Leonhard Euler is commonly regarded, and rightfully so, as one of greatest mathematicians to ever walk the face of the earth. The list of theorems, equations, numbers, etc. named after him is unmatched. There are so many mathematical topics named after him that if I were refer to Euler’s formula, I would have to specify which one. For now let’s consider one particular identity/equation of his*, namely:

Euler’s identity (aka Euler’s equation)

*As it turns out Euler’s identity was most likely discovered by an earlier mathematician, Rodger Coats, who died while Euler was still a boy. Don’t let this deceive though, Euler still had plenty of math to contribute to this and many other topics.

A usual response to Euler’s identity. (Relevant xkcd)

Veteran mathematicians and fledgling calculus students alike gawk at Euler’s identity. It has been described as “the most beautiful equation in mathematics,” and for good reason. It relates the 5 most fundamental constants of mathematics:

π — pi

e — Euler’s number

i — Imaginary Unit (Square root of -1)

1 — One (Unity)

0 — Zero (Nothing)

While this equation is certainly beautiful, or at least significant, it is only one case of a more general formula:

Euler’s formula

To me this is much more beautiful than Euler’s identity. It manages to relate exponentiation, complex exponentiation nonetheless, to the trigonometric functions. And as a result, this equation serves as a bridge between several topics relating to complex numbers including: complex logarithms, complex numbers in polar form, complex/imaginary angles, and so on. I go into more detail on the application of Euler's formula in another post.

But how does one arrive at such a magnificent (if not a bit intimidating) formula? What does a number to the power of an imaginary number even mean? Can we really say these two expressions equal each other?

In this article I will attempt to show how to get to Euler’s formula assuming you know three things:

What an imaginary number is (i = √ -1)

is (i = -1) A very basic idea of differentiation , i.e. you know what a derivative is (this is really all the calculus you’ll need).

, i.e. you know what a is (this is really all the calculus you’ll need). And what a factorial is, e.g. 5 factorial is written as 5! and equals 5*4*3*2*1=120.

There are a few sub-points, or lemmas, to explain before we finally put them all together and prove Euler’s Formula. If you already happen to know a certain lemma then you can of course skip over it.

Lemma 1: The Powers of i are Cyclical

The first important fact we must establish is that as you raise i to the 0th, 1st, 2nd, 3rd, and so on powers, a pattern starts to emerge:

The powers of i from 0 to 7

As you can see the powers repeat, or cycle, every 4th number. This makes calculating an arbitrary integer power of i simple.

To find iⁿ, just divide n by 4 and look at the remainder. If the remainder is:

0 (i.e no remainder) then iⁿ = 1

1 then iⁿ = i

2 then iⁿ = -1

3 then iⁿ = -i

Put another way:

For any integer n [lemma 1]

Where (n mod 4) means “the remainder after n has been divided by 4.” Keep in mind that this is only true when n is an integer (…-3,-2,-1,0,1,2,3…) and not if it is a fractional or irrational number (3/5, π, √2, etc).

**If it’s unclear why the powers of i do this, keep in mind that:

Any number to the power of 0 is 1,

Any number to the power of 1 is that same number,

And i² is -1 because the definition of i is that it is √-1.

Once you know these three things the higher powers are just repeated multiplication of i.

Lemma 2: The Derivatives of Sine & Cosine

The second piece of proving Euler’s formula is knowing the derivative of the sine and cosine functions: