racodelitusmn

2022-07-06

Lets say we have irrational numbers ${\alpha }_{1},...,{\alpha }_{n}$ in the interval $\left(0,1\right)$. Represent each αi as a binary expansion $0.{a}_{i}^{1}{a}_{i}^{2}...$ where each a${a}_{i}^{j}\in \left\{0,1\right\}$. Define the "dovetail" of the ${\alpha }_{i}$ to be the number with binary expansion $0.{a}_{1}^{1}{a}_{2}^{1}{a}_{3}^{1}...{a}_{n}^{1}{a}_{1}^{2}{a}_{2}^{2}{a}_{3}^{2}...{a}_{n}^{2}{a}_{1}^{3}{a}_{2}^{3}{a}_{3}^{3}...$
So that the dovetail is irrational?

isscacabby17

Expert

It's never rational. This follows from the fact that a real number is rational if and only if its binary expansion is eventually periodic (possibly with endless $0$s).