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 ?
Multiplication 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 180o 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 90o 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!