somewhere on your Meridian making it noon for you the technical name for this noon for you is local apparent solar noon the clock on your wrist and the clock on your phone don't tell you your local apparent solar time because long ago we realized that if every meridian had its own time a person just a few kilometers away seeing different shadows than you did would disagree with you on what time it was so town's adopted their own time now later on this trick was standardized and time zones as we know them today came about but that's not all we didn't like about shadow based sundial time to explore deeper we have to begin by asking what's a day I mean obviously it's just the time it takes the earth to turn around once right but according to what everything else in space is moving in some way to the universe doesn't include a convenient sheet of graph paper at Absolute rest we can trace paths on the best we can do on that front is to look at very very far far far away stars so far away like distant features of the landscape out the window of a moving car they barely move as Earth does now to them a Meridian on earth completes a trip around about once every twenty three point nine hours this is called a side aerial day side aerial means pertaining to the Stars even though the sidereal day seems pretty clear it's not what our calendars and clocks are based on because there's a nearer star whose position relative to us has a bigger effect on our lives the Sun looking down on the north pole at Earth's counterclockwise spin the earth also moves counterclockwise around the Sun after a sidereal day the earth has moved a bit along its orbit so some more rotation is required for the same meridian to point back towards the Sun again this longer definition of one rotation is what the modern calendar and clock is based on it is called the solar day but here's the thing exactly how long the earth has to rotate to complete a solar day changes day to day our clocks are just based on the average amount of time this takes so throughout the year they fall ahead and behind the Sun this is a solar graph a picture of the sun's path across the sky every single day if our clocks actually told us local apparent solar time if you took a picture of the sky every day at noon you should get a line of Suns but this is what really happens over the course of a year it will appear as though your clock is running slow and then fast and then slow again and then fast again this problem was known since at least ancient times even if its cause was it in order to reconcile the two the equation of time was constructed in this sense equation means to reconcile the equation of time was applied to what a clock said in order to compute the real time the solar time a sundial would show now some fancy clocks called equation clocks were made that would do this for you but eventually we gave up we gave up and just said no the real time isn't what the Sun says it's what our inventions say now this transition was a big one it was humanity growing up it was like the first time you realize you're stronger than your parents we realized our timepieces were more regular and turned our backs on the timepieces nature had but what causes this disagreement in the first place as it turns out the answer revolves around revolving the way the Earth revolves around the Sun if the Equator faced the Sun all the time and the earth always orbited at the same speed the subsolar point would just stay right there on the equator throughout the year and the amount of extra time spent rotating the earth needed to do to finish a solar day would always be the same but those two things aren't the case first of all the Earth's orbit is slightly elliptical so its speed varies throughout the year when it's moving around the Sun faster around the beginning of January the amount of extra turning time needed to complete the solar day is longer than when it's further away from the Sun and moving more slowly there's more because the earth is tilted the subsolar point is dragged throughout the year in a circle around Earth that's not the equator so it changes directly moving northeast then leveling out and going southeast before leveling out and going northeast again during times of the year when the subsolar point is being dragged by Earth's orbit mostly east it gains against Earth's spin faster more time is required for the day to finish now by coincidence we are alive at a time when both of these phenomena lengthen and shorten days at roughly the same time so they add up making September 18th almost a minute shorter than the longest day of the year December 22nd for northern hemisphere Ian's December has the shortest periods of daylight but the whole solar day from sunrise to sunrise is for everyone on earth the longest of the year on December 22nd people in the North just spent most of it in darkness earth's tilt it doesn't just affect how long a day is it also affects how long a year is this is because the Earth's tilt is what causes the seasons for the half of the earth tilted towards the Sun the same amount of solar radiation is spread across less space than it is on the other half so there is more heat energy laid down per area this causes what we call summer and winter for the other half the amount of time from one of these seasonal orientations of the earth to its occurrence again is called a solar year or a tropical year it's a very useful way to define a year because it contains every single season exactly since it's based on the very orientations that caused them but the problem is this the number of solar days that occur in a solar year is not a whole number it's almost 365 but after that many solar days about 1/4 of a day more happens before the solar year starts again this makes designing a calendar more like designing a calendar if your calendar only ever has 365 days in a year over time those dates will drift from the seasonal positions they used to occur during unless hmm this extra quarter of a turn adds up to a full day after four years see March 1st is coming a day too soon now so if we just delay March by adding an extra day at the end of February every four years a leap day we're back on track leap days do not add days to your life you're still going to live the same number of them they just change what we call them but really who cares about being one day earlier every four years I mean one day isn't much you hardly notice it but over time well if America's founders had declared not only independence from Britain but also from leap days and abolished them from happening today 240 years later their calendar would be a full two months ahead of Earth's position putting America's coldest winter days in April and its hottest summer days in October adding a day every four years is what the famous Julian calendar does introduced in 46 BC by Julius Caesar it was the defacto standardized Western calendar for a very long time more than a millennium but it's not perfect look closely leap days actually move the calendar just slightly too far each time because I lied 365 and a quarter solar days don't occur within one solar year the real number is slightly less and fluctuates year-to-year based on long-term changes to earth and the sun's movements which means adding one day every four years is just a teensy weensy too many by 1582 julian calendar dates were ten days behind the seasons compared to where they used to be that's not bad ten days and more than a millennium and a half but the Catholic Church cared because they wanted Easter to occur exactly when it used to centuries ago astronomers at the time realized that if leap days pushed the calendar too far behind the seasons we would just need to celebrate fewer of them to fix the problem specifically we would need three fewer leap days every four centuries the rule they wrote to achieve this stated that every four years would continue to be a leap year except if it was divisible by 100 unless it was also evenly divisible by 400 this removes three every 400 years on October 4th Pope Gregory the 13th introduced this new calendar it took his name the Gregorian calendar he also undid the drift that had occurred since the early days of the Julian and declared on October 4th that tomorrow would be October 15th October 5th to the 14th never happened in 1582 in countries that listened to the Pope it took the rest of the world centuries to hop on board England and its colonies like the soon-to-be United States of America adopted the Gregorian in September of 1750 – by which point their Julian dates were off from the seasons by 11 days hence the disappearing of the 3rd through 13th when adopted the 1st of the year was also moved from March to January first this explains why George Washington's birthday has two answers although more closely hitched to the seasons than the Julian the Gregorian calendar still isn't perfect its difference causes dates to become one day off from the seasons every 3,000 216 years other calendars have been proposed like the one stand-up maths calculated that drifts off even more slowly his video is a great watch by the way but enough of all of this let's sit back and enjoy Earth's movement without trying to divide it up and name it as a caveat keep in mind that Earth's oceans and liquid insides and other celestial bodies are always pulling and tugging and sloshing around minutely changing Earth's movements their effect is measurable but difficult to notice at big scales and also don't look like much in the short term short like the length of a human life looking from above the North Pole the equator spin counterclockwise at about sixteen seventy kilometers per hour relative to the Sun Earth orbits counterclockwise at one hundred and eight thousand kilometers per hour along a path tilted 23.4 degrees to it's spin within our local neighborhood of stars our entire solar system is drifting seventy thousand kilometers per hour roughly in the direction of the bright star Vega in the constellation of Lyra and our solar system is part of a giant galaxy called the Milky Way on a plane tilted about sixty degrees approximately like the windshield of a car looking from above Earth's North Pole our entire solar system races clockwise around the galactic center at about seven hundred and ninety two thousand km/h our whole galaxy is also moving through the universe we know this because when the universe was very young it was so hot electrons and protons jumped around and photons of light scattered constantly they couldn't travel very far before scattering again so the universe was opaque but then around 380,000 years after the Big Bang the universe cooled just enough for electrons and protons to form hydrogen suddenly abruptly photons decoupled from this obstacle course and could travel relatively unencumbered the universe became transparent to light since that moment those early photons have been propagating through space every day ancient photons that last scattered off this opaque fog at the moment of decoupling a light date further from Earth currently is reach us they are part of the cosmic microwave background radiation it is visible in every direction microwave because although they used to be more energetic the universe's expansion has redshifted them now some parts of this radiation are more redshifted than others because of our own movement through the universe controlling for the movements we've already talked about relative to this infinite cooling baby picture of the universe the first and oldest detectable light we are headed riding along in the Milky Way in the direction that the constellations of Leo and Virgo are to us at a speed of 2.1 million kilometers per hour towards a thing we don't fully understand yet simply called the Great Attractor this is how you on Earth's surface are moving through the universe aboard Spaceship Earth okay now stop this is roughly 100 years of Earth's movement through space this path we trace for where we began here is the path you will take through the universe in your lifetime you didn't buy a ticket for this ride your parents signed you up without asking but nonetheless it is quite literally the ride of your life and as always thanks for watching it is now time for a major Vsauce announcement you know how for the last couple of years we've released a holiday box well introducing the curiosity box by Vsauce we created this thing as a quarterly box this comes to you four times a year and it comes packed with amazing exclusive Vsauce merchandise and incredible geeky toys picked by myself Kevin and Jake plus as always a portion of the proceeds from this box goes to funding Alzheimer's research you can get your own by subscribing at the curiosity box com I'm incredibly proud of this go check it out and as always thanks for watching

The Map of Mathematics



Views:3996653|Rating:4.94|View Time:11:6Minutes|Likes:142687|Dislikes:1768
The entire field of mathematics summarised in a single map! This shows how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.

If you would like to buy a poster of this map, they are available here:
North America:
Everywhere else:

I have also made a version available for educational use which you can find here:

To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes. This is the errata where I correct my silly mistakes. My goal is to one day do a video with no errors!

1. The number one is not a prime number. The definition of a prime number is a number can be divided evenly only by 1, or itself. And it must be a whole number GREATER than 1. (This last bit is the bit I forgot).

2. In the trigonometry section I drew cos(theta) = opposite / adjacent. This is the kind of thing you learn in high school and guess what. I got it wrong! Dummy. It should be cos(theta) = adjacent / hypotenuse.

3. My drawing of dice is slightly wrong. Most dice have their opposite sides adding up to 7, so when I drew 3 and 4 next to each other that is incorrect.

4. I said that the Gödel Incompleteness Theorems implied that mathematics is made up by humans, but that is wrong, just ignore that statement. I have learned more about it now, here is a good video explaining it:

5. In the animation about imaginary numbers I drew the real axis as vertical and the imaginary axis as horizontal which is opposite to the conventional way it is done.

Thanks so much to my supporters on Patreon. I hope to make money from my videos one day, but I’m not there yet! If you enjoy my videos and would like to help me make more this is the best way and I appreciate it very much.

Here are links to some of the sources I used in this video.

Links:
Summary of mathematics:
Earliest human counting:
First use of zero:
First use of negative numbers:
Renaissance science:
History of complex numbers:
Proof that pi is irrational:
and

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Matematika që mësojmë në shkollë nuk i bën drejtësi mjaftueshëm fushës së matematikës. Ne shohim vetëm një kënd të saj, por matematika si tërësi është lëndë e madhe dhe me një llojllojshmëri të mahnitshme. Qëllimi im me këtë video është të ju tregoj juve të gjitha ato gjëra mahnitëse. Ne do të fillojmë që nga më e para pikë e fillimit. Origjina e matematikës bazohet në numërim. Në fakt, numërimi nuk është vetëm aftësi e njeriut, kafshë të tjera kanë aftësi të numërojnë gjithashtu dhe dëshmi për numërimin e njeriut shohim që në kohët pre-historike me shenja të bëra në eshtra. Përgjatë viteve ka pasur risi të ndryshme, me ekuacionin e parë nga Egjiptianët, Grekët e lashtë bënë hapa të mëdha në shumë fusha si gjeometria dhe numerologjia, dhe numrat negativ ishin zbuluar në Kinë. Dhe zeroja si një numër ishte përdorur për herë të parë në Indi. Pastaj në Kohën e Artë të Islamit matematicientët Persian bënë hapa të mëdha mëtutje me librin e parë të shkruar për algjebër. Pastaj matematika lulëzoj në renesancë përgjatë shkencave të tjera. Tani, ka më shumë për historinë e matematikës se sa ajo çka unë sapo thashë, por do të kërcej në kohën moderne dhe matematikën ashtu si ne e njohim. Matematika moderne për së gjeri mund të ndahet në dy fusha, matematika e pastër: shkenca e matematikës për hir të vetes, dhe matematika e aplikuar: kur ju zhvilloni matematikën në ndihmë të zgjidhjes së disa problemeve reale në botë. Por, ka shumë mbikalime. Në fakt, shumë herë në histori dikush ka kaluar përgjatë shkretisë së matematikës motivuar thjesht nga kurioziteti dhe sikur i drejtuar nga një sens estetik. Dhe pastaj ata kanë krijuar një tërësi të re të matematikës që ishte e mire dhe interesante por nuk sjell diçka që mund të jetë e dobishme. Por pastaj, le të themi pas qindra vjetësh, dikush do të punoj në ndonjë problem në këndin e fundit të fizikës apo shkencës kompjuterike dhe do të zbulojnë që kjo teori e vjetër në matematikën e pastër është pikërisht ajo çka ju nevojitet për të zgjidhur problemin e tyre real të botës! Që është e mahnitshme, mendoj! Dhe kjo lloj gjëje ka ndodhur kaq shumë herë përgjatë shekujve të fundit. Është interesante sa shpesh diçka kaq abstrakte përfundon të jetë shumë e dobishme. Por do duhej të përmendja dhe se matematika e pastër në vetvete është ende një gjë shumë me vlerë për tu bërë ngase mund të jetë interesante dhe në vetvete të ketë bukuri dhe elegancë të vërtetë që gati bëhet si art. Në rregull mjaft me këtë shpjegim të entuziazmuar, le ti hyjmë punës. Matematika e pastër është e përbërë nga disa pjesë. Studimi i numrave fillon me numrat natyral dhe çfarë mund të bësh me ata me operacione aritmetike. Dhe pastaj shikon lloj të tjerë të numrave si numrat e plotë, që përmbajnë numrat negativ, numrat racional me thyesa, numrat real që përfshijnë numrat si pi që vazhdojnë në pika të pafundme decimale, dhe pastaj numrat kompleks dhe një tërësi tjetër. Disa numra kanë veti interesante sikur janë numrat e thjeshtë, ose numri pi ose eksponenciali (treguesi). Ka edhe veti të këtyre sistemeve numerike, si për shembull, edhe pse ka një sasi të pafundme të të dyjave numrave të plotë dhe real, ka më shumë numra real se sa ka numra të plotë. Kështu disa pafundësi janë më të mëdha se të tjerat. Studimi i strukturës ka të bëjë me ku ti fillon të marrësh numrat dhe ti vendosësh në ekuacione në formën e ndryshoreve. Algjebra përmban rregullat se si pastaj manipulon këto barazime. Këtu do të gjesh dhe vektorë dhe matrica që janë numra shumë dimensional, dhe rregullat se si këto bashkëveprojnë me njëra-tjetrën është kapur në algjebrën lineare. Teoria e numrave studion karakteristikat e të gjithave në pjesën e fundit në numrat si vetitë e numrave të thjeshtë. Kombinatorika shikon vetitë e strukturave të veçanta si pemët, grafet, dhe gjëra të tjera që janë të bëra nga copa diskrete që mund ti numërosh. Teoritë grupore shikojnë tek objektet që janë në bashkëveprim me njëra-tjetrën pra në grupe. Shembull i njohur është kubi i Rubikut që është një shembull i grupit të permutacioneve. Dhe teoria e renditjes heton se si objektet renditen duke ndjekur rregulla të caktuara, si diçka është sasi më e madhe se sa diçka tjetër. Numrat natyral janë një shembull e një strukture objektesh të renditura, por gjithçka me çfarëdo lidhje binare mund të renditet. Një pjesë tjetër e matematikës së pastër shikon tek figurat dhe si ato sillen në hapësirë. Origjina është në gjeometri që përfshin Pitagorën, dhe është e afërt me trigonometrinë, me të cilat jemi të njoftuar në shkolla. Gjithashtu ka gjëra argëtuese sikur gjeometria fraktale që janë modele matematikore me shkallë të pandryshueshme, që do të thotë mund të zmadhoni modelin pafundësisht dhe gjithmonë do të duket e njëjtë. Topologjia shikon tek pjesë të ndryshme të hapësirave ku jeni të lejuar që vazhdimisht ti deformoni ato por jo të i ndani apo ti bashkoni ato pjesë. Për shembull rripi i Mobiusit ka vetëm një sipërfaqe dhe një kënd pavarësisht çka i bëni. Dhe gotat e kafes dhe petullat në formë gjevreku janë e njëjta gjë – në aspekt topologjik. Teoria e matjeve është mënyra e caktimit të vlerave në hapësira apo struktura të lidhura së bashku me numrat dhe hapësirën. Dhe së fundmi, gjeometria diferenciale shikon vetitë e formave në sipërfaqet e lakuara, për shembull trekëndëshat kanë kënde të ndryshme në sipërfaqe të lakuar, dhe kjo na sjell neve në pjesën e radhës, që është ndryshimet. Studimi i ndryshimeve përmban kalkulusin që përfshin integralet dhe diferencialet që shikon tek fusha e hapur nga funksionet apo sjellja e shkallëve të funksioneve. Dhe kalkulusi vektorial shikon të njëjtat gjëra për vektorët. Këtu gjejmë dhe një tërësi të fushave të tjera si sistemet dinamike që shikojnë sistemte që evukojnë gjatë kohës nga një gjendje në tjetrën, sikur rrjedhja e lëngut apo gjërave me cikleve me reagim sikur ekosistemet. Dhe teoria e kaosit që studion sistemet dinamike që janë shumë të ndjeshme ndaj kushteve iniciuese. Së fundi analiza komplekse shikon vetitë e funksioneve me numra kompleks. Kjo na sjell në matematikën e aplikuar. Në këtë pikë është me vend të përmendim që çdo gjë këtu është më shumë e ndërlidhur se sa që unë e kam vizatuar. Në realitet kjo hartë duhet të duket më shumë si një rrjetë që lidh të gjitha subjektet e ndryshme por ju mund të bëni kaq shumë vetëm në një rrafsh dy dimensional, kështu i kam shpërndarë këto më së mirti si kam mundur. Në rregull do të fillojmë me fizikën, që përdor gati gjithçka në anën e djathtë në një shkallë. Fizika matematikore dhe teorike ka një lidhje shumë të ngushtë me matematikën e pastër. Matematika gjithashtu përdoret në shkenca të tjera natyrale me kiminë matematikore dhe biomatematikën që merren me shumë gjëra prej modelimit të molekulave tek biologjia evolucionare. Matematika është po ashtu e përdorur dukshëm në inxhinieri, ndërtimi i gjërave ka marr shumë matematikë që nga koha e Egjiptianëve dhe Babilonasëve. Sisteme elektrike shumë komplekse sikur avioni apo rrjeti i energjisë përdorin metoda në sisteme dinamike të quajtura teori të kontrollit. Analiza numerike është mjet matematikor zakonisht i përdorur në vende ku matematika bëhet shumë komplekse për tu zgjidhur plotësisht. Kështu që në vend të saj, ti përdor shumë
të vlerave të përafërta dhe i kombinon ato së bashku për të marr përgjigje të përafërta të mira. Për shembull, nëse ti e vendos një rreth brenda një katrori, hedh shigjeta në të, dhe pastaj e krahason numrin e shigjetave në pjesët e rrethit dhe katrorit, ti mund të përafrojsh vlerën e numrit pi. Por në botën reale analiza numerike bëhet në kompjutera të mëdhenj. Teoria e lojërave merret me se cilat janë zgjedhjet më të mira në rastin e një strukture rregullash dhe lojëtarëve racional dhe përdoret në ekonomi ku lojëtarët mund të jenë inteligjent, por jo gjithmonë, dhe fusha të tjera si psikologji, dhe biologji. Probabiliteti është shkenca e ngjarjeve të rëndomta sikur hedhja e monedhave apo zaret apo njerëzit, dhe statistika është shkenca e koleksioneve të mëdha të proceseve të rëndomta apo organizimi dhe analizimi i të dhënave. Kjo natyrisht është e lidhur me financa matematikore, ku ju doni sisteme të modeleve financiare dhe të marrë një avantazh për të fituar të gjitha ato shtresa të trasha. Lidhur me këtë është optimizimi, ku ju provoni të kalkuloni zgjedhjen më të mirë përgjatë një strukture të shumë opsioneve të ndryshme, të cilat ju mund ti vizualizoni normalisht duke provuar të gjeni pikën më të lartë apo më të ulët e një funksioni. Problemet optimizuese janë të natyrës së dytë tek ne njerëzit, ne i bëjmë ato gjatë gjithë kohës: duke provuar të marrim vlerën më të mirë për para, apo të zmadhojmë harenë tonë në një mënyrë. Tjetër fushë që është shumë e lidhur me matematikën e pastër është shkenca kompjuterike, dhe rregullat e shkencës kompjuterike në fakt janë derivuar në matematikën e pastër dhe është një shembull tjetër i diçkaje që është punuar shumë përpara ndërtimit të kompjuterëve programues. Mësimi i makinës: krijimi i sistemeve kompjuterike inteligjente përdor shumë fusha në matematikë sikur algjebra lineare, optimizimi, sistemet dinamike dhe propabiliteti. Dhe në fund teoria e kriptografisë është shumë e rëndësishme në kompjutim dhe përdor shumë matematikë të pastër sikur kombinatorikën dhe teorinë e numërave. Pra kjo mbulon pjesët bazike të matematikës së pastër dhe të aplikuar, por nuk mund të përfundoj pa shikuar në themelet e matematikës. Kjo fushë provon të punoj vetitë e matematikës vet, dhe pyet se çka është baza e të gjitha rregullave të matematikës. A është ndonjë strukturë komplete e rregullave bazike, të quajtura aksiomë, nga e cila vjen e gjithë matematika? Dhe a mund të vërtetojmë që e gjitha është e qëndrueshme në vetvete? Logjika matematike, teoria e vendosur dhe teoria e kategorisë mundohen ti përgjigjen kësaj dhe një rezultat i famshëm në logjikën matematike janë teorema e pakompletuar e Godel, që për shumë njerëz, do të thotë që Matematika nuk ka një strukturë të kompletuar dhe të qëndrueshme të aksiomave, që tregon që e gjitha disi është e bërë nga ne njerëzit. Që është e çuditshme duke parë që matematika shpjegon kaq shumë gjëra në Univers kaq mirë. Pse do duhej një gjë e bërë nga njerëzit të ketë aftësi të bëjë një gjë të tillë? Po aty, kjo është një mister i thellë. Gjithashtu kemi teorinë e kompjutimeve që merret me modele të ndryshme të kompjutimit dhe se sa me efikasitet ata mund të zgjidhin probleme dhe përmban teori të kompleksitetit që merret me atë se çka është dhe nuk është e llogaritshme dhe sa memorie dhe kohë do duhej, që për shumicën e problemeve interesante, është një sasi e çmendur. Kështu ajo është harta e matematikës. Tani gjëja që kam dashur më së shumti kur kam mësuar matematikë është ajo ndjenja që ti e ke kur diçka që dukej kaq konfuze më në fund klikon në trurin tënd dhe gjithçka ka kuptim: sikur një moment pagëzimi, disi sikur të shihje nëpërmjet matricës. Në fakt disa nga momentet më të kënaqshme intelektuale kanë qenë kuptimi i disa pjesëve të matematikës dhe pastaj të ndjerit sikur unë kisha një shikim të shkurtër në natyrën themelore të Universit në të gjitha çuditë e saj simetrike. Është e mrekullueshme, e dua atë. Punimi i hartës së matematikës ka qenë kërkesa më e popullarizuar që kam marrë, për të cilën isha shumë i lumtur sepse e dua matematikën dhe është e mrekullueshme të shikosh kaq shumë interesim për të. Kështu që shpresoj që ju ka pëlqyer. Natyrisht është vetëm kaq shumë sa mund të fus në këtë pjesë kohe, por me shpresë që i kam bërë drejtësi subjektit dhe se ju e keni vlerësuar të dobishme. Kështu do ketë më shumë video që do vijnë nga unë shpejt, këtu janë të gjitha gjërat e zakonshme dhe ishte një kënaqësi, shihemi herën tjetër.