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

Lecture 1 | Introduction to Robotics



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Help us caption and translate this video on Amara.org: Lecture by Professor Oussama Khatib for Introduction to Robotics …

Lecture 2 | Machine Learning (Stanford)



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

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

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

Introduction to computer Networks



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Algebra – Ch. 4: Exponents & Scientific Notation (33 of 33) Simplify Exponential Expressions 10



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In this video I will simplify exponential expression: 10.

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Algebra – Ch. 4: Exponents & Scientific Notation (31 of 33) Ignore Please, Editing Problems



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In this video I will simplify exponential expression: 8.

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LU Decomposition – Shortcut Method



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This video explains how to find the LU Decomposition of a square matrix using a shortcut involving the opposite of multipliers used when performing row operations.
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welcome back for a second lesson on matrix Lu decomposition in this video we'll use Gaussian elimination and a shortcut to determine the Lu decomposition the goal of Lu decomposition is to write a square matrix a as the product of matrix L and matrix U where matrix L is a lower triangular matrix and matrix U is an upper triangular matrix for the shortcut method our lower triangular matrix L will have a main diagonal consisting of ones and all elements above the main diagonal must be 0 for the upper triangular matrix all the elements below the main diagonal must be 0 notice for the upper triangular matrix the main diagonal does not have to consist of once in order for this decomposition to be possible matrix a must be able to be reduced to row echelon form which would be matrix you without interchanging any rows also matrix L and matrix u are not unique and because we're using the shortcut method we're going to be using the opposites of the multipliers used in the row operations to obtain you to build matrix ill so let's take a look at an example we want to determine the Lu decomposition of the gibbon 3×3 square matrix so we want to perform row operations on this matrix to obtain a upper triangular matrix which means we need a 0 in this position this position and this position let's start by obtaining a 0 in this position here we want to add some multiple of another row to Row 2 to obtain a 0 in this position here so we can replace Row 2 with 2 times Row 1 plus Row 2 to obtain a 0 in this position let's go ahead and do that first row stays the same second row would be 2 times 1 plus negative 2 that's 0 two times four plus eight that's 16 and two times negative three plus five is negative one third row stays the same now to perform this row operation we say our multiplier was this positive to this positive two helped us to obtain a zero in Row two column one of this matrix well if we look at our lower triangular matrix here's Row two column one this element is going to be the opposite of our multiplier well the opposite of positive two is negative two this is how we're going to build our lower triangular matrix we'll always use the opposite of the multiplier used to obtain the zero in the specific position now that we have this matrix let's obtain a zero in this position here so again we want to add a multiple of another row to row three to obtain a zero here let's replace Row 3 with negative three times Row 1 plus Row three first two rows stay the same the third row will have negative three times one plus three that's zero here we have negative 3 times 4 that's negative 12 plus 4 that's negative 8 and then here we would have negative 3 times negative 3 that's positive 9 plus 7 that's 16 for this row operation our multiplier was negative 3 negative 3 helped us obtain a zero in Row 3 column 1 here's Row 3 column 1 of our lower triangular matrix the opposite of negative 3 is positive 3 so this element is positive 3 now let's obtain a 0 in this position here which is Row 3 column 2 we need to add a multiple of Row two to Row 3 to obtain a 0 well half of 16 would be positive 8 so let's replace Row 3 with 1/2 times Row 2 plus Row 3 again our multiplier is a multiple of another row that we have to add to the given row to obtain our 0 so here the first two rows stay the same 1/2 times Row 2 plus Row 3 1/2 times 0 plus 0 is still 0 1/2 times 16 is 8 8 plus negative 8 is 0 now here we have 1/2 times negative 1 that's negative 1/2 plus 16 that would be 15 and 1/2 or let's just say 15 point 5 notice how we now have an upper triangular matrix we'll call this matrix U and then to complete our lower triangular matrix notice how our multiplier in this case was positive 1/2 to obtain a 0 in Row 3 column 2 or this position here the opposite of positive 1/2 is negative 1/2 so now we have our Lu decomposition here's matrix L and here's matrix u so they put the pieces together we know a equals L times u here is matrix a the given matrix here's the lower triangular matrix and here is our upper triangular matrix let's go ahead and take a look at a second example let's start by obtaining a 0 in this position here again we need to add a multiple of another row to this row to obtain a 0 here so I'm going to replace Row 2 with negative 1/2 times Row 1 plus Row 2 so the first row stays the same for the second row negative 1/2 times 2 that's negative 1 plus 1 that's 0 negative 1/2 times 4 is negative 2 plus negative 4 that's negative 6 negative 1/2 times negative 4 is positive 2 plus 3 that's 5 third row stays the same our multiplier is negative 1/2 so we're going to put a positive 1/2 in row 2 column 1 of our lower triangular matrix which is here so this element is positive 1/2 to obtain a 0 here let's replace Row 3 with 3 times Row 1 plus Row 3 first two rows stay the same then for the third row we have 3 times 2 plus negative 6 that's 0 3 times 4 plus negative 9 or 12 plus negative 9 is positive 3 and 3 times negative 4 negative 12 plus 5 is negative 7 our multiplier is positive 3 which helped us to obtain a 0 in Row 3 column 1 so this position here in the lower triangular matrix would be the opposite of positive 3 or negative 3 and then for the last step to obtain a 0 here we'll need to add half of Row 2 to row three first two rows stay the same so 1/2 of 0 plus 0 is 0 1/2 of negative 6 is negative 3 plus positive 3 that's 0 and then we'll have 1/2 times five that's five halves plus negative 7 which is negative 14 halves which would be negative 9 halves so this element here is negative 9 halves notice our multiplier here was positive 1/2 to obtain a zero in Row three column two so Row three column two of our lower triangular matrix will have an element that's the opposite of positive 1/2 or negative 1/2 so here's our upper triangular matrix and here's our lower triangular matrix for our Lu decomposition so putting the pieces together we know that a equals L times U so a is the given matrix L was our lower triangular matrix which we built using the shortcuts and u was the upper triangular matrix which we found by performing row operations ok I hope you found this helpful

Compound Interest Formula Explained, Investment, Monthly & Continuously, Word Problems, Algebra



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This algebra & precalculus video tutorial explains how to use the compound interest formula to solve investment word problems. This video contains plenty of examples and practice problems for you to work on.

Here is a list of topics:
1. Compound Interest Explained – Formula & Equations
2. Compounded Monthly, Semi Annually, Quarterly, Daily, Weekly and Compounded Continuously
3. Compound Interest Word Problems – Investment, Mutual Funds, Savings Account, and Index Annuity
4. Logarithms – Solve for t
5. Compound Interest – Solve for r using e
6. Future Value vs Present Value – Math Problems

in this video we're going to work on some word problems using the compound interest formula there's two equations that you need to know a is equal to P times 1 plus R divided by n raised to the N times T now in this formula P is basically the principal that's how much you would deposit in a savings account or a checking account on something like that a represents the future of value of that amount in the account after it's been credited for interests over a period of time so you can think of P as the present value how much you put in in the present and a is like how much it will be worth 10 or 20 years later r is the annual interest ring so let's say if the annual interest rate is 8% you need to plug in point zero eight for our you need to convert eight percent into a decimal to do that you would divide by 100 n is how many times you receive interest in a given year so let's say if it's compounded monthly that's 12 months in a year so n would be 12 for muffle now what about for weekly what is the value of n there's 52 weeks in years so n will be 52 daily and it's 365 quarterly n is 4 semi-annually and is 2 and annually and is 1 and T is basically the time in years now there's one more equation that you need and it's this one a is equal to P times e raised to the RT a and P are basically the same as the last equation P is the principle that's the amount that you put in to an account a is the future value after some time maybe 10 or 20 years later he is basically the inverse of the natural log function so if you have your calculator to find e you may have to type in shift from natural log or second Ln something like that R is the annual interest rate as a decimal and T is the time in years so when do we use this equation compared to the other equation now if you hear the key word compounded continuously use this equation if it's compounded times something else let's say monthly daily weekly quarterly you would use the other equation but the only time you would use this equation if the problem says it's compounded continuously so let's work on some problems susan puts 20,000 in a savings account paying 8% annual interest compounded monthly at this rate how much money will be in the account after 40 years so it's not compounding continuously therefore we need to use this equation so P is the principle she puts in twenty thousand in the account R is the annual interest rate which is eight percent if we divide that by 100 that's point zero eight and she receives that total eight percent annual interest in twelve months so basically that eight percent is divided into twelve so her account is credited with interest every month so we're going to divide this point zero eight by n which is twelve and we want to find out how much money will be in the account after forty years T is forty point zero eight divided by 12 plus one is basically one point zero zero six repeating twelve times 40 is 480 so if you type this in a calculator you should get a value of around four hundred eighty five thousand four hundred sixty seven dollars and seventy nine cents so that's how much money will be in the account after forty years so as you can see it pays to save early here's another problem you can pause the video and work on it john wants to have two million for e time in forty five years he invests in a mutual fund pain an average of nine point five percent each year compounded quarterly how much should he deposit into his mutual fund so we need to use this equation a equals P one plus R divided by n raised to the NT so we know the future value he wants to have two million in his account so he needs to decide how much he should put in now to get to that level so we're looking for paying the problem R is the annual interest rate nine point five divided by 100 is 0.09 five and it's compounded quarterly that is four times a year so four times a year his account is credited with interest so we're going to divide it by four and then it's raised to the NT or four times forty five T is the time in years so you can type it in exactly the way you see it let's find out what this value is equal to first point zero nine five divided by four is point zero two three seventy five and let's add one to it so that's one point zero two three seven five and four times 45 is 180 so this is going to be P times sixty eight point three seven six one five two so to solve for P one needs to divide both sides by this number so P is 2 million divided by sixty eight point three seven six one five two so that's going to be about twenty nine thousand two hundred forty nine and ninety six cents so if he invests about 29,000 let's round it to two fifteen if he invest that much and if he finds an account pain and an interest rate of 9.5% compounded quarterly then in forty five years he should have two million in his retirement so if he starts invested in the famous twenties by his mid or upper 60s he can have that much in savings so as you can see due to the effect of compound interest it pays to save early Sara wishes to turn her ten thousand investment into a hundred thousand and twenty years how much interest does she need to receive compounded annually to reach her goal so in this problem we need to solve for R so let's use this equation again so a is the value in twenty years she wants a hundred thousand P is her initial deposit the principal which is ten thousand R is the annual interest rate which we're looking for and n is one since its compounded annually which means that she receives interest once per year and T is 20 so the first thing we should do in order to solve for R is divide both sides by ten thousand a hundred thousand divided by ten thousand is ten so basically she wants to multiply her investment by a factor of ten so now what can we do to solve our how can we get rid of this exponent in order to open the parentheses we need to turn the 20 into a 1 to do that raise both sides to the reciprocal of 20 or run over 20 20 times 1 over 20 is 1 so we have is 10 raised to the 1 over 20 equals 1 plus our so to solve for our we need to subtract both sides by one so it's 10 raised to the 1 over 20 minus 1 10 raised to the 1 over 20 is about one point one to two and subtracted by 1 this is equal to point one to two now to turn into a percentage multiplied by one hundred percent so R is 12 point two percent so if she wants to multiply her investment by a factor of 10 she needs an account that is paying 12 point two annual interest if he could find that than in 20 years she's going to multiply her investment by a factor of 10 so if she invests a hundred thousand and twenty years is going to be a million if she invests two hundred thousand 20 years is going to be 2 million Mary invests 50 thousand dollars into an index annuity that's averaging 8.4 percent per year compounded semi-annually at this ring how many years will it take for her account to reach 1 million so let's write the equation a is equal to P times 1 plus R divided by n raised to the NT so her goal is to reach a million that's d that's the a value so to speak her investment the principle is 50,000 the interest rate is 8 point 4 percent which is point zero eight four and it's compounded semi-annually which means she receives interest twice a year so n is two so what we need to do is solve for T so first let's divide both sides by 50,000 so what's the 1 million / 50,000 that's equal to 20 so she wants to multiply her investment by 20 point zero eight four divided by two plus one is one point zero four two in order to solve 14 we need to use logarithms so let's take the log of both sides so on the Left we're going to have a log 20 and on the right we're going to have a log one point zero 4 2 raised to the 2t a property of logs allows us to take the exponent and move it to the front so therefore we now have is log 20 is equal to 2t times log 1.0 for tune now to get T by itself let's divide by 2 log 1.04 to both sides so therefore T is equal to I'm going to take it one step at a time log 20 is about one point three zero one zero three log one point zero four two times two is point zero three five seven three five four if you divide these two numbers you should get thirty six point four years if I typed it in correctly mistakes do happen but this is how long it's going to take her to multiply her investment by a factor of twenty so in thirty six point four years if she can find an account that is averaging eight point four percent per year in interest she could turn this fifty thousand investment to a million Juliette invests a hundred thousand in an account paying seven point two percent interest compounded continuously how much money will be in her account after 30 years now anytime you see this key expression compounding continuously this is the equation that you need so we're looking for the future value of her account 30 years from now so we're solvent for a we have her principle investment it's 100,000 and the interest that she's receiving is 7.2 percent or point zero seven two as a decimal and her account will be active for 30 years point zero seven two times 30 is basically two point sixteen and E which is the inverse of the natural log function e raise to two point one six is about like eight point six seven one one three seven six something times 100,000 so her investment is going to be worth eight hundred sixty seven thousand one hundred thirteen dollars and seventy seven cents mark wants to have 1.5 million in 50 years how much should he invest now in an account paying 12% interest compounded continuously so here is our key expression which means we need to use this equation again so we have the future value the value in 50 years so that's a and this problem we're looking for ping we need to know how much he should deposit into his account in order to reach this goal R is 12% or 0.12 and the time is 50 years so first let's multiply point 12 times 15 and that's equal to 6 now to get P by itself let's divide both sides by e to the 6 so P is going to be 1.5 million divided by e to the 6 and basically this is equal to three thousand seven hundred and eighteen dollars and thirteen cents which seems very very small but the reason why this small amount turns into this large amount is because of the time 50 years is a long time that's one and two the interest rate is much higher than the interest of the other problems which were like seven 8% a 12% interest rate compounded continuously will greatly increase this account value over a long period of time as you can see a small investment was greatly multiplied over 50 years John invest five million in an account paying eleven percent interest compounded continuously how long will it take for his investment to turn into two million so let's try this problem so we have the same formula a is equal to pert and we have the future value of two million and his deposit of five thousand R is point 1 1 but this problem will looking for T so let's begin by dividing both sides by 5000 so if you wish to do this in your head you can get rid of three zeros so you have 2000 divided by 5 mm is basically 20 times 120 divided by 5 is 4 4 times 100 is 400 so if you take 2 million and divided by 5,000 it will give you 400 so we have 400 is equal to e raised to the point 11 T now instead of using log we're going to use natural log the reason being is the natural log of E is equal to 1 so natural log 400 is equal to the natural log e raised to the point xi e so whenever you have a variable in the exponent you can use the log function or the natural log function but when you're dealing with E is easier to deal with or use the natural log function so what should we do now once you get to this part take the exponent and move it to the front so we have the natural log of 400 and that's equal to point 11 T times the natural log of e now the natural log of e as you mentioned is 1 so that's just going to disappear so our last step is simply to divide both sides by point 11 natural log of 400 is about 5.99 146 if we divide that by point 11 this is going to be 54 point four seven so that's how long he needs to invest if he wants to have 2 million now 5,000 is a small investment but if he invests early that's the key he can take the advantage of the effective compound or compound interest his money will grow to 2 million if he does it in if he invest early let's say 54 years early so as you can see whenever you invest early and you can use time to help multiply your investment now granted this effect will be greatly increased if you can find account our savings account or checking account that's paying a very high interest rate typically in mutual funds and indexed annuities is probably the best place where you can get such high interest rates but that is it for this video thanks for watching and have a great day

Linear Programming



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**DOH! There is a STUPID arithmetic mistake by me at the very end!** Sorry!
Linear Programming. I do a complete example!
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okay in this video I'm going to do an example of what's called linear programming and the basic procedure for linear programming there's something you try to maximize or minimize usually subject to a couple of constraints or inequalities and what you'll do is you'll graph the inequalities you'll graph the region bounded by all the inequalities find the vertices which are basically just the corners of the region you'll basically plug those corner points into the function whatever the largest value is that you're maximum the smallest is your minimum so here's basically just a little example of what's going to happen you'll graph your region your inequalities you'll find some little region here it's in the green again you'll basically just find the vertices the corners plug those in and determine which one's a max and which one's a minimum okay so here's our actual problem that we'll do here so suppose you have 240 acres of land and you're going to make forty dollars per acre for the corn 30 dollars per acre of oats that you plant but it takes our to heart to harvest the stuff so suppose you have 320 hours available and it takes two hours of labor per acre for the corn and oats only take one hour of labor so how many acres of each should be planted to maximize profits okay so intuitively you know you'd probably like to plant a lot of well I don't know it's trade-off right if you plant corn you get you get more money but you know you can do two hours you can harvest two acres of oats for every one acre of corn but you actually double your money then so but then of course you're taking up extra space so alright so there's there's some issues here so let's see if we can't figure it out so I'm going to let X equal the number of acres of corn that we'll plant and then the same thing we'll let y equal the number of acres of oats that we plant ok the thing that we're trying to maximize here in this case we want to maximize our profit well what are what would our profit be well for every acre of corn you get basically $40 so we'll take 40 and multiply it by the number of acres of corn and this will represent the total income the total income from the corn and then we'll add to that 30 times y and that will represent simply the total income that'll be the total income from the oats if we plant why acres okay so we also have some other constraints here we only have 240 acres of land well so that's where we're going to get some of our constraints here okay so we've taken care of the profit okay so we only have 240 acres of land well certainly as well notice will have the inequalities X has to be greater than equal to zero Y has to be greater than or equal to zero because you can't you know you can either play 0 or some positive number of acres notice also one of our inequalities is that X plus y ok so we're planning X acres of corn Y a KERS of oats this is represents the total number of acres and we know that the total number of acres has to be under less than or equal to 240 likewise we only have 320 hours available man-hours to to accomplish our task so it takes 2 hours per acre of corn plus 1 hour per acre of ODEs and we know that that has to be less than or equal to 320 okay because that's how much time we have available so these are going to be our systems of inequalities here that we're going to have to graph here so let me erase this part okay so this is what we're trying to maximize again our profit equals 40 X plus 30 Y and now let me move these over to the side so X has to be greater than or equal to 0 Y has to be greater than or equal to 0 X plus y has to be less than or equal to 240 and then we have our other inequality 2x plus y has to be less than or equal to 320 so now all we're going to do is graph these regions find the points of intersection and basically plug them back into our profit equation ok so actually let me I'm going to redo this graph slightly we really don't even need anything except for the top right quadrant so there's X and there's Y right because if X has to be greater than or equal to 0 and Y has to be greater than or equal to 0 that only happens in the top right quadrant if we graph the inequality I'm going to write it well you could almost think the x and y-intercepts of this first inequality if you plug in x equals 0 we'll get y equals 240 if we let y equals 0 we'll get x equals 240 since it's less than or equal to we'll make it a solid line likewise we could graph so this will be the first region likewise we could graph the second region by finding X and y intercepts so if we let X equal 0 if we turn this into an equation we get 2 times 0 plus y equals 320 or equivalently just Y is 320 so we'll make that a little higher and then if we let y equals 0 we'll get 2x equals 320 divide both sides by 2 we'll get x equals 160 okay so not quite to scale so we'll graph so this is okay so this is the first line over here when we graph the second inequality it would look something like this and the region that satisfies both of these inequalities is going to be the area underneath so one part would say okay that would be the area or it's this the inequality the region we would shade underneath the first line if we look at the second line the region underneath would be that part and we're looking basically for the overlap so we're just going to keep the stuff in between so I'm even going to get rid of the other the other lines okay so now we have our little our region here all we have to do is basically figure out the corner points which really we already have done for the most part we know that this is going to be the point 160 comma 0 we know that the point up here is simply 0 comma 240 we've got the origin which is just 0 0 so the last thing we would have to do is figure out this other point and if we set the lines equal to each other so I'm going to solve for x and y here I'm basically going to turn my inequalities into equations so X plus y equals 240 2x plus y equals 320 and what I'm going to do is I'm going to actually rewrite the first one underneath so X plus y equals 240 and I'm going to use elimination by addition here to solve my system of inequalities or excuse me my system of equations now so multiply both sides of the first excuse me the second one by negative 1 we'll get 2x plus y equals 320 and the denominator you'll get negative x or not the denominator but the second equation negative X minus y equals negative 240 if we add these up the x's will get 1x the Y's will cancel 320 minus 240 is x equals 80 and once we know that x equals 80 we can plug that into either one of our equations to figure out the Y value so if we plug it into the first one or excuse me the second one we'll get 80 plus y equals 240 simply subtract 80 we'll get y equals 160 so x is 80 and Y is 160 okay now I have my 4 corner points okay so we've got the points 0 0 0 comma 2 40 80 comma 160 and last but not least we have 160 comma 0 okay so I'm just picking the corner points here and again all we have to do now is just simply plug those into our function C which is the biggest which is the smallest so let's see if we can't do this real quick so certainly if you plug in 0 your profits not going to be anything you'll get your profit of 0 so certainly that wouldn't make sense if we plug in 0 for x and 240 so we'll make it 244 why so okay so 40 times zero is certainly just equal to zero and then we'll have to add on 30 times 240 well 30 times 240 is going to give us 7200 or profit of seventy-two hundred dollars let's plug in our 80 and 160 so our profit will equal 40 times 80 plus 30 times 160 40 times 80 is thirty-two hundred thirty times 160 is going to be forty eight hundred if we add these together we get seven thousand and last but not least we can plug in our 160 value so it says our prophet will be 160 times 40 plus 30 times zero which is going to give us a profit of sixty-four hundred dollars so it looks like in this case it was actually certainly the smart thing to do kind of what we said was to plant all the oats because well you can basically for every acre of oats even though there's the time constraint you'll actually make twice as much money so so in case so in this problem our maximum the farmer if you're the farmer in this case should plant all oats and you'll make as much money as possible

Algebra Basics: What Are Functions? – Math Antics



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hi I'm Rob welcome to math antics in this algebra basics lesson we're going to learn about functions outside of the realm of math the word function simply refers to what something does but in math the word function has a more specific meaning in math a function is basically something that relates or connects one set to another set in a particular way a set is just a group or collection of things often it's a collection of numbers but it doesn't have to be a set could be a collection of other things like letters names or just about anything sets are sometimes shown visually like this but more often you'll see sets written using a common math notation where some or all of the members of the set are put inside curly brackets with commas between them like this a set can have a finite or an infinite number of elements for example a set containing all the letters of the alphabet has only 26 elements while a set of all integers has an infinite number of elements okay so a set is just a collection of things and a function relates one set to another but how exactly does it do that well to understand how functions work it will help if we start by naming the two sets the input set and the output set a function is something that takes each value from an input set and relates it or map's it to a value in an output set and you'll often hear these input and output sets referred to by special math names the input set is usually called the domain and the output set is usually called the range and it's really common to see some or all of the functions inputs and outputs listed in what we call a function table a function table normally has two columns one on the left for the input values and one on the right for the corresponding output values the function itself is often written above the function table and in the form of some sort of mathematical rule or procedure for example let's say that the input set of a function is a list of common polygon names like triangle square Pentagon hexagon and octagon the function itself could be a simple rule that says output the number of sites that means if we input triangle into the function the output will be 3 and if we input square the output will be 4 if we input Pentagon the output will be 5 and so on so this function simply relates the name of a polygon to its number of sides that's cool but most of the functions that you'll encounter in algebra will be a little more abstract than that they'll usually just relate one variable to another variable in the form of an equation like this one y equals 2x in this equation if we treat X as the set of numbers that we can input the domain and y as the set of numbers that we get as outputs the range what we have is a very simple algebraic function and just like the polygon example we can make a function table to show some of the possible input/output combinations for this function we can choose any number at all for the value of x but to keep things simple let's just try M putting 1 2 & 3 as values of X and see what outputs we get for our table if we input the value 1 in other words if we substitute the value 1 for the X in our equation then we get y equals 2 times 1 which simplifies to y equals 2 and since Y is our output variable we put a 2 in the output column next if we input the value 2 into our function we get y equals 2 times 2 which means y equals 4 so the output value is 4 and last if we input the value 3 into our function we get y equals 2 times 3 which means y equals 6 so the output value is 6 let's see the pattern for each input value the output value is twice as big which is what we would expect because the original equation says that Y the output is equal to 2 times X the input okay so we've seen some examples of functions that relate inputs to outputs but there's an important limitation about functions that we need to know to understand what that limitation is let's try to make a function table for the equation y squared equals x again the next variable in this equation will be our set of inputs and the y variable will be our set of outputs since Y is our output variable it will help if we first solve this equation for y and we do that by taking the square root of both sides but because of negative numbers we need to take both the positive and negative root of x since there are two possible solutions to our equation but won't that mess up our function table if we input an x value of 4 the positive or principal route would be to but we also have the negative route as a solution if x equals four then y equals two and y equals negative two are both possible solutions to the equation y squared equals x so in this case for each value of x that we have put into the equation we'll get two values of Y as outputs can a function do that you see functions aren't allowed to have what we call one-to-many relations where one particular input value could result in many different output values when too many relations certainly do exist as we can see from this example but we don't call them functions for something to be called a function it has to produce only one output value for each input value so a function doesn't just relate a set of inputs to a set of outputs a function relates a member of an input set to exactly one member of an output set the equation y equals to X qualifies as a function because no matter what number you put in you'll always get just one number as an output but the equation y squared equals x does not qualify as a function because a single input can produce more than one output let's look at another simple algebraic equation to see if it's a function y equals x plus 1 again the X values will be inputs the domain and the y values will be the outputs the range let's quickly generate a function table for a few possible input values like the integers negative 3 through positive 3 if you watched our last video about graphing on the coordinate plane you may notice that each row of this function table is basically just an ordered pair it's an x-value followed by a y-value we can even rewrite all the inputs and outputs an ordered pair form if we wanted to and that means you can also graph all of these pairs of inputs and outputs on the coordinate plane you can graph a function here are the points from our function table plot it on the coordinate plane and here's the resulting graph we get if we connect those points it forms a straight line and it's an example of what is called a linear function in algebra there are lots of different kinds of functions that have interesting graphs quadratic functions cubic functions trig functions and many more these graphs may look like just a bunch of squiggly lines but they're all functions and we can tell their functions just by looking at their graphs because they all pass the vertical line test remember how functions aren't allowed to have more than one output value for a particular input value well the vertical line test helps us see if a graph has any of those one-to-many relations that would disqualify it as a function here's how it works imagine that a vertical line is drawn on the same coordinate plane as the graph that you want to test then imagine moving that vertical line left and right across the domain paying close attention to the point where the vertical line intersects with the graph if that vertical line only intersects the graph at exactly one point for every possible value of x in the domain then that means that there's only one output value for each input value there's only one y value for each x value so the graph qualifies as a function okay so all of these graphs pass the vertical line test and our functions but what's an example of a graph that doesn't pass the vertical line test well here's one it's the graph of our equation y squared equals x the domain of this equation doesn't include any negative input values so there's some places where a vertical line wouldn't intersect the graph at all and that's okay and there's one place where the vertical line would intersect the graph at just one point which is also okay but as we move to the right on the x axis you can see that our vertical wind is now intersecting the curve in two places that means this equation has given us two possible outputs for some of its inputs which means that it's not considered a function ok now before we wrap up we need to talk briefly about some common function notation that can be pretty confusing the first time you see it in math books so far we've been writing functions like this y equals two x and y equals x plus 1 but you'll often see these exact same functions written like this instead but why why did the variable Y get replaced with that F parenthesis X thingy and what does that even mean well it turns out that a really common way to represent a function is this this notation simply means that a function named F takes an input value named X and gives an output value named why and you say it like this a function of X equals y or f of X equals y for short the problem with this notation is that you could easily misinterpret it as a variable f being multiplied implicitly by a variable X to give an answer of Y but that's not what this means in this case and is not the name of the variable and it's not being multiplied instead f is the name of a function it would be a lot more clear if mathematicians just use the entire word function as the name and then use the names input and output instead of x and y these two notations mean exactly the same thing but the first one uses an abbreviation for the function name and standard variable names for the input and output these are the most common names but you could use others if you wanted to okay so that's the basic notation but how did the equation get changed to f of X instead of Y well it comes from the idea that if two things are equal in math you can substitute one thing for the other since we've agreed on this general notation for a function f of x equals y that means that you can use f of X or Y interchangeably either one can represent the output set of a function but if they're interchangeable why would you use the more complicated f of X when you could just use Y instead well using f of X highlights the fact that you're dealing with a function with a specific input variable and not just an equation and it gives us a handy notation for evaluating functions for specific values for example you could start off by saying let the function f of X equal 3x plus 2 then you could ask someone to evaluate the function for the input value 4 by saying what is f of 4 that means you'll substitute a 4 in place of any X's that are in the function for this function that would mean F of 4 equals 14 and you could do this for other values to f of 5 equals 17 and f of 6 equals 20 pretty easy huh all right so that's what functions are in math there are things that relate an input value to exactly one output a and the set of all input values is called the domain well the set of all output values is usually called the range from algebra function typically come in the form of equations that can be graphed on the coordinate plane by treating the input and output values as ordered pairs of course there's a lot more to learn about functions but this basic introduction should help you get started working with them in algebra don't forget to practice using what you've learned in this video by doing some exercises as always thanks for watching math antics and I'll see you next time learn more at math antics com

Algebra – Ch. 3: Formula, Inequalities, Absolute Value (30 of 33) Inequal. w/ Abs. Values Ex 1



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In this video I will find and graph the solutions of x given |3x-5| “less than” 7.

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welcome to electro line here's a first example of how to solve inequalities that contain absolute value signs we have the quantity 3x minus 5 absolute value signs around that is less than 7 so here we have the example case like this where the thing between F of L you signs is less than the constant which then implies that that quantity inside the absolute value signs must be between negative and 5 and 5 with the inequality symbols pointing to the left so what we can then say is that negative 7i must therefore be smaller than 3x minus 5 which must then be smaller than a positive 7 and now we have it in a format that we can easily solve the way we solved it before the first thing we need to do is get rid of the negative 5 so we're going to add a positive 5 to all three parts of that inequality that means we end up at negative 7 plus 5 it's less than 3x minus 5 plus 5 which then in turn is less than 7 plus 5 so hopefully see here's we're simply going to add plus 5 everywhere to get rid of the minus 5 in the middle and then we get minus 2 is less than 3x which is less than 12 now the next step is divide everything by 3 to get rid of the numerical coefficient in front of X so we divide by 3 divided by 3 divided by 3 and remember since we're not dividing by negative number we don't have to change the direction of the inequality signs and so finally we have negative 2/3 is less than X which is less than 4 and then if want to show what this looks like graphically we have a negative 2/3 we have 0 we have 2 4 over here notice that the endpoints are not included and we realize it must be the value all in-between like this and so there we have the graphical solution for our initial problem now if you want to check to see if we did it correctly let's put in an arbitrary value let's go ahead and check when we let X is equal to seven so we're right led because it's not equal to zero not seven zero so let X equal zero we're going to do a check like this we plug that back in the original equation so or inequality three times zero minus five take the absolute value that should be less than seven notice three times 0 0 the absolute value of -5 is 5 and 5 will be less than 7 so that checks so that means that this is most likely the correct answer and that is how it's done

Algebra – Ch. 4: Exponents & Scientific Notation (2 of 35) The Rules of Exponents: Summary



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In this video I will summarize 15 rules of exponents.

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welkom twee lecteur online nou wonden most challenging hardwick spannends zat tussen ferry number rules and sometimes je forget to choose which or you not sure what will try me zo'n situatie en dat soort mix het hart zowel begon te duur views hoe ontwikkel ik het vers de sanrio all the rules trek het denken met de mannen en den bond x-plane each one seperately wonnen de time and then at some point logan de show you some examples for wie op de single barrel clubpie to simplify het artikelen kan een probleem worden artikel account of expression nou citroensap we het exponent sometimes i get to december zult het volgende deel van harde vloeren devonshire rol zo is not necessary to volwassen stud way of doing het oor simpel van tanks biljetten dat leder bijna alle just simply sam bryce al de vs wil zeide lief poesje be familiar to defer school is er ivf extrude en times teksten en de muziek op de echte die en plas en bronnen word je voor motor ply-en de bases er dus een ben wie andyax monnens dat wonder we one dat spel de moslim per rol voor exponent and consequently is voor de white one by the other ik studeer aan de final bij fcm dan is het raak ik spannend om een mantelpluim heerlijk spannend want we divine wisseldrank het spoor mensen gaan we must make sure dat de bases dus een minuut kees den de turbo zoals gram per mol is al voor jouw experiment power en de holding re studie henpower zo we hebben een exponent nam een exponentieel de speak bamboe multiplied ik spannend echter de en times en de first real sorry wie de foundation of ondersteuning exponent dan dus de number for this en rol ze kan een wheelie handy guide sample we hebben x meest door een exponent en de sikkel toch man over extra padrick spannend or one over extra magic spannen risico teksten de pauze exponent soort al is hij niet een oh dat is wel u kent de simpel vijf tanks om me heftig something that were using any number meeste deserve our physical te lang en zo deze mather eens even teksten desmarets 3 minus 5 momenten nummer is wees er dus europower je zult won leer one will take you can see what she wrote a girl power is petazzie special event dan beseffen waarom hebt u numbers mond de put together we hebben die haar ex en by and together dwdd en powernode dat de exponent de student zelf over each of the factors in site nodig dat is must be multiplied na het hebben al deze geleden toen mijn motto time riding tanks name bij engelse tracking tim heeft u very careful to show you some examples leren weddings dan word ik voor nat multiplier dividing keerwanden divining apps by my en de holding is b student houwer ik kende en de student zelf over boten ex en die en benoemde nummer in de noemer en de number in de de naam en ether wat interest in curry dat kennis die is tools combined je via ex over wij wisten we negatieve exponent dat sikkel toe die inversie dat op en afrika tout compris daad resoluut pauze reeks ponens zal what can i make sure the dutch private zo nodig dat we can gather de negatief edx plannen bij simpele flipping de fraction teken in brussel de fractie rondom het en extra man over en power risico studie en troep op dat nummer apps whatever access and fresh ekster de en over enpower tenminste noemen te de naam leder dat kan bieden roet en de naam in de noemer is konden bieden exponent of de nummer insider medical zak studie en over en is die cultuur en group of teksten die en invloed hebben negatief nummer een front dat we extra for power this does not apply to do not give saai de simpel meer negatief de kronen die extra for power een soort negatief eckstein seks camsex camsex voor factors noden zit dus na een prive en de negatief saai if you want to brighten up sign me niet de papier indices van de apple talk about the tail zodat leder then we konden talk about anti fog notation waar we konden zien de nummer henry's toen ik spannende la vernauwen sufficient to realize hebben de nummert en resem exponent dat nummer tab simpel in de kind en numbers euro's behine de want zo tenten de fit power is 100.000 is de one big five girls maxwell is van begin een tube vreselijks bonus ex te zeuren plant voor wat is army of complete look getijdenstuk en word uit in twee fraction zo point for the buttons for over ten wickham het in de stuw over vijf en demy de zeeuw brewdog us open er is de vuur of exterus en power and if something like this extra one point for power dan mee op de realize a one point for this one plus point for en dan me take this role in bewust extra en plus en risico de ekster in time sekse en zo moeilijk over hier extra oneplus point for legal de ekster one-time sectoren point for and then of course extra point for can read and this extratone vetste slag we deden voor hier en deze cirkel te dus één ding de folie wax koord dan ex times that simply means ex times the radical de visbrug of ik square zozo de basic rules in expan een exponent snow zusters 15 of de pixel had toekennen membranes doos de winkel traditionele time explain why did you we they duur en dan de truc is dan richting een warboel to buy one i kept their had de simpel visonic specials wit exponent all over the place will get a dat hij was show je had u die je foto's kan een paar sommen get there this is how it's done

Algebra – Ch. 4: Exponents & Scientific Notation (3 of 35) Exponent Rule 1



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In this video I will explain work out examples of Rule 1 of exponents: If the bases are the same and they are MULTIPLED, then ADD the exponents.

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welcome to Elektra online the most important rule in exponents is the rule over here rule number one where we have X to the nth power times X to the M power notice the bases are the same which means we're going to add the exponents but we have to be very careful because along with this rule come among the most common mistakes that we end up making when we're not careful and here are three probably the most common three mistakes we make using exponents notice that if instead we have a multiplication we have an addition we cannot use that rule you cannot say that X to the N plus X to the M is equal to X to the n plus M that is not the case because it's an addition instead of a multiplication for that rule to be able to be exercised and used it must be X to the M times X am it must be multiplication the next mistake that we make often is when we have X to the N times y to the M that is not equal to x times y to the n plus M because here the bases are not the same remember those are the rules in order for us to be able to use that rule the basis must be the same and we must multiply the two key items in that to make that rule work otherwise you cannot use that rule and thirdly something we'll get into a little bit more when we have the combination of X plus y raised to the nth power that is not equal to X to the N plus y to the N again a very common mistake you can again look at it because the addition here we cannot use that rule either and we'll talk about that rule some more later three very common stakes try not to make those here we have some more examples for example X to the third power times X to the fourth power is equal to X to the three plus four which adds up to seven so it's equal to X to the 7 power again the base are the same and we're multiplying so we're able to use that rule here we have two to the second power times two to the fourth power which is two to the two plus four which is two to the sixth power basically two multiplied by itself six times there are six factors of two when you multiply them together you get 64 another way of looking at that's very same problem is you can say well 2 to the second that's two times two and two to the fourth power that's two times two times two times two four factors so that's 2 times 2 which is 4 times 2 times 2 times 2 times 2 which is 16 4 times 16 is 64 you get the very same result obviously for the rule to work you should get the same result but notice you can look at the very same problem in two different ways and it's exactly the same thing what if you have three numbers multiplied together each with exponents like the number a it can represent any number and notice we have a times a times a but it's raised to the fourth power to the second power to the fifth power the rule is the same when we have more than two bases when you multiply you can simply add all the exponents 4 plus 2 plus 5 which is 11 so this is equal to 8 to 11th and finally what if we add or 1 if we multiply do bases together and one of the exponents is negative doesn't matter we still employed the very same rule Y to the fourth power times y to the negative 2 power is equal to Y to the 4 plus a negative 2 4 plus an 82 is 2 so this becomes y squared so Y to the 2nd power we also say Y squared y to the third power we say Y cubed 2 those are some of the easy ways of expressing an exponent when it's a 2 or 3 anyway that's the major rule the first rule in exponents really understand what you're allowed to do and what you're not allowed to do with that rule and that way you prevent from making those mistakes and that is how it's done

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.

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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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TRANSCRIPT

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.

Mathematical Economics | গাণিতিক অর্থনীতি | আয় ব্যয় ও উৎপাদন



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Mathematical Economics | গাণিতিক অর্থনীতি | আয় ব্যয় ও উৎপাদন

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বীজগণিত ৪.১ অধ্যায়ের ৪, ৬, ৮ নম্বর অংক

Algebra – Understanding Quadratic Equations



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Visit for more math and science lectures!

In this lecture I will take the mystery out of the quadratic equation and help you understand what it represents.

today we're going to take the mystery out of quadratic equations so the lecture today says understanding quadratic equation so what are quadratic equations well what they are is a relationship between two variables typically Y and X and it typically will look something like this y equals ax squared plus BX plus C now that's the standard form as we call it a B and C are simple constants any sort of numbers X is the independent variable and then Y is the dependent variable so as an example let's write an equation such as y equals let's say x squared minus 6x plus 5 okay in this case a would equal 1 that's the coefficient from the x squared term B is equal to minus 6 and C is equal to 5 all right now one of the things we want to be able to do with one of these quadratic equations is learn how to graph it another one is learn how to find the roots but before we get into all that let's kind of get a general feeling of what acuatic equation is it's again a relationship between x and y so we should be able to graph it on an XY plane so there's our y axis there's our x axis and typically a chaotic equation will be graphed either it will look like this or it will look like that and the way you can tell if it's going to open upward or open downward as we call it is if the number in front the x squared term is positive it'd look like this that's a plus a and if the number if found the x squared term is negative then it looks like this so think of it as a minus 8 all right that helps us a little bit and I also realize that if you graph these that this parabola as we call them these graphs for correct equations called parabolas and it could potentially for it to graph it it could end up looking like this or can end up looking like this or can up looking like that or maybe it looks like this and notice that in some cases the parabola will cross the x-axis like this particular parabola will cross it in two places this particular parabola will cross it in two places and these two parabolas do not cross the x-axis at all the places where one of these parabolas which is a graphical representation of a quadratic equation like this where the parabola crosses the x axis those are called the roots of the quadratic equation so we call these roots and here again these would be considered the roots of the quadratic equation notice that these two do not have roots so sometimes when they ask you to solve a quadratic equation they're basically asking you find the roots of the parabola and of course if the parabola does not cross the x-axis like again this is here the x axis this here is the y axis like in this case or in this case then there's no solution there are no roots to the quadratic equation all right so to help us figure out what it actually is and how to graph these quadratic equations let's go ahead and factor this this happens to be a factorable equation so we can write this as y is equal to the product of two binomials we write the X and the X since there's a negative here and a positive there that means one must be plus and one must be negative or no in that case I'm sorry actually they both must be negative maybe because the only way you can get a positive number there is either that you have two positive numbers here are two negative numbers and since this is negative you have to have two negative numbers so now we're looking for two numbers when they you multiply them you get a 5 when you add them you get a 6 so looks like a 5 and a 1 because if I multiply x times the negative 1 and multiply negative 5 times a an X and add them together I get negative 6x my middle term alright now if we want to solve a quadratic equation to find the roots realize that the roots are the points on on the x-axis that means that that location the y-value is equal to zero so any point on the x-axis my y-value is zero which means if I'm going to find the roots I'm going to set my y equal to zero if I do that this equation now becomes zero is equal to X minus 5 times X minus 1 and if I solve this for X then I will find the points where the equation crosses the x-axis all right now in fact two quantities multiplied together and the solution is zero that means either one or the other must be 0 because the only way you can multiply two things together and get 0 is if either X minus 5 equals 0 or the X minus 1 equals 0 and of course if x equals X minus 5 equals 0 then X must equal 5 or if X minus 1 equals 0 then X must equal 1 and those are the locations in this particular example where the parabola will cross the x-axis so if I'm going to graph what I have here my example if I'm going to graph this example on my XY axis here I know that one of the roots or one of the points where the the graph will cross the x axis x equals 5 1 2 3 4 5 so right there and the other place where it crosses the x axis will be at x equals 1 which is right here and those are considered the two roots of my chorionic equation all right now there's another thing about a quadratic equation that's very important it's called the axis of symmetry if I find the midway point between those two numbers so this is 1 2 3 4 5 and so the midway between 1 and 5 is the number 3 if I now draw a dashed line vertical line through the point x equals 3 on the x axis that is now called the axis of symmetry and again if you look at my examples over here if you draw a line halfway between here and halfway between right there and right in the middle there right in the middle there right in the middle there and right in the middle there notice those lines those dashed lines are the exact line that divides the parabola in two equal parts therefore that's called the axis of symmetry now notice that the number from the x squared term is positive it's a positive 1 that means the parabola opens upward that means I'm going to have a problem that looks kind of like this to find a few more details if I now plug in to my equation the value of x where the axis of symmetry goes right through in this case the number 3 so I'm going to solve for my equation when x equals 3 that's again the point where the axis is similarly goes right through let's see what I get so I'm taking my equation and instead of X I'm going to write a 3 so I get a 3 squared minus 6 times 3 plus 5 so notice I took my original example instead of an X I write at 3 so this xri 2 3 and if I work that out I get 9 minus 6 times 3 is 18 plus 5 and so that's a plus 14 minus 18 that's equal to minus 4 so that means when x equals 3 my Y will be negative 4 so 1 2 3 4 that's negative 4 right here so x equals 3 y equals negative 3 that is the bottom or top point of my parabola so if my parabola ups upward the lowest point of my parabola is this point right there if I my parabola opens downward then my lowest or in this case my highest point right there will be that point right there so in this case since a is a positive number my parabola opens upward so this will be my lowest point on my parabola also known as the vertex so the vertex of my parabola now there's one more special point about the parabola sometimes the parabola will cross the y axis like over here in this case it doesn't look like it but if it goes on if you keep going long enough again eventually you will cross the y axis right here this parabola crosses the y axis over here and if the probability goes on long enough eventually you can see way down here somewhere the parabola will so cross the y axis how do we find that point well remember anytime you cross the y axis that means that the x value must equal 0 so to find that particular point we take our initial equation again and plug in 0 for X to see where the equation or where the parabola crosses the y axis so I'm going to solve for y when x is equal to 0 again I take my equation right there plug in a 0 for every X that I find so I get 0 squared minus 6 times 0 plus 5 or y when x equals 0 is equal to 5 so going to my example here my graph when x equals 0 my Y value should be 5 so it's 1 2 3 4 5 and so my problem across that point as well now notice I have four points I know my equation can be graphed like a parabola here's my lowest point or the vertex there's my two roots there's the point where the parabola crosses the y axis if I now carefully connect all those dots with a free hand like that I have now drawn a graphical representation of my example right here my quadratic equation and you can see how that looks like a nice parabola so that's what a parabola is that's what a quadratic equation represents and if you want to look at the points very carefully again notice you have the lowest point on your parabola called the vertex you have the two points where the parabola will cross the x axis those are also known as the roots so that's a route that's a route right there also notice that most rabbits will cross the y-axis and when they do that's the point you can find you can find that point by plugging in 0 for X in your equation and then of course the axis of symmetry runs right midpoint between the two roots or also goes right through the vertex so there you get a pretty good feel for what a parabola is and for what a quadratic equation is alright now we're going to show you some examples of actual how to solve these quadratic equations and how to graph them in a little more systematic fashion

Mathematical Economics | বাজার ভারসাম্য | Part 1



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Mathematical Economics | বাজার ভারসাম্য | Part 1

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Economics । অর্থনীতি | প্রচুরক নির্ণয়
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বীজগণিত ৪.১ অধ্যায়ের 13, 15 নম্বর অংক.

সাধারণ জ্ঞান ভাষা আন্দোলন | সাধারণ জ্ঞান

(Cell & structure of cell | কোষ ও কোষের গঠন)

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

Most US College Students Cannot Solve This Basic Math Problem. The Working Together Riddle



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To complete a job, it takes: Alice and Bob 2 hours, Alice and Charlie 3 hours, and Bob and Charlie 4 hours. How long will the job take if all 3 work together?

Mathematical Economics | জাতীয় আয় | National Income | Part 1



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Mathematical Economics | জাতীয় আয় | National Income | Part 1

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বীজগণিত ৪.১ অধ্যায়ের ৪, ৬, ৮ নম্বর অংক

The Map of Mathematics



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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.

A Brief History of Pi



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Note that there’s a huge amount about pi that I didn’t cover in this video due to time – I didn’t even mention proofs of it being irrational and transcendental, or why we call it pi! I chose to focus on the development of its approximation as a hook to teach the broader history of mathematics, rather than make this video an exhaustive list of facts. The wiki is a great place to learn more about the rest of the number’s history and applications in maths and physics:

A few nitpicky things:
– I made mistakes distinguishing between ‘digits of pi calculated’ and ‘decimal points of pi calculated’ in some sections, so it is possible that this error is elsewhere without me knowing.
– Archimedes didn’t do his approximation with squares, he started with hexagons and then increased the order of the polygon. I chose to present the zeroth order version of his algorithm using squares for simplicity, but note that this is not what he did.
– Something which got lost from the final version of this video is my argument that during the Age of Enlightenment pi shifted from being a physical (measured) constant to a purely logical (theoretical) one. This then embodies the philosophical shift in society at the time. This is hinted at but not fully explained, so I thought I’d put this here.
– Lastly, I am truly sorry for the pronunciations which I doubtlessly completely murdered in this video. At least I spared you my attempt at Chinese.

I am hugely indebted to Alex Bellos and his excellent book Alex’s Adventures in Numberland for the inspiration to make this video. There is an entire chapter of the book devoted to a broader but shallower discussion of pi and its history, which I highly recommend.

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this video is brought to you by Squarespace whether you need a domain website or online store make it with Squarespace take any circle measure its circumference and its diameter the ratio of these two numbers is a mathematical constant we call pi while this definition is simple pi has been studied for thousands of years and history of our understanding not just of the value of pi but also what it means forms a history of all of mathematics it takes us from the Middle East to Europe to China to India and even America it's a history which involves revolutions murder and the infinite maths is as old as civilization older even there's evidence of counting going back thirty thousand years and two of the very earliest civilizations the ancient Egyptians and Babylonians both investigated pi around 4000 years ago the Babylonians estimated PI to be 3 and 1/8 or 3.125 now that's the first of a few estimates you're gonna hear in this video so for reference remember that the first few digits of pi are 3.1415926 there are more that means that the Babylonian estimate of Pi is accurate to 1% of its true value which is kind of astonishing when you remember that this is a time in human history when iron was first being used and the last mammoths went extinct the ancient Egyptians on the other hand estimated PI slightly less accurately as 3.16 but how do you even estimate the value of pi you have to count it by definition measure a curved surface which is super tricky to do accurately well one way of doing it is to cheat and actually use a square compare a square and a circle well it's quite a little bit like a circle but that was much like a circle as a Pentagon which has one more side than a square and a Pentagon doesn't look quite as much like a circle as a hexagon which has one more side again and a hexagon doesn't look quite as much like a circle as a heptagon and so on you can think of a circle as a regular polygon just want with an extremely large number of sides so many sides in fact that each individual one is infinitesimally small meaning that the circle looks round this was exactly the thinking that legendary ancient Greek mathematician Archimedes used when estimating pi around 220 BC in fact it was probably the very last thing he ever did to approximate pi he reasoned why not measure the perimeter of a square adding up the lengths of all of its edges and then dividing that number by the squares diameter but what is the diameter of a square is it the length of its diagonal or the length of one of its edges why not both said Archimedes draw one square with its corners just touching the perimeter of a circle another square with its faces just touching the perimeter of that same circle add up the lengths of the sides of each square divided by their effective diameters and you have two estimates for the value of pi the true value of which lie somewhere between those two numbers but here's the really clever part because the difference between those two values is pretty big if you're using squares because a square isn't much like a circle but replace those squares with Pentagon's and you shrink the difference between those two numbers meaning that there's a smaller range of values that PI could be your estimation just got more accurate and if you replace those Pentagon's with hexagons you'll get an even more accurate estimates keep increasing the number of faces on the shape that you're drawing inside and outside the circle and your estimate will get more and more accurate as long as you have the time and patience to draw said shapes there is a reason why this thing was called the method of exhaustion Archimedes got up to a 96 sided shape which incidentally is called an a neocon Turki hexagon I really hope I said that right giving an estimate of Pi between three point one four zero eight and three point one four to nine so accurate to two decimal places as I mentioned earlier this was likely his final contribution to science because into 1/2 BC he was killed by Roman soldiers who invaded his hometown Zaira Q's he was apparently performing this calculation at the time allegedly his final words were don't disturb my circles European progress in the study of pi died with Archimedes for well over a thousand years fortunately however there was plenty of the world which was not in Europe a mathematicians here were also interested in PI in particular three mathematical superpowers of the first millennium ad were China India and Persia ideas when these three nations were soon to change the world first off Chinese mathematicians used a method of exhaustion similar to our comedians but instead of considering the parameters of shapes they considered their areas and this dude no I'm not going to try and pronounce his name because I'll only get it wrong used to polygon with 3072 sides to obtain pi to five decimal places 200 years later a father-and-son team used a polygon with over 12,000 sides to extend that record to six decimal places and that was a world record which stood for 800 years the problem was it was just difficult to do these calculations they weren't especially hard to understand it was just awkward to write down what you were doing to physically do the calculation and this was something that would only be resolved by the introduction of two world changing ideas from India and Persia say that you want to do a calculation you know that you and your friends together weigh a hundred and twenty-five kilos and you also know that you weigh 70 kilos the question is how much does your friend weigh mathematically we'd write this as X plus 70 equals 125 where X is your friends weight in kilos subtract 17 from both sides and you get the answer 55 kilos now in that simple example I just used two ideas which were revolutionary to the classical world firstly I wrote large numbers like 125 and 70 using a simple notation we take it for granted these days but the ability to write any number using just ten symbols and a place value notation where the position of a symbol in a number determines its size massively simplifies arithmetic to see what I mean try and do that calculation only using Roman numerals our modern decimal notation was first developed in India some time before 400 AD and then rapidly spread to Persia where the second key idea came from the second key idea was representing your friend's weight using some symbol X and then manipulating both sides of the equation this of course is algebra originally developed by Babylonian and ancient mathematicians but truly established by Persian mathematician and all-round very influential dude Mohammed eben Musa al-khwarizmi using decimal notation and algebra allowed for much easier calculations across all of maths and mathematicians working on calculating PI used it to turbocharge their work after the Renaissance and a renewed interest in mathematics along with crucially new tools from the east Europe was back in the game and in 1630 the most accurate estimate of Pi using the polygon method was achieved by Austrian astronomer Christiaan grind Berger who used a shape with 10 to the 40 sides yes really to calculate pi to 38 decimal places and then because mathematicians are sensible people with lives to lead they decided that was accurate enough and they'd leave it there oh wait the adoption of algebra by European mathematicians triggered a whole new way of looking at the world a change in thinking generally grouped under the title the Scientific Revolution which itself went on to inspire the Age of Enlightenment with thinkers like Rene Descartes and John Locke amongst other ideas the Enlightenment movement emphasized the value of Reason over tradition and new mathematical ideas were held up as Paragons of this they were pure reason the change in how 17th century European mathematicians calculated PI is arguably a perfect example of the shift from following what the ancients did to new rational theoretical approaches because while the ancients like Archimedes may have measured the perimeters of shapes increasingly similar to circles now European mathematicians were using a method based entirely on reason a method based on infinite series an infinite series is just an expression made up of things added together one after the other after the other after the other and so on until forever if those contributions keep getting smaller as you go on then the series converges to a particular value sometimes you can work out what that value will be using logical arguments but sometimes you just have to keep calculating term after term after term until you reach an accuracy that you're happy with the method of using infinite series to calculate pi was first used not in Europe but again in India you could kind of argue that what Archimedes did was an infinite series but the first person to write a mathematical function as an infinite series was Indian mathematician math hava of Sangamo grammar in the 14th century he wrote down expressions for the sine cosine and tangent of an angle as well as the inverse tangent quick refresher if you write the expression y equals tan of X the expansion for the tangent would tell you what y equals if you already know what X is while the expansion of the inverse tangent would tell you what X is if you already know what Y is by its definition the function tan of X precisely equals 1 when x equals 1/4 pi that means that if you have an expression for the inverse tangent then if you plug 1 into that expression and keep calculating terms you'll end up with an increasingly accurate estimate of 1/4 pi madhava did this and calculated PI to 11 digits but then his method seems to have been forgotten only to be apparently independently rediscovered in 17th century Europe by Scott James Gregory and German Gottfried Wilhelm lightness and at this point everything kicked off the new decimal notation and algebraic technique allowed for record calculations of Pi in 1699 it was calculated to 271 digits by abraham sharp who was beaten in 1706 when John machen reached a hundred digits who was in turn beaten by thomas von tete de l'année I hope that's how you say his name in 1719 with 112 digits it wasn't just the case that each of those mathematicians had more spare time than the previous one they were competing with each other using different infinite series which converged on PI faster instead of just using the inverse tangent infinite series they might use a combination of different inverse tangent values or something completely different the competition then became less about which mathematician had done the most calculations and instead which mathematician had the fastest converging infinite series development of increasingly efficient infinite series continued well into the 20th century with the technique kind of coming full circle as the current infinite series of choice was developed by Indian prodigy mathematician Srinivasa Ramanujan of course by the 20th century mechanical computers had been invented making it much easier to calculate pi you basically just used one until he got bored in 1949 Americans D F Ferguson and John wrench calculated PI to 1120 digits but they were bringing a knife to a gunfight because that very same year the first calculation of Pi by an electronic computer was done nearly doubling their record with two thousand and thirty seven digits from here the history of Pi is basically a list of increasingly powerful computers running for a long time and spitting out increasingly absurd numbers of digits at the time of recording the world record for digits of pi calculated is held by peter trib with a shade under twenty two and a half trillion digits calculated the question of course is if we know that pi is going to keep going on forever it's a transcendental number why should anybody bother calculating anymore dishes well for one thing calculating pi is actually a really good way of making sure that your brand new shiny computer is working properly calculating pi uses up a lot of mental brainpower for the computer you have an answer that you can check yours against and also if you keep going just a little bit longer than the previous person you can have a casual world record secondly pi is actually a really good random number generator if you look at the first two hundred billion digits of pi you'll find the number zero occurs almost precisely 20 billion times and the same goes for the other digits 1 through 9 that means that if you were to pick a random digit in those 200 billion there's an almost exactly 10 percent chance of it being one under almost exactly 10% chance being to and so on this makes calculating PI to a large number of digits very valuable to people that want to generate random numbers people working in cryptography for example but lastly and arguably most importantly people keep calculating more digits of pi for the same reason that why people memorize tens of thousands of digits of pi and the same reason why people climb mountains and swim oceans and invent the double luge because they can humans are weird we like to understand the world around us and as our civilization has developed we've built increasingly complex tools to help us understand the world it wasn't essential for our survival that we did that we just did it because of the way we're wired because we could pie is a thread that's gone through all of human history because it's a microcosm of how we interact with the natural world from the ancients to present day through revolutions in Thor and across the world as long as there are people there's always going to be somebody who just wanders what's the next digit long may that continue I'd like to close out this video with two quick announcements announcement the first I finally launched a website go to the very shiny and new Simon ox fist calm for a hub for everything I do online including this YouTube channel sponge in electric and the wiki cast there's a page there detailing the gear that I used to make these videos and coming soon there's also going to be a page detailing all the advice that I give out to people who are interested in applying to the universities of Oxford and Cambridge and then announcement number two if you'd like a look at my website and why wouldn't you I built it with Squarespace who very kindly sponsored this video before making this website I didn't have any experience with web development at all but it was super easy guys so easy so quick to use one of their templates customize it to look the way I want and then just fill in my stuff and then bang it's done I never have to patch or upgrade or install anything it's all taken care of brilliant also it was super easy to set up for selling merchandise which um may be happening soon you you'd like to build your own website for your next project then definitely use Squarespace for it go to Squarespace comm for a free trial and when you're ready to hit the big red button and build the website go to Squarespace com forward slash Simon Clark to get 10% off your first purchase thanks to Squarespace for sponsoring this video thank you for watching it if you enjoyed it please do give it a share pop it alike maybe leave a comment and thank you again I'll see you next time