# Euler’s Identity

One of the most beautiful mathematical equations ever discovered is Euler’s identity. It simply gave the relationship between two transcendental numbers which is *e*, the base of natural logarithm, *π* or *pi*, the ratio of circumference to the diameter of a circle and* i* which is an imaginary number. A number that we less expect which is equal to

This identity is a special case of Euler’s formula which is shown below.

This formula is true to any value of x. Let *x = π*

In basic trigonometry, we know that

and

By simple substitution:

Now come to think of it, irrational number is raised to the product of an imaginary and another irrational number results to an integer value?

Another proof by *De Moivre’s Theorem*:

We know that

By substitution:

Let *n = 1, x= π*

### Dan

#### Latest posts by Dan (see all)

- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015