# Euler's Identity

You may have heard of the formula , or . Intriguing? Or mystifying? Read on...

## Identity Crisis - The "What"

Euler's identity is well renowned because it incorporates 3 mysterious constants integral to pure mathematics, as well as 1 & 0, or -1. These are:

**e**: The "base of all natural logarithms" - the integral part, and the first 20 decimal places, of*e*are 2.71828 18284 59045 23536... See this Wikipedia article for more information.**π (pi)**: The ratio of a circle's circumference to its diameter - to 20 decimal places*π*= 3.14159 26535 89793 23846. See this Wikipedia article for more information.**i**: Also called the "imaginary number",*i*= √-1. There is no 'decimal expansion' for*i*, hence its name. Its understanding extended traditional mathematics into "complex mathematics". Again, see the Wikipedia article.

## Go Figure - The "Why"

Now, Euler's *Identity* is a "special case" of Euler's *Formula* (this means that if a certain set of conditions are satisfied, Euler's *Formula* turns into Euler's *Identity*). Consequently, we first need to prove Euler's Formula.

Now, this involves **differentiation**. If you're not happy with that, I strongly suggest you start with some basic background reading on this subject. WikiBooks makes a great refernce and guide for many subjects. Calculus is no exception, so see the WikiBooks *Calculus* book's section on differentiation.

When you're happy, particularly with regard to differentiation of the basic trigonometric function, you can begin to analyse Euler's Formula:

### Euler's Formula

- Firstly, define the function f(
*x*), where This is 'legal' to differentiate, since it has a value at all points: implies that e^{ix}at no point equals 0, which would cause a division by zero, making the funcion impossible to differentiate. - Now, differentiate this function to obtain f'(x), making use of the quotient rule:
- Put to simplify the writing.
- Then
- Therefore:

- If the function differentiates to 0, then it has a rate of change of 0, and so doesn't change. Hence, we can see that for any
*x*. - So .
- Now, if , clearly we have: QED

This derivation is known as Euler's Formula, and uses no property of *i*, except that *i* × *i* = -1, and that it is a constant.

### Euler's Identity

The conditions for our "special case" are simple: x = π.

Simple evaluating Euler's Formula at this point give us the desired result:

Therefore and .