Pythagorean Triples and Complex Numbers

# Pythagorean Triples and Complex Numbers

A Pythagorean triple is a solution to the Pythagorean Theory $a^{2}+b^{2}=c^{2}$ such that $\{a,&space;b,&space;c\}&space;\in&space;\mathbb{N}$. For example, {3, 4, 5} is a Pythagorean triple because $3^{2}+4^{2}=5^{2}$.

This becomes important when discussing complex numbers of the form $x+yi$. The reason is that the Pythagorean theorem applies to complex numbers, too, because x and y above are the legs of a right triangle. The length of the hypotenuse of such a triangle is equal to $\sqrt{x^{2}+y^{2}}$. This is also called the "magnitude" or "absolute value" of the complex number, and it represents the distance of the complex number from 0. It is frequently abbreviated $\left&space;|&space;z&space;\right&space;|$.

Now, if you take a Pythagorean triple such as $3+4i$, and multiply it by another pythagorean triple, say $5+12i$ and multiply them together:

$z=(3+4i)$
$w=(5+12i)$
$(3+4i)(5+12i)$
$=15-48+20i+36i$
$=-33+56i$

Here's what's amazing: 33 and 56 are themselves legs of a Pythagorean triple!

$-33^{2}+56^{2}=65^{2}$

The reason this works blew my mind (#CLICKBAIT).

Multiplying two complex numbers also multiplies their magnitudes, so:

$\left&space;|&space;z&space;\right&space;|\cdot&space;\left&space;|&space;w&space;\right&space;|=\left&space;|z&space;\cdot&space;w&space;\right&space;|$

This is true of all complex numbers. Now, let's say that $\left&space;|&space;z&space;\right&space;|\in&space;\mathbb{N}$ and $\left&space;|&space;w&space;\right&space;|\in&space;\mathbb{N}$ then since $\mathbb{N}$ is algebraically closed for multiplication, it makes sense that $\left&space;|&space;z&space;\cdot&space;w&space;\right&space;|&space;\in&space;\mathbb{N}$.

Also, since $\{Re(z),\&space;Im(z),\&space;Re(w),\&space;Im(w)\}&space;\in\mathbb{N}$ then it follows that $\{Re(z\cdot&space;w),\&space;Im(z&space;\cdot&space;w)\}&space;\in\mathbb{N}$.

So, in the new triangle, we have both legs of integer length and a hypotenuse of integer length. This is the very definition of a Pythagorean triple.

## Thomas Davidson

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