# Generalizing the notion of "radix" to rational, irrational, or even imaginary numbers

I know to covert any natural number to any natural Base. For eg., $$10=1\cdot2^3+0\cdot2^2+1\cdot2^1+0\cdot2^0$$ Hence ten in binary is $$1010$$.

But what if I want to expand this limit of the radix from natural numbers to, say, rational numbers or, say, to irrational numbers(!), or maybe, even to imaginary numbers?

Does this notion of "Radix" even make any sense? Or will this open a new "field" of mathematics?

What will be the pros and cons of generalizing the base of numbers like this? Is this be practical?

• You can also write rational numbers in binary for example, $\frac 34 = \frac 12 + \frac 1{2^2}$ so in binary it is $0.11$ Jul 25, 2021 at 9:21
• I'm honestly a bit careful about this question, you see, all my previous questions have negative votes, and I don't want the same with this one too,. So can you please point out if there is any mistake in mistake in this question which I can work upon? Your suggestions are welcome : ) Jul 25, 2021 at 10:12
• @StackpackedKar what about en.wikipedia.org/wiki/P-adic_number ?? Does this look like something you need? Jul 25, 2021 at 10:55
• thanks everybody. Jul 25, 2021 at 11:01
• how much time should I wait before accepting an answer? Jul 28, 2021 at 15:08

The problem of representations not being unique also arises (in a more tame form) with integral bases too -- in base ten we need to accept that $$0.9999\ldots$$ and $$1.000\ldots$$ represent the same real number, which occasionally confuses students a lot. But at least in an integral base, each number as at most two representations. In an non-integral basis, a number might have infinitely many different representations, and it would be very difficult to determine if two representations denote the same number -- or if they don't, then which of them is larger.