# Determine base required to accurately represent number

If you have an expression like 1/3, that can't be accurately represented in base 10. It would look like 0.3333333.... However, it can be represented in base 3 as 0.1. Is there a way to mathematically find a base that can accurately represent the results of a given expression?

• You mean any rational number? Then yes, there is some base in which that number has a finite expansion. This is not the case for an irrational number (because if there were such a finite expansion in some base, then the number would be rational.) – Simon S Apr 7 '15 at 21:31
• I'm not sure if there always is a way. What if the number is not rational. I do find it interesting though that if $x = \frac{1}{3} = 0.33333....$ then $10x = 3.3333....$ so $10x-x=3 => 9x = 3 => x = \frac{1}{3}$ so the base $10$ can be used in some way but that also only applies to recurring decimals – StephanCasey Apr 7 '15 at 21:34