Pythagorean Triples and Complex Numbers

Pythagorean Triples and Complex Numbers

A Pythagorean triple is a solution to the Pythagorean Theory such that . For example, {3, 4, 5} is a Pythagorean triple because .

This becomes important when discussing complex numbers of the form . 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 . 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 .

Now, if you take a Pythagorean triple such as , and multiply it by another pythagorean triple, say and multiply them together:

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

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

Multiplying two complex numbers also multiplies their magnitudes, so:

This is true of all complex numbers. Now, let's say that and then since is algebraically closed for multiplication, it makes sense that .

Also, since then it follows that .

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.

Mind blown

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