Decimal to Fixed Point Conversion
Let $x$ be the real number you wish to convert. We will assume without loss of generality that $0 \le x &lt; 1$. Form the Dedekind-completeness of the order on $\mathbb{R}$, it follows that $x$ can be (almost) uniquely written as

$$
x = \sum_{h=1}^{\infty} 2^{-h} \alpha_h
$$

where $a_h \in \{ 0, 1\}$ for all $h$. Now observe that:

$$
2x = \sum_{h=1}^{\infty} 2^{-h+1} \alpha_h = \alpha_1 +\sum_{h=1}^{\infty} 2^{-h} \alpha_{h+1}
$$

where the rightmost sum is nonnegative and less than $1$. Multiplying by $2$, therefore, allowed us to discover a single binary digit of $x$, namely $\alpha_1$. The same procedure can be iterated to yield as many digits as you wish.