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

I am assuming that you mean to ask which rational numbers can be expressed in binary using a finite representation. I am deducing this from your example of 1/3 in decimal. The fact is that every rational number can be expressed in binary if you allow infinite representations. But this question is only interesting from a computer science perspective if you only permit finite representations. I am further assuming that you are not asking about specific computer representations (say, IEEE 754) but rather merely asking about general positional representations.

A rational number p/q with (p, q) = 1 can be expressed a finite representation in base b if and only if every prime factor of q divides b. No irrational numbers have a finite representation in any base.

In particular, a rational number p/q with (p, q) = 1 can be expressed as a finite representation in binary if and only if every prime factor of q divides 2. That is, the only rational numbers p/q with (p, q) = 1 that have a finite representation in binary are those where q = 2^k for some nonnegative integer k. Moreover, all such rational numbers can be expressed in a finite representation in binary. These numbers are known as dyadic rationals.