I know that when converting a decimal number from base 10 to base 2, the result might be not terminating, even though the number is terminating in base 10.
For instance, 0.2 -> 0.0011 0011 0011 …
Is the contrary possible ? That means, is there a non-terminating decimal number which, converted to base 2, is terminating ? I can't find any !
If yes, is there a method to find them ?
And by the way, what about irrational numbers ? Could an irrational number in base 10 become rational in base 2 or in another base ?
Best Answer
Following TMM's comment, for an expansion to terminate in base $10$, you need to be able to write it as $\frac a{10^n}=\frac a{(2\cdot 5)^n}$ for some naturals $a, n$ For it to terminate in base $2$, you need to be able to write it as $\frac b{2^m}$ for some naturals $b,m$ Given that it terminates in base $2$ with some $b,m$, you can take $n=m, a=2^mb$ to show it terminates in base $10$. If it terminates in base $10$, with some $a,n$, you would like to set $m=n, b=\frac a{5^n}$, but the last may not be a natural and these are the ones that will not terminate. Your example of $0.2_{10}$ fits this exactly.