Proof of Euler’s formula

Three proofs, just to be sure.

In this article we will see various ways to prove Euler’s formula and Euler’s identity.

Euler’s formula

Euler’s formula is:

Where θ is measured in radians.

Euler’s identity

If we substitute a value of π for θ in we get Euler’s formula:

Since cos π is -1, and sin π is 0, this leads to Euler’s identity:

If we prove Euler’s formula, this will also prove Euler’s identity.

What do we mean by proof?

Quite often in mathematics, we might have an equation where the meaning of the equation is extremely clear, and we simply need to prove that it is true. This applies, for example, to Pythagoras’ theorem — we know what a triangle is, and we know how to square and add numbers, so all we need to do is prove that the formula is correct.

With Euler’s formula, things aren’t quite so clear-cut. We are raising e to the power of an imaginary number. To take the naive definition of powers, we are multiplying e by itself an imaginary number of times. What does that even mean?

An alternative way to look at this is to say that we have decided to define Euler’s formula to be true…