When one is dealing with complex numbers, it is many a times useful to

think of them as transformations. The problem at hand is to find the n

roots of unity. i.e

As is common knowledge z = 1 is always a solution.

## Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

**Now what does finding the n roots of unity mean? **

If

you start at 1 and perform n equal rotations(because multiplication is nothing but rotation + scaling), you should again end up

at 1.We just need to find the complex numbers that do this.i.e

This implies that :

And therefore :

**Take a circle, slice it into n equal parts and voila you have your n roots of unity.**

## Okay, but what does this imply ?

**Multiplication by 1 is a 360 ^{o} / 0^{o }rotation.**

When

you say that you are multiplying a positive real number(say 1) with 1 ,

we get a number(1) that is on the same positive real axis.

**Multiplication by (-1) is a 180 ^{o} rotation.**

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis

The act of multiplying 1 by (-1) has resulted in a 180^{o} transformation. And doing it again gets us back to 1.

**Multiplication by i is a 90 ^{o} rotation.**

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90^{o} rotation.

This applies to -i as well.

so on and so forth,

Have a great day!