Does the word "integer" only make sense in base 10?
I've always wondered this and have never seen it really discussed anywhere.
We all understand the typical definition of an irrational number, where a number can't be expressed as a ratio a/b where a and b are integers and b is nonzero.
So what if we were working in base pi? Then we could simply write pi as 10/1 or something.
However I don't think we can remove the irrationality of a number by simply changing bases, but I wanted to see if there were maybe other definitions that might help here.
Best Answer
The naturals come from adding 1 a finite number of times to 0. The integers include these and their negatives. Neither of these statements refer to the number base needed to express them as numerals. Then the rationals again are ratios of integers without reference to base.
The equivalence of this definition of rational to terminating or repeating numeral expansions is dependent upon the base being a rational. If you want to write numbers in base $\pi$, it would be natural to write $\pi=10_\pi$, but then $4$ and $9$ which will have "irrational looking" expansions. This is an artifact of using an irrational base.