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 nth roots of unity. i.e

## 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 ?

M**ultiplication by 1 is a 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 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 is a 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.

That’s about it! – That’s what the nth roots of unity mean geometrically. Have a good one!