This quickly became one of my favourite derivations. And with the launch of my blog, I couldn’t wait to write about it.
First, an introduction. This is Euler’s formula:
When we set x = pi, this formula evaluates to what is called Euler’s identity,
Now, before any further explanation and before I actually present the proof (skip to the bottom, if you wish), I should like to pause and take a few moments to wax eloquently. Consider these words as part of the dramatic build up. Because what we have before us is, in my opinion, one of the most breathtaking examples of mathematical beauty.
If on first look Euler’s identity means nothing to you, that’s ok. In a few moments I’ll hopefully be able to assist in understanding why the theorem is so remarkable and deep. Meanwhile, we can preface the actual derivation by highlighting that the formula itself is to math and science what Mozart or Beethoven are to music. Physicist and one of my personal idols, Richard Feynman, once called the equation “our jewel” and “the most remarkable formula in mathematics”. Mathematics professor Keith Devlin has been quoted as describing Euler’s identity, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”.
This might seem like hyperbole, but Devlin’s description really does capture the essence of what is so significant about the theorem as possibly the most beautiful to have been discovered by human beings. It is so profound and important that, at least for me, it is possibly one of the genuinely deep historical human artifacts. I would place it alongside Maxwell’s equations and others.
What I would personally say, is that this particular theorem and its derivation (below) is an especially historical and immediately intuitive proof, which does offer greater depth to humanity and the project of human civilization. In that it links five fundamental mathematical constants, capturing the relationship between calculus (series expressions), trigonometry (sines and cosines), and the algebra of complex numbers (considering the exponentiation of a complex number), I would describe it as historical in significance because it represents a moment in which human beings might glimpse at some deeper truth of nature.
Indeed, it is so profound that there are even some people who, perhaps misguidedly, argue that the theorem proves the existence of god! Not that I am religious; nor am I a participant in religious cognition. It seems that human beings can find evidence for god or some other deity in almost anything, including a piece of toast. That said, it is not unreasonable to suggest that the mathematical conclusion that we’re about to explore suggests some order to mathematics and also to nature, especially if you think about the derivation in all its nuance. This is a purely objective suggestion.
In our limited human minds, it may only be a scratch at the surface. But this theorem would seem a remarkable fact to be discovered, much like any other law of nature. It certainly would seem to offer yet another example in the history of science and mathematics that highlights the objective mathematical character of reality. And if you don’t believe this, I highly suggest you study it for yourself.
Euler’s identity proof (Taylor series)
There are a number of ways to derive Euler’s identity. Some derivations are more elegant than others. The one I present is well known. And while not necessarily the nicest or most elegant or most rigorous, I think it is the one proof that is most deep and historical and dramatic. It also seems the most accessible, which is a good thing because I think more people need to appreciate the significance of the follow result.
For the sake of saving space, I am not going to be overly rigourous in this blog. I am not going to offer a complete and rigorous proof, and so one will have to trust some of my basic assumptions. I am also not going to explain some of the basic tools and concepts used. When I have some spare time, I would like to make a video on this derivation and lay it out in detail. Meanwhile, if at any point you don’t believe me, or wish for a fuller treatment, there are many articles and papers and videos freely available online. Sal Kahn already offers a decent seven-part series that serves as an accessible introduction.
To start, we should look to the Maclaurin series. The Maclaurin series is a special case of the Taylor series, which itself is a specific example of a power series. In simpler terms, one could think of the Taylor series as basically an extension of the logic of approximating functions with polynomials at any point (i.e., taking n derivatives).
This is the Maclaurin series, which I’ve written in latex:
Assuming one has a basic understanding of the above, we can begin by working out the series for sin x, cos x and e^z. Here are the basic expansions, which I’ve written using my digital blackboard:
You will have to take my word on the expansions. If you would like to understand how we arrive at these expansions, it is fairly easy to research on the web.
In looking at the image, notice the expansion for sin(x) and cos(x). Sin(x) shows x^n / n! for all the odd values of x. For the cos(x) expansion, we have x^n / n! for all the even values of x. And, for the expansion of e^x we see both odd and even values for x. So what is going on?
Let’s analyse by adding sin(x) to cos(x), which I’ve highlighted in magenta:
What you may have noticed is how sin(x) + cos(x) is similar to the expansion for e^x. In fact, it is clear that they have all the same terms. The only difference is the signs.
In sin(x) + cos(x) we see the pattern: positive positive, negative negative, positive positive, and so on. In the exponential expansion, each term is positive.
To make it more clear, see the following:
It’s really very peculiar that this pattern has emerged. Sin(x) and cos(x) have no direct connection to e^x. The former come from trigonometry. The latter from exponentiation. When we add the polynomial representations of the two fundamental trigonometric functions together, we arrive almost exactly at the polynomial representation of e^x.
Needless to say there is an indication here that something significant is at work, that we’re beginning to touch on something deep. And it’s around this point, when I first learned this derivation, that I began to get excited.
So why the curiously similar pattern? Let’s take another step forward.
To do that, we’re now going to consider the expansion of the function e^ix, where i is the imaginary unit. All we need to do here is substitute into our expansion for e^x, so that every time x appears in the expansion we substitute ix. The series expansion looks like this:
And now we can simplify, understanding the value of the imaginary unit i when raised to a power. And it turns out, there is yet another pattern in the signs. Notice how every time the imaginary unit appears (when we raise i to a power), it patterns the series for sin(x) in terms of the placement of positive and negative terms. In considering the final line in the expansion of e^ix below, now also notice the pattern of the signs when considering all of the series.
But we’re still not done. Because we can re-write e^ix by separating out the imaginary terms and the real terms.
Notice that for the real terms, this is actually the Maclaurin representation of cos(x). And notice, too, that for the imaginary terms, this is the Maclaurin representation for sin(x).
Very cool! But we’re still not done. Because here comes the moment – the awesome and awe-inspiring conclusion.
If we agree that what we end up with here is the series expansions for cos(x) and sin(x), which we discussed at the outset. And in understanding that like the series for cos(x) and sin(x), our real terms that we separated and our imaginary terms that we just separated out also have an infinite amount of terms, we can say that all the real terms converge with cos(x). Similarly, all the imaginary terms converge with sin(x).
Therefore, we can re-write what we have as follows:
And there we have it! Euler’s formula. An incredibly useful formula that, among other things, helps relate real numbers to imaginary numbers. It also has many useful applications. But with all that aside, this really is an amazing result. Not only have we found a relationship between e and the two fundamental trigonometric functions, but we’ve also found a relationship with the imaginary unit i. In short, we found a relationship with some of our most important mathematical constants.
But if that is not enough, there’s one more thing we can do. And it involves incorporating pi. In other words, we can also now raise e to the power of i times pi. The conclusion is magnificent.
It’s a breathtaking result. What we see here, among other things, are the most fundamental mathematical constants – each from completely different areas – all linked together! And while more rigorous proofs exist – in other words, while we can certainly prove the above result – understanding it and explaining it is a completely different thing. This is, really, one of the most astonishing mathematical results.
In the future, when I have some spare time, I’ll work through more formal and rigorous proofs. But for now, this is plenty to savor.