Euler's Identity - $e^{\mathrm{\pi i}}-1=0$

Introduction

e^πi + 1 = 0 is one of the more interesting mathematical equations since it takes 3 different number concepts, the natural log raised to pi times i, the imaginary constant. Simplifying this expression leads to a simple answer of -1. Now I am pretty nerdy, but this seems pretty amazing to me.

• Three of the most important numbers can be put in an expression to equal -1
• How a number raised to a power can be a negative
• How the proof is done with elementary functions

Because of all this, the concept of Math has intrigued the last few days, so I thought I would put my own explanation up on here and spread some math knowledge to all.

Background

First off, I am going to explain what each of the numbers involved is.

e is the base of the natural logarithm. e can be expressed two different ways. The typical definition is the following:

However, it can also be defined as:

Both of these will lead to a series which goes on to infinity and will lead to an irrational number:

π is the ratio of a circle's circumference to it's diameter. This is an irrational number and is equal to:

i is defined as the following:

Because you cannot typically take the square root of a negative number, the idea of i or the imaginary digit was introduced. For example, the square root of -9 would then be 3i.

Proof

Using the definition:

we get:

Now using the definition of i:

Rearraning the equation, we get:

Using the following substitutions, from the Taylor Series:

We get: