# Explain euler's formula and eulerian form of complex numbers.

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Euler's formula says:

`e^(i theta) = "cos" theta + i "sin" theta`

It's like the difference between Cartesian coordinates and polar coordinates.

One way, you tell how many units left/right, and how many units up/down

The other, you say how far away you are, and then the angle.

So, for example

`4 e^(i pi / 3) = 4("cos" pi /3) + 4*i("sin" pi/3) = 2 + i 2 sqrt(3)`

So, you can think of this as 4 away from the origin, with angle `pi / 3` from the positive real axis

Or you can think of this as: 2 in the real axis direction, then `2sqrt(3)` in the imaginary axis direction.

Here is how to go back and forth between `x+iy` and `re^(i theta)`

To go from `x+iy` to `re^(i theta)`

`r = sqrt(x^2 + y^2)`

`"tan" theta = (y)/(x)`

Example: `3+7i`

`r^2 = 3^2 + 7^2 = 58`

`r = sqrt(58) = 7.6158`

`"tan" theta = (7)/(3)`

`theta = 1.1659`

`3+7i = (7.6158) e^(i 1.1659)`

To go from `re^(i theta)` to `x+iy`

`x= r "cos" theta`

`y= r "sin" theta`

Example: `5 e^(i pi / 4)`

`x = 5 "cos" pi/4 = (5 sqrt(2))/(2)`

`y = 5 "cos" pi/4 = (5 sqrt(2))/(2)`

`5 e^(i pi / 4) = (5 sqrt(2))/(2) + i (5 sqrt(2))/(2) `