Mathematics is replete with connections between seemingly unrelated objects. They pop up in the most curious places and fill us with the sense that we are being allowed a sneak peek at a small part of a vast cosmic secret, a structure too large and mysterious to have been created by the small minds of men. Such is the case with Leonhard Euler’s infamous ‘identity’, taking two irrational numbers, a complex number, 0 and 1, and presents them is a neat and suspiciously simple package.



Let me flesh this out a little for the uninitiated. The number e equals 2.71828… and is, like, π irrational. That simply means that it cannot be written as a fraction. e springs up everywhere in mathematics, from continuously compounded interest to probability to the growth of populations, but (for our purposes) its most helpful definition is as the sum of an infinites series:



Where 4!, for example, represents 4 * 3 * 2 * 1 = 4. This is called the ‘factoral’ of 4. Replace x with 1 and you have e^x = e = 2.71828…

The number x should be familiar to everyone, if only from high school geometry: it is the ratio of a circle’s diameter to its circumference. Lastly, i is the symbol used to represent √-1 . Since -1 has no square root on the line, i is called an imaginary number.

Even with a basic understanding of the components of Euler’s Identity, it still evades the power of intuition. How can an irrational number, e, multiplied by itself π times equal anything reasonable like 0 or 1? And what does it even mean to take a number to an imaginary power? These questions have answers, but to understand them we must turn to, of all places, trigonometry.

The infamous trigonometric functions sin x and cos x are usually associated with the ratios of sides of triangles, but it turns out that they, too, can be represented as sums of infinites series. This, by the way, is how a calculator approximates trigonometric ratios:



and



You may recall that Zeno dismissed the idea of finding the sum of an infinites series as absurd – as you have probably noticed, most mathematicians since the Enlightenment don’t share his anxiety! This is fortunate for us, because things are about to get very interesting.

First, notice how the expansions of e^x, sin x and cos x are related but distinct. This is our first clue to understanding Euler’s Identity. The cosine expansion includes all even powers and exponents, sine function includes all odd powers and exponents, and ex includes both. If we add sin x and cos x, we will have something that looks similar to e^x, but with the sign all wrong:



This is intriguing, but here comes Euler’s real insight: if we multiply sin x by i, all the odd powers will be multiplied by i, hence we obtain



Similarly, if we take the expansion of e^x and replace every x with ix we have



However, i = √-1 implies that i² = -1, so anytime we have i² in the formula above we can replace it with -1. This gives us



which, as the first few terms indicate, is exactly the same as isin x + cos x! If you can follow this little bit of math magic, the rest is easy. We now have the identity



Replacing every instance of x with π, we obtain



But sin x = 0 and cos π = -1, so



Add 1 to both side and voila!