The identity of Euler and Euler's Identity

Leonhard Euler (1707-1783) was surely one of the most prolific of great mathematicians. Among his contributions, we need to mention two, each of which comes down to us as a single letter: the letter e and the letter i.

 Euler was born in Basel, Switzerland, so his life and work might fittingly be considered a riposte to the old anti-Swiss jibe (originally from The Third Man) that Switzerland has produced nothing for all its years of peace and democracy more than the humble cuckoo clock. 

Since Euler’s day and because of his work, i stands for the simplest of the numbers that Descartes had called “imaginary.” This i refers to the square root of -1. We don’t need to bother ourselves further with the question “what is the square root of -1?” It is simply i, by stipulation. We don't end there, of course, but we can start from there and build something new and important.  

Also since Euler’s day and because of his work, e stands for perhaps the most remarkable of irrational real numbers. This e is a constant that shows up whenever non-mathematicians try to use mathematics to model continuous growth. For example, suppose biologists expect that the bacteria in a particular petrie dish will double over the course of a particular unit of time, t.

How do they model this? Conceivably, they could imagine that there are 100 bacteria in the dish all through period t0, and that at the instant that period t1 begins, presto! There are 200. But that just seems wrong, even as a working hypothesis.

Suppose they divide t0 into ten parts: t0.1, t0.2, t0.3 and so forth. They might expect that the number would grow by one-tenth during each of these units. Does this give us the desired doubling when we get to t1?

No. we get more than a doubling. After all, if the number increases from 0.0 to 0.1 by ten percent, we go from 100 bacteria to 110. But then if there is another 10 percent increase from 0.1 to 0.2, we get not just another 10 bacteria – we get another eleven. The growth rate compounds.

In fact, when we get at last to t1 on this plan, increasing the size of the population by ten percent at each step, we’ll have 2,593 or 2,594 bacteria, because of the compounding effect.

But why should we believe that the size of the population remains the same within any of the subdivided periods? We can always break it down further, after all. We can break the tenths of t into tenths, and we can break those again into tenths, dividing t into a thousand parts. The more and smaller the parts, the closer to continuous growth we’ve gotten. Call the number of  units within t, n.  How do our biologists model that?

They’ll end up using e, which is the limit of this simple expression

(1 + 1/n)ⁿ

… as n approaches infinity.

The neat thing about that expression is that as you increase n, two very different consequences would seem likely to result. It is as if you’re hitting the brakes and the accelerator at the same time. The increase of the denominator in the fraction 1/n lowers its value, thus lowering the value of the expression inside of the parentheses.

If n is ten, the expression inside the parentheses above is 1.1. If n is twenty, this falls to 1.05. If n is 100, to 1.01. And so forth.

On the other hand, whatever n is inside the parentheses, n is the same thing outside the parentheses, as the power to which the expression will be raised. And this is the accelerator. So the value of the whole expression doesn't change very quickly at all as n rises. The two effects cancel each other out.

Let’s chart the value of that expression as a whole.

If n is 2                                    Then (1 + 1/n)ⁿ = 2.25

 

If n is 10                                  Then (1+1/n)ⁿ  = 2.593742

 

…        20                                                        = 2.653298

 

…        100                                                      = 2.704814.


The limit of this process as n approaches infinity, Euler’s constant, or just plain e, is an irrational number that begins: 2.7182818284590452353602874.

That is a wonderfully important number, that shows up (as our bacterial example was intended to show) in a variety of fields of applied matehematics wherever anyone is attempting to understand a process of continuous growth. Such as, in finance, continuously compounded interest.

One more point before we say goodbye to Euler: One neat cap upon his immortality is that he proved an identity, appropriately called Euler’s Identity, a simple equation that incorporates each of the five most important numbers in the world. 

If you're in search of an image to tattoo on your arm, consider that as a candidate. This identity draws together our discussion of irrationals (like both e and pi) and of imaginary numbers (paradigmatically i) with one another, and with the two binary numbers 1 and 0.