Tom's Math Class: 2017

A Pythagorean triple is a solution to the Pythagorean Theory $a^{2}+b^{2}=c^{2}$ such that $\{a, b, c\} \in \mathbb{N}$ . 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 $\left | z \right |$ . 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: $\left | z \right |\cdot \left | w \right |=\left |z \cdot w \right |$ This is true of all complex numbers. Now, let's say that $\left | z \right |\in \mathbb{N}$ and $\left | w \right |\in \mathbb{N}$ then since $\mathbb{N}$ is algebraically closed for multiplication, it makes sense that $\left | z \cdot w \right | \in \mathbb{N}$ . Also, since $\{Re(z),\ Im(z),\ Re(z),\ Im(z)\} \in\mathbb{N}$ then it follows that $\{Re(z\cdot w),\ Im(z \cdot w)\} \in\mathbb{N}$ . 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.

Tom's Math Class

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Saturday, November 4, 2017