Yet again, Euler's identity has hit the headlines as a “beautiful" equation. In a paper in the journal Frontiers in Human Neuroscience, scientists showed a variety of equations to 15 postgraduate or postdoctoral mathematicians and measured their brain activity. The part of the brain associated with “beauty" was active, with Euler's equation provoking the most reaction.
Ramanujan scored lowest.
Personally, I think Euler's identity is pretty ugly. This is intended to clarify my opinion on it.
2 Beauty is in the Eye of the Equator
The full list of equations is available from the journal website. It's well worth a look. A sheet with the equations and descriptions was given to the participants in advance of the scanning. I won't go in to detail on every equation, but some things strike me from that list of equations and their descriptions.
First, the ridiculous. Equation 2, “The Pythagorean Identity". This gets my immediate vote as “ugly". Not because of the equation, but because they wrote and . Similarly the series expansion of the exponential family is rather than . Why is this important? Because the researchers showed the equations to non-mathematical people to see if the form of them was in any way beautiful so appearance was in their minds when designing the experiment.
Now, the more subtle. The Gaussian integral gets the description “ubiquitous in mathematical physics" while Ramanujan's formula is just “Equation expressing the inverse value of as an infinite sum". I don't know much about number theory, but I'd be willing to bet that Ramanujan's formula is far more amazing than that sentence conveys. And while the Gaussian integral is “ubiquitous in mathematical physics", the actual equation is simply the value of the full integral. Now, I consider this to be quite something – why should it involve , for example? – but the weight of the description seems a little heavy.
We can see this again in the inequality relating the size of a set to the size of its power set. (Let's ignore the fact that these were meant to be equations, it is a neuroscience article after all so the authors can't be expected to know about mathematical terminology; we'll also ignore the presence of Russell's paradox in Equations 50 and 51.) This simple inequality, which could be written , is loaded down with a description linking it to the continuum hypothesis.
Going through the “equations" one finds similar oddities. Equation 31's description has the phrase “triumphantly" in it, whilst Equation 10 is just “An identity for Euler's number ".
So I'm a little unhappy with the descriptions. But I'm even more unhappy with the choice of equations.
3 “De Jure" or “De Facto"
This gets to the heart of what I dislike about the hype about Euler's Equation (or Identity). Let's look at its description:
Euler's identity links 5 fundamental mathematical constants with three basic arithmetic operations each occurring once.
It's hard to argue with the five constants: , , , , and . I might quibble at the fundamental, particularly with regard to , but let's not be too pendantic. What I do want to be picky about is the “three basic arithmetic operations". Presumably, these are addition (of and ), multiplication (of and ) and taking powers (of to the power ).
It's that last that I want to look at carefully. What is the “basic arithmetic operation" of taking powers? We teach it as “repeated multiplication". However, this only works for positive numbers: is multiplied by itself times.
So for any other exponent we have to make a rule as to how to handle it. The rules work fairly intuitively based on how we extend from natural numbers to first integers, then rationals, then reals.
To go from natural numbers to integers, we put in negative numbers. The number is “What we add to to get back where we started". So is “What we multiply by to get back where we started". Thus . Note that we've used the fact that here.
Similarly, for rational numbers we start from the observation that and deduce that is the th root of (we may have to be careful about what values we allow for , but that isn't important here).
To get to real numbers, we remember that real numbers can be approximated by rational numbers so we approximate by . That this works needs a little effort to show, but it does work.
We're already a little way away from a “basic arithmetic operation". Certainly, you couldn't compute this by hand, even if you knew the numbers involved precisely.
But there's worse to come. We don't want for (or even ). We want . How are we going to get in there? The only thing that we know about is that . But this is useless information: it says that , but this doesn't help us much. With the other extensions we had “The unknown operation undoes a known operation" whereas this says “Doing the unknown operation twice gets us somewhere known". This doesn't give us much information on where we were in the middle, though.
There are a variety of solutions, but they all are based on the same principle: find some other characterisation of powers that does extend in an obvious way.
Let us look at a few of the more common suggestions. They all hinge on defining for by some means other than taking powers.
This appears (for ) as Equation 10 in the list. One way to think of it is to see it as saying that an appropriate limit of binomial distributions is a Poisson distribution. That is certainly fundamental, but I question whether it is basic.
This is Equation 8 (with the malformed , as well as other poor spacing). Justifying the use of this for the complex exponential function usually involves invoking power series and foreshadows analytic continuation. Again fundamental, but not really basic.
is the unique solution of the differential equation with initial condition .
In slightly different form, this is Equation 22 on the list. To use this, we have to know about existence and uniqueness of solutions of differential equations. I'd consider that fundamental, but again I'd hesitate at calling it basic.
When I teach about then I do so in the context of a course that also deals with differential equations. So the third on the list is my preferred way. I also feel that differential equations with their relationship to Physics provides a way for less mathematically inclined students to gain a glimpse of what is going on.
Let's quickly run through how this works. We define as the unique solution of the differential equation with initial condition . Putting we get that is defined to be the unique solution of the differential equation with initial condition . Let's ignore the fact that we're assuming, not very tacitly, that such a solution exists and is unique (again, not a trivial result!). Even with that knowledge, this still does not help us much. The proof that solutions exist and are unique does not go far to tell us how to find them. So we need to look for a way to convert the knowledge we have, but can't use, into knowledge that we can use. The difficulty is, as always, with the . The one thing we know about is that it tends to disappear if we get two of it. In this case, one appeared under differentiation so let's differentiate again to see if that deals with it. When we do this, we find that .
Here we have a function that when differentiated twice results in its negative. Such functions are known to any student of trigonometry: and and any linear combination thereof. Thus for some and . The catch is that as we've allowed to appear in one place, we have to allow for it to pop up all over the place and in particular and can be complex. Substituting in to the initial conditions, we see that . To find out what is, we have to be a bit smart and realise that holds for all and so . This gives and so the solution is .
Now comes the important point. We have defined to be the solution of with . From that definition we have deduced that . From that deduction, we substitute in to get . So this equation holds de jure: “because we said so". It all started with us defining in a particular way that, to be blunt, was a long way from powers in the sense that .
So how should we describe ? Well, our chosen route to is that of differential equations and our intuition about differential equations is likely to be built up from their use in describing motion. The differential equation for is . When drawn on the complex plane, this describes motion whereby the velocity vector is always at right angles to the position vector. It is not hard to deduce that when this happens the motion is constrained to lie on a circle. Once we have established that then we can deduce that our speed (the magnitude of the velocity) is unchanging so we are moving around this circle at a constant rate. That rate is equal to our radius, so if we start at then we are moving around the circle at “arc length" speed. After seconds, therefore, we have gone a distance of around the circle leaving us ... half way around.
Thus, I submit, a more honest description of would be:
This equation relates the fact that if you go halfway around a circle then you end up at a point diametrically opposite where you started.
That's a little less likely to end up on anyone's top ten favourite equations.
I find these “favourite equation" polls do tend to miss the distinction between de facto and de jure. Before declaring an equation (or inequality) to be favourite or beautiful we should consider why it is true. Is it de jure: it is true because we have declared it to be so. Or is it the, to me, more profound de facto: we have discovered it to be true.
The point of this was to explain why I thought that Euler's identity was not so beautiful. One could argue that this simply shows that Euler's identity contains even more than is seen at first. The difficulty with this argument is that Euler's identity is quoted as being “beautiful" because of its simplicity, not because of its connections.
Beauty in mathematics is a difficult thing to pin down, not least because the very word “beauty" is loaded with meaning from general use and that meaning varies considerably from person to person: “beauty is in the eye of the beholder", as we are often told.
I prefer the word “profound" for a piece of mathematics as it includes a measure of understanding. And I think that if you pinned down a mathematician as to what they actually meant when they declared a piece of mathematics to be beautiful then the word “profound" would be a better fit.
The understanding is important. We are judging ideas, not scribblings on paper. You cannot look at an equation and declare: “I don't know much about mathematics, but I know what I like." and expect to be taken seriously when deciding which equations are beautiful or not. If I do not understand something, I disqualify myself from any discussion as to its relative profundity, and equally the more I come to understand it the more I can appreciate just how profound it really is.
So I would not be surprised to learn that to those who really understand it, Ramanujan's identity is far more profound than Euler's. And when I actually think of all that is involved in Euler's identity, then I do find that I have an appreciation of its profundity. So much so that in the end, I feel that it is actually undersold by being promoted as “The Most Beautiful Equation". Beauty is, after all, only skin deep. Sadly, this can lead to beautiful things being dismissed as being shallow and leaving their depths unexplored. If we forget that Euler's identity is meant to be beautiful and actually examine it for what it is, then we might find the truth in the saying that:
All that is gold does not glitter
and search for the value in its hidden depths instead of barely skimming the surface.