A Binary Expansion

Number Theory Level 5

$\Large \alpha = 0.10110111011110 \cdots$

Let $\alpha$ be a real number written in base 10, that is, the $n$ th digit of $\alpha$ is 1, unless $n$ is of the form $\dfrac{k(k + 1)}{2} - 1$ , in which case it is 0. Choose all correct statements from below.

$\alpha$ is an irrational number.
For every integer q $\ge$ 2, there exists an integer r $\ge$ 1, such that $\dfrac{r}{q} < \alpha < \dfrac{r + 1}{q}$ .
$\alpha$ has no periodic decimal expansion.
$\alpha$ has periodic decimal expansion.

1,3 2,3 2,4 1,4 1,2,3 1,2,4

1 solution

Miles Koumouris
Dec 22, 2017

The decimal expansion is not periodic, so 4 is incorrect, and 3 is correct. All rational numbers have periodic decimal expansions, and vice versa, so 1 is correct. For $q\leq 9$ , there exists no integer $r\geq 1$ satisfying the inequality $\tfrac rq<\alpha$ . Hence, the correct statements are $\boxed{1,3}$ .

For interest's sake, $\alpha =\dfrac{100}{9}-5\sqrt[8]{10}\cdot \vartheta_2\left(0,\dfrac{1}{\sqrt{10}}\right),$ where $\vartheta_a(x,q)$ denotes the Theta function .

A Binary Expansion

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