Binary representation of real numbers without dots

Question

How can I represent a real number using only 0's and 1's? I do not want to use any extra symbol like '.' to separate the integer part and the mantissa.

Answer 1

Well, the function

$$f(x) = \frac{\arctan x + \pi/2}{\pi}$$

is a bijection from $\mathbb{R} \to (0,1)$. So when you take a number $x$ and compute $f(x)$, you obtain a number in the form of $0. [\dots]$; then any number may be represented only by the mantissa, since the integer part would be $0$ for all of them anyway, and since $f$ is a bijection, each mantissa uniquely identifies a real number.

Hence, you may just pick a random $x \in \mathbb{R}$, find its binary representation, and then use it to compute $f(x)$ in binary representation as well; the result will once again be a number whose integer part is $0$ as before, but this time the mantissa will only consist of $0$'s and $1$'s because of the binary representation; and for the reasons stated above, this mantissa can be the representation of $x$ you need for your purposes.