# Beautiful Proofs(#3): Area under a sine curve !

So, I read this post on the the area of the sine curve some time ago and in the bottom was this equally amazing comment :  $\sum sin(\theta)d\theta =$  Diameter of the circle/ The distance covered along the x axis starting from $0$ and ending up at $\pi$.

And therefore by the same logic, it is extremely intuitive to see why: $\int\limits_{0}^{2\pi} sin/cos(x) dx = 0$

Because if a dude starts at $0$ and ends at $0/ 2\pi/ 4\pi \hdots$, the effective distance that he covers is 0. If you still have trouble understanding, follow the blue point in the above gif and hopefully things become clearer.

# nth roots of unity : A geometric approach

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 $z^n = 1$

## Multiplication as a Transformation

Multiplication in the complex plane is mere rotation and scaling. i.e $z_{1} = r_{1}e^{i\theta_{1}}, z_{2} = r_{2}e^{i\theta_{2}}$ $z_{1}z_{2} = \underbrace{r_{1} r_{2}}_{scaling} \underbrace{e^{i(\theta_{1} + \theta_{2})}}_{rotation}$

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 $z^n = 1$ $\underbrace{zz \hdots z}_{n} = 1$ $z = re^{i\theta}$ $r^{n}e^{i(\theta + \theta + \hdots \theta)} = 1e^{2\pi k i}$ $r^{n}e^{in\theta} =1e^{2\pi k i}$

This implies that : $\theta = \frac{2\pi k}{n}, r = 1$

And therefore : $z = e^{\frac{2\pi k i}{n}}$

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 180o 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 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!