Monday, August 18, 2014

3-4-5 complex number has Infinite order

This is a good exercise/challenge with complex numbers.

Consider the complex number  $Z = \large \frac35 + \frac45 i$. (Where $\large i = \sqrt{-1}$.)

Prove that $Z^n$ is never equal to 1 for any positive whole number $n = 1, 2, 3, 4, \dots $.

This complex number $Z$ comes from the familiar 3-4-5 right triangle that you all know: $3^2 + 4^2 = 5^2$.

In math we sometimes say that an object $X$ has "infinite order" when no positive power of it can be the identity (1, in this multiplicative case). For example, $i$ itself has finite order 4 since $i^4 = 1$, while 2 has infinite order since no positive power of 2 can be equal to 1. The distinct feature of $Z$ above is that it has modulus 1, so is on the unit circle $\mathbb T$ in the complex plane.