What number in binary can only be represented as an approximation?

Question

In decimal (base 10), 1/3 can only be approximated to 0.33333 repeating.

What number is the equivalent in binary that can only be represented as an app

Answer 1

It's every number that can't be expressed as k/2^n for integer k and whole number n.

The easy way to find all these numbers is to write down some prime numbers that do not include 2. 3, 5, 7, 11, 13, 17 and 19 are good examples of prime numbers that don't include 2.

Start multiplying. 1/3, 2/3, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, etc.

if you do this -- and you avoid numbers of the form k/2^n -- you'll enumerate every possible fraction that cannot be exactly represented in binary.

You should probably stop enumerating when you get to numbers for which the left-most 64-bits are all identical.