Basis Expansion of Irrational Number

Theorem
A basis expansion of an irrational number never terminates and does not recur.

Proof

 * Suppose $$x \in \R$$ were to terminate in some number base $$b$$.

Then (using the notation of that definition):
 * $$\exists k \in \N: f_k = 0$$

and so we can express $$x$$ precisely as:
 * $$x = \left[{s . d_1 d_2 d_3 \ldots d_{k-1}}\right]_b$$

This means:
 * $$x = s + \frac {d_1}{b} + \frac {d_2}{b^2} + \frac {d_3}{b^3} + \cdots + \frac {d_{k-1}}{b^{k-1}}$$

This is the same as:
 * $$x = \frac {s b^{k-1} + d_1 b^{k-2} + d_2 b^{k-3} + \cdots + d_{k-1}} {b^{k-1}}$$

Both top and bottom are integers and so $$x$$ is a rational number.


 * Now suppose $$x \in \R$$ were to recur in some number base $$b$$.