Question

Question #d632b

Answer 1

https://en.wikipedia.org/wiki/Euler%27s_formula#Applications_in_complex_number_theory

Explanation:

Too many to list here but wikipedia goes into this topic in sufficient detail.

Answer 2

A very cool formula indeed...but it's implications are not algebraic.

Explanation:

Euler's formula takes the most common constants of mathematics along with the only two numbers that are neither prime nor composite and relates them together:
`e^(pi*i)+1=0`

However, the reason this is so important and not just a clever way to use the numbers lies in trigonometry and the complex plane.

Around 1740, Euler was able to compare the series expansions of the trigonometric functions and the exponential functions. Building off the work of others he came up with the following relationship:
`e^(x*i)=cos(x) + i* sin(x)`

And when you evaluate it at `pi`

`e^(pi*i)=cos(pi) + i* sin(pi)`
`e^(pi*i) = -1` (a pure real from an imaginary exponential!)
`e^(pi*i)+1=0`

a most amazing result that shows us that the complex exponential is a period function in the complex plane.