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Join us for #YourDailyEquation with Brian Greene. Every Mon – Fri at 3pm EDT, Brian Greene will offer brief and breezy discussions of pivotal equations. Even if your math is a bit rusty, tune in for accessible and exciting stories of nature and numbers that will allow you to see the universe in a new way.
Episode 11 #YourDailyEquation: Euler’s Identity is widely viewed as the most beautiful of all mathematical equations, combining a handful of disparate fundamental quantities into a single mathematical formula. In this episode of Your Daily Equation, Brian derives the identity, one that is second to none in capturing the poetry of mathematics.
Brian Greene is a professor of physics and mathematics at Columbia University, and is recognized for a number of groundbreaking discoveries in his field of superstring theory. His books, The Elegant Universe, The Fabric of the Cosmos, and The Hidden Reality, have collectively spent 65 weeks on The New York Times bestseller list.Read More
Hey everyone, welcome to Your Daily Equation. Hope you’ve had a good day, that you’re feeling okay. I’ve had a pretty good day today. I’ve been working actually on an article for the New York Times on, of all subjects, the question, why art matters? And yeah, obviously from the perspective of a physicist, mathematician, not someone who is an artist. But it’s kind of fortuitous because the equation that I want to talk about today is often described, and I certainly would describe it this way, as one of the most beautiful or perhaps the most beautiful of all mathematical equations. And so this idea of art and aesthetics and beauty and elegance, it kind of all comes together in this mathematical formula, which makes it quite an appealing subject to write about, to think about, and also a wonderful little encapsulation of really what we physicists, what mathematicians mean when they talk about beauty in mathematics.
As you’ll see in the equation when we get to it, it just puts together in such a compact, elegant, economical equation, different aspects of the mathematical world and tying disparate things together into a novel pattern, a beautiful pattern, a pattern that just fills you with wonder when you look at it is what we mean when we talk about the beauty of mathematics. So let’s jump into the equation. And for this one I’ll need to do a lot of writing. So let me immediately just bring my iPad up to here and let me bring this up on to the screen. Okay, good.
All right, so the formula that I’m going to be talking about, it is known as Euler’s formula or often Euler’s identity. In that we have this guy Euler in the title here. Let me actually just say a couple words about him. I could show you an image, but it’s kind of even more fun. Let me just swap right back over here. So these images, clearly they’re stamps, right? So this is a stamp from the Soviet Union from, I guess it’s the mid 1950s. I think it was the 250th birthday of Euler. And then we see this picture as well. This other stamp from, I think it’s from Germany on the 200th anniversary of, may have been the death of Euler. So clearly he’s a big deal if he’s on stamps in Russia and in Germany. So who is he? So Leonard Euler was a Swiss mathematician who lived in the 1700s, and he was one of those grand thinkers that even mathematicians and other scientists would look to as the epitome of mathematical achievements. Sort of the epitome of creative thought in the mathematical sciences.
I don’t know the exact number, but he was so prolific. Euler left behind something like, I don’t know, 90 or 100 volumes of mathematical insight. And there’s a quote, I’ll probably get this wrong, but I think it was LaPlace again, one of the great thinkers, who would tell people that you had to read Euler if you really want to know what mathematics was about, because Euler was the master mathematician. And that’s coming from the perspective of somebody else who was a master mathematician, a master physicist. So let’s get to this formula here and let me bring my iPad back up. It’s not coming up. Okay, now it’s back up. All right, good. Okay, so to get there … And look. In deriving this beautiful little formula, there are many ways to go about it. And the route that you follow depends upon the background that you have, sort of where you are in your educational process.
And look, there’s so many different people who watch this that I don’t know the best way in for any of you. So I’m going to take one approach. It’s going to assume a little knowledge of calculus, but I’ll kind of try to motivate at least the parts that I can motivate. And the other ingredients, if you aren’t familiar with them, either just let it wash over you and just enjoy the beauty of the symbols or perhaps use the discussion that we’re having as motivation to fill in some of the details. And look, if I was to do an infinite number of these Your Daily Equations, we’d cover everything. I can’t, so I have to sort of start somewhere.
Where am I going to begin is a famous little theorem that you learn when you take calculus, which is known as Taylor’s theorem. And how does this go? It goes as follows. It says, look, if you have some function, let me give it a name, have some function called F of X. Right? And Taylor’s theorem is a way of expressing F of X in terms of the value of the function at say a nearby point that I’m going to call X sub zero, nearby to X. You express it in terms of the value of the function at that nearby location. Now it won’t be an exact equality because X can differ from X not.
So how do you capture the difference in the value of the function at those two distinct locations? Well, Taylor tells us that you can get at the answer if you know some calculus by looking at the derivative of the function evaluated at X not, times the difference between X and X not. That won’t be the exact answer in general. Rather, Taylor says you have to go to the second derivative evaluated at X not times X minus X not squared. And this one you have to divide through by two factorial. And just to make it all look kind of uniform, I can divide this one by one factorial if I’d like. And you just keep on going. You go to the third derivative at X not times X minus X not cubed, over three factorial and on it goes. And if you’re being careful about this, you have to worry about the convergence of this series that I’ve written, which in principle would go on to infinity. I’m not going to worry about those sort of important details. I’m just going to assume that everything will work and the subtleties will not come and sort of bite us in a way that will invalidate any of the analysis that we’re carrying out.
Okay, so what I’d like to do now is take this general formula, which in principle applies for any function that’s appropriately behave, that it can be differentiated arbitrarily many times, and we’re going to apply it to two familiar functions, which is cosine of X and sine of X. And again, I know that if you don’t know what sine and cosine are, then you’re probably not going to be able to follow everything that I’m talking about. But just to kind of have everything written down in a complete looking manner, let me just remind you that if I have a nice triangle like this, it really needs to meet up there at the top. And let’s say this angle is X and let’s say this hypotenuse here is equal to one, then cosine X will be the length of that horizontal side and sine X will be the length of that vertical side.
So that’s what we mean by cosine and sine. And if you take a course in calculus and learn some of the details, you will learn know that the derivative of cosine X with respect to X is equal to the minus sine of X, and the derivative of sine of X with respect to X is equal to cosine of X. And that’s nice because with that knowledge we can now go back up here to Taylor’s theorem and we can apply it to cosine and sine. So why don’t we do that? Let me change colors here so we can make this pop out a little bit more. So let’s look at cosine of X and let’s choose X not, the nearby location, to be the value zero. So that will just be most useful. That special case will be most useful to us. So just plugging away into Taylor’s theorem, we should look at cosine of zero, which is equal to one when this angle X is equal to zero. You see that the horizontal part of the triangle will exactly equal the hypotenuse. So it will be equal to one.
And now let’s keep on going, but to avoid writing down things that will vanish, notice that since the derivative of cosine to sine and sine of zero up here is equal to zero, that first order term will vanish. I’m not even going to bother writing it. Instead I’m going to go right over to the second order term and if the first root of cosine of sine, then derivative of sine will give us the second order term which will, if I include the sine, will be minus cosine. And cosine of zero is equal to one. So the coefficient that we have over here will just be minus one over two factorial. And upstairs, in fact, let me even just put it immediately upstairs. Upstairs I will have X squared and again if I go then to the third order term, I will have a sine coming in from the derivative of the cosine from the second order term. Evaluated at zero will give us zero so that term will go away. I’ll have to go to the fourth order term and if I do that again, the coefficient will be equal to one. I’ll get X to the fourth over four factorial and on it will go.
So I only get these even powers in the expansion and the coefficients just come from the even factorials. Okay, so that’s cool. That’s for cosine. Let me do the same thing for sine X. And again it’s a matter of just plugging in. Same kind of thing. In this particular case when I’m expanding my X not equal to zero, the first order term will give us sine of zero which is zero so it drops out. So I have to go to this guy over here. The zero-th order term, I should say, drops out. So go to the first order term. The derivative in this case will give me cosine. Evaluating that at zero gives me a coefficient of one so I will just get X for my first term. Similarly, I will skip the next term because its derivative will give me the term that vantages at zero, so I have to go onto the third order term and if I do that and I keep track of the sines, I will get minus X cubed over three factorial. Then the next term will drop out by the same reasoning and I get X to the fifth over five factorial. So you see that the sine, and that’s of course a one there implicitly, the sine gets the odd exponentials and the cosine gets the even one.
So it’s very nice. A very simple Taylor series expansion for sine and cosine. Fantastic. Now keep those results in the back of your mind and now I want to turn to another function that at first sight will seem to have no connection to anything that I’m talking about so far. So let me introduce a completely different color, I don’t know, maybe a dark green to distinguish it, not just intellectually but also from the standpoint of the color palette that I’m using. And to introduce this, well, the function itself will be the function E to the X. I should say a few words about what E is since it’s pretty important in that formula. There are many ways to define this number called E. Again, it depends on where you’re coming from. One nice way is to consider the following. Consider the limit as n goes to infinity of one plus one over and raised to the nth power.
Now first off, just note that this definition that we have here has nothing to do with triangles, cosine, sine. Again, that’s what I mean by it looks completely different, but let me give you some motivation for why in the world you would ever consider this particular combination, this particular limit, this number as n goes to infinity. Why would you ever think about that? Well, imagine that I give you a dollar. Okay? I give you $1. And I say, “Hey, if you give me that dollar back, I’ll consider it a loan, and I’m going to pay you interest on that.” And let’s say I tell you I’m going to over the course of one year, give you 100% interest. Then how much money will you actually have at the end of that year? If I’m the bank, right? How much money will you have in the bank account?
Well, you started with one dollar, okay. And then 100% interest means that you’ll get another dollar. In a minute, I’m going to stop writing down these dollar signs. So you’d have $2. That’s pretty good. Pretty good interest, right? 100%. But then imagine you say, “Hey, maybe you want to pay me that interest rate, but not all at once. Maybe you want to pay me half of that interest in six months and then six months later give the other half of the interest rate.” Now that’s interesting because that gives you compound interest, right? So in that particular case, you’d start with $1, okay. At the end of six months, I’d give you half a dollar more, and then six months later I’d have to pay you interest on this. Which again, if I’m giving you that 50% interest, if you will, every six months, then this is the amount of money that I’d owe you.
As you see, you’re getting interest on the interest in this particular case. That’s why it’s compound interest. So this gives me three halves times three, that gives me nine quarters, which is say $2.25. So clearly it’s a little bit better if you get the interest compounded. Instead of $2 you get $2.25. But then you start thinking, Hey, what if the bank gives you the interest every four months, three times a year. What would happen in that case? Well now I’d have to give you one plus one third of the interest in the first third of the year. Then I’d have to give you again one third, that 33 1/3% interest, and the second … Oh, I’m running out of … What if my iPad dies before I’m done? That would be so painful. Root for me to get through this.
Okay, I’m going to write more quickly. So one plus one third. So in this case you’d get what, so that’s four thirds cube? So that would be 64 over 27 which is about $2.26 or so. A little bit more than you had before. And again, right, you can keep on going. So I don’t have to write it all out. If you were doing quarterly compounded interest, then you’d have one plus one quarter to the fourth power. Aha, look, it’s one plus one of our n to the n for n equal to four. And in this particular case, if you were to work this out, let’s see, so this would give us five to the fourth over four to the fourth, that’d be 625 over 256, and that is $2 and I think 44 cents, something like that. Anyway, you can imagine, keep on going.
And if you did this as the exponent goes to infinity, that is your compounding interest. You infinite quickly, but you get one over that amount of the total yearly interest in each of those installments. How much money would you get? And that then is the limit as n goes to infinity of one plus one over n to the nth power. And you can work this out and the answer is well money-wise, you’d get about $2.72 or if you’re not going to limit it to the just accuracy of pennies, the actual number that you get is a number that goes on forever. 2.71828 … It’s like pi in that it goes on forever. Transcendental number. And this is the definition of E. Okay, so E is a number and you can then ask yourself, what happens if you take that number and you raise it to a power called X? And that’s your function, F of X.
And you will learn, again in a calculus class, is the beautiful fact. And this is another way of defining this number E, that the derivative E to the X with respect to X is just itself, E to the X. And this has all sorts of deep ramifications, right? If the rate of change of a function at a given value, of a given argument X, is equal to the value of the function at X, then its rate of growth is proportional to its own value and that is what we mean by exponential growth. E, exponential growth, and this is E to the X exponential growth. So all these ideas come together. Now given this fact, we can now, if I just scroll back and I hope my iPad is not going to die, it’s acting up. I can feel it. Oh, come on. Would you scroll with me? Ah, good, maybe having too many fingers on it or something. I can now use Taylor’s theorem but apply it to the function F of X equals E to the X.
And since I have all the derivatives, it’s straightforward for me to work it out. Again, I’ll expand it about X not equal to zero. So I can write then E to the X, if X not is equal to zero, E to the zero, anything to the zero is one, and that will occur over and over again because all of the derivatives are just E to the X, they all get evaluated at X not equal to zero, so all of those derivatives in that infinite expansion are all equal to one. So all I get then is X over one factorial plus X squared over two factorial plus X three over three factorial and on it goes. That is the expansion of E to the X. Okay.
Now one more ingredient before we can get to the beautiful finale, the beautiful Euler identity. I now want to just introduce a little change, not E to the X but E to the iX. Do you remember what i is? i is equal to the square root of minus one. Right? Usually you can’t take the square root of a negative number but you can define it to be this new quantity called i, which means that i squared is equal to minus one which means that i cubed is equal to minus i, which means that i to the fourth is equal to one. And that’s all useful because when I plug in to E to the iX in these expressions, I need to take various powers not only of X but also of i, this little table gives us the result that I will have. So let’s just do that. So E to the iX is equal to one plus iX over one factorial. Now X squared will involve i squared, that’s minus one. So I get minus X squared over two factorial.
Okay. X cubed will involve i cubed. I would get minus i times X cubed over three factorial. And X to the fourth, the term I haven’t actually written down there, but that will just give me i to the fourth is equal to one. So I’ll get X to the fourth over four factorial and on that will continue to go. Now let me play a little game and pull out all the terms that have no i in it and those terms that do have an i in it. So the terms that don’t have an i gives me one … In fact, I’m going to risk changing colors here, please iPad do not die on me. So I will get one minus X squared over two factorial plus X to the fourth over four factorial and keeps on going. Okay, and that’s one term. Plus, and let me just change colors again, let me pull out an i and I will get this first term is X, and then the next term will be minus x cubed over three factorial from this guy over here, and then plus X to the fifth over five factorial. Haven’t written that down, but it’s there, and on and on it goes.
Now what do you notice about this? If I can scroll up, you will notice that cosine of X and sine of X, these expansions that we had earlier. If I now reflect on what I have here, this is just equal to cosine X plus i times sine X. Holy smokes, E to the iX, something that doesn’t seem to have any connection to cosines and sines. I mean, it’s compound interest after all, has this beautiful relationship. Let me see if I can bring this back … with cosine and sine. Okay now for the finale, right? Let’s let X equal to the value pi. Then the special case gives us E to the iPI is equal to cosine of pi plus i sine of pi. And sine of pi is equal to zero, cosine pi is equal to minus one.
So we get this fantastically beautiful formula, E to the iPI equals minus one, but I’ll write that as E to the iPI plus one equals to zero. And at this point the trumpets really should be blaring. Everybody should be on their feet cheering, mouth wide open, because this is such a wondrous formula. Look what it has in it. It has in it the beautiful number pi that comes in with our understanding of circles. It has this strange number i, square root of minus one. It has this curious number E coming from this definition that I gave before, and it has the number one and it has the number zero. It has all of the ingredients that are kind of the fundamental numbers of mathematics. Zero, one, i, pi, E. They all come together into this spectacularly beautiful, spectacularly elegant formula.
And that’s what we mean when we talk about beauty and elegance in mathematics. Taking these disparate ingredients that come from our attempt to understand circles, our attempt to make sense of the weirdness of the square root of a negative number, our attempt to make sense of this limiting process that gives us this weird number E, and of course, the number zero. How could there be anything more fundamental than that? And it all comes together in this beautiful formula, this beautiful Euler identity. So stare at that formula, paint it on your wall, tattoo it on your arm. It is just a spectacular realization that these ingredients can come together in such a profound, yet simple looking, elegant, mathematical form that is mathematical beauty. Okay. That’s all I wanted to say today. Until next time, take care. This is Your Daily Equation.