eli5 Euler’s Theorem?

Question

I’ve heard this is “the most beautiful and useful equation in all of math. I’ve watched videos, but still don’t understand what it does.

In: 10

Answer 1

Step 1:

Here’s how you derive the form of the Taylor power series expansion for a function:
https://math.stackexchange.com/questions/481661/simplest-proof-of-taylors-theorem.

The Taylor series is particularly useful because it allows you to write any non-polynomial function as an infinite series of powers of a linear term (ie (x-a)) with some constant in front of each term in the series (you can also Taylor-expand polynomials, but you’ll just get back the same polynomial with the terms grouped differently as a finite series).

Of course, functions like e^(x), with a being a constant, and sin(x) and cos (x) are non-polynomials and that’s why Taylor series are applied onto them.

Step 2:

https://proofwiki.org/wiki/Power_Series_Expansion_for_Sine_Function

Compute the Taylor series for e^(z), sin(x), and cos(x). Use the cases that at some values for x, sin or cos are 0 or 1 or -1 (ie cos(0) = 1, sin(0) = 1) and that the derivative of cos or sin leads you to the other or the negative of the other.

Use the case that derivative of e^(z) = e^(z), being itself.

Step 3:

Note that if you set z = i •x, the Taylor power series of e^(z) gives you a mix of real and imaginary terms that you can match perfectly with the Taylor series of cos(x), sin(x) in the form e^(I • x) = cos(x) + i • sin(x)

Step 4:

Set x = pi. cos(pi) = -1. sin(pi) = 0, which zeroes the term i • sin(pi). Since the right side of the equation is -1 and the right side equals the left side, e^(i • pi) = -1.