rmd1228887e

2022-07-15

Is there a 'far' irrational number?

Alexia Hart

Expert

Consider $x=\sum _{j=1}^{\mathrm{\infty }}{a}_{j}/{4}^{j}$ where each ${a}_{j}$ is either $1$ or $2$. Thus the binary expansion of $x$ consists of two-digit blocks which are either $10$ or $01$. Then x is far. But there are uncountably many choices, so all but countably many of them are irrational.