n roots of unity

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

image

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

image
image

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

image
image
image
image
image

This implies that :

image

And therefore :

image

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

image


Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o 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.

image


Multiplication by (-1) is a 180o 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 180o transformation. And doing it again gets us back to 1.

image



Multiplication by i is a 90o rotation.

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

This applies to -i as well.

image

so on and so forth,

Have a great day!

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