### What is behind the most beautiful equation?

17/04/2020Recovered from Medium

### The intuition

Letâ€™s see negative numbers as vectors, where the negative stands for the opposite direction to one. This concretely is having a positive one with a rotation angle of 180Â°.

So, multiplying a negative number twice is equivalent to making half a revolution twice, hence it results in an entire revolution giving again a positive number direction.

### The proof

This is like a trick, but it is mathematically valid? Yes, thanks to Eulerâ€™s identity. Any product of complex numbers, including positive and negatives, can be described as a product of Eulerâ€™s formulas, where making use of its exponential properties allows us the addition of angles (see â€śrecallâ€ť annotation on the image):

While this works for the product, addition and subtraction are ruled by head to the tail method of vectors.

### Conclusion context

In general, multiplying any number is the process of multiplying the absolute value and adding the anglesâ€¦ This is true for negative numbers whose absolute valueâ€™s product is 1 and adding the angles (twice 180Â°) is 360Â° which is the positive direction again.

If a kid asks you, why? You could answer by associating negatives number with half revolution. Turning them back twice would be a funny analogy.

This even works as a complex number introduction as the imaginary number i is one with an angle of 90Â°, and it just works the same as negative numbers with 180Â°. đź™Ś