Euler’s identity is the equation:

\[

e^{i \pi} +1=0.

\]

If you have any taste for mathematics at all, it is hard not to smile at this. In one equation we have each of five “special” numbers (*e*, *i*, pi, 1 and 0), along with one instance of each of three basic arithmetic operations (addition, multiplication and exponentiation.) Not too shabby!

But why is the equation true? Well, the first thing to notice is that we have an imaginary number in the exponent. That means that appreciating this equation requires an understanding of how we deal with such things. Towards that end, we have the following definition:

\[

e^{i \theta}=\cos \theta+ i \sin \theta,

\]

where theta is some real number.

Given this definition, it follows that

\[

e^{i \pi}=\cos \pi+ i \sin \pi.

\]

If you remember your basic trigonometry, then you know that

\[

\cos \pi = -1 \phantom{xxx} \textrm{and} \phantom{xxx} \sin \pi=0.

\]

Euler’s equation now follows immediately.

But now you might be thinking that the fix is in. Anyone can whip up a bizarre definition and then discover that, as a result, one arbitrary collection of symbols is equal to another. Why, though, would you summon forth such a bizarre definition in the first place? Who said you could drag trig into this?

Appreciating the beauty of Euler’s identity requires understanding why that triggy definition is the appropriate one to use. And understanding *that* requires thinking for a moment about what we want out of a definition.

So let’s go back to the natural numbers, by which I mean

\[

1, 2, 3, 4, 5, 6, 7, \dots

\]

Useful stuff, no question about it. You can use these numbers to label collections of objects. You can even do arithmetic with them, so long as you are content with addition and multiplication. Subtraction, alas, is more difficult. You are fine if the number you are subtracting is smaller than the number being subtracted from, (if the subtrahend is smaller than the minuend, in the jargon) but you are SOL if the smaller number is on the left.

It would be nice if we could embed the natural numbers within a larger set. If we play our cards right we would then have a set that preserves all of the good stuff about natural numbers, but wold also give us some added functionality. There is such a set, of course, and it is called the integers. It is the set you obtain when you take the natural numbers, and throw in these symbols:

\[

0, -1, -2, -3, -4, -5, -6, \dots

\]

The trouble is that you can’t just throw in symbols willy-nilly. You have to verify that your new symbols play well with others. Can I do arithmetic with my new numbers in a way that is consistent with everything else I know about arithmetic? The answer is yes, but only if I define things carefully. For example, the somewhat awkward seeming definition that the product of two negative numbers is positive turns out to be logically required. I’ll save the details for another post.

The process continues. The integers are nice, but in general you can’t divide with them. So we embed them inside the rational numbers (the term “rational” indicating that we are talking about fractions, or ratios), and now we can divide. In the rational numbers we can add, subtract, multiply and divide, but they still lack certain desirable properties related to the convergence of sequences. So we embed the rationals inside the real numbers.. The real numbers are pretty good, you can even do calculus with them, but tit turns out that certain polynomials do not have solutions. So the real numbers get embedded within the complex numbers.

Which brings us back to complex exponentiation. We will take for granted that we know how to raise a real number like *e* to a real number exponent. We shall then insist that our definition of complex exponentiation be consistent with that prior definition for real exponentiation. At a minimum, if we take *x+iy* to denote an arbitrary complex number, then we better have:

\[

e^{x+iy}=e^x e^{iy},

\]

since this is a consequence of one of our standard rules for manipulating exponents. And since we already know what the first term in the product means, our problem has been reduced to considering what we might call “pure imaginary numbers.”

Now what? For real exponentials we started with positive integer exponents. Raising a number to such a power meant multiplying the number by itself the prescribed number of times, a simple and common sensical notion. It turns out that the definitions for negative, rational and real number exponents are then forced on us logically, just like we were forced to define the product of two negative numbers to be a positive number. The details of how *that* is done require a post of its own.

But now we seem to have hit a wall. How on earth are we going to extend this definition to imaginary exponents? There’s a story in that, but we shall save it for the dramatic conclusion…