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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?

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    $\begingroup$ You can also write rational numbers in binary for example, $\frac 34 = \frac 12 + \frac 1{2^2} $ so in binary it is $0.11$ $\endgroup$ Jul 25, 2021 at 9:21
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    $\begingroup$ 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 : ) $\endgroup$ Jul 25, 2021 at 10:12
  • $\begingroup$ @StackpackedKar what about en.wikipedia.org/wiki/P-adic_number ?? Does this look like something you need? $\endgroup$
    – Spectre
    Jul 25, 2021 at 10:55
  • $\begingroup$ thanks everybody. $\endgroup$ Jul 25, 2021 at 11:01
  • $\begingroup$ how much time should I wait before accepting an answer? $\endgroup$ Jul 28, 2021 at 15:08

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There are positional notations with bases that are not positive integers:

  • In systems with negative bases, both positive and negative numbers can be expressed with a single sequence of digits without any prepended sign for negative numbers.

  • Systems with complex bases extends this to being able to can express arbitrary complex numbers with a single sequence of digits.

These systems work well enough mathematically -- the reason they're mostly considered just intellectual curiosities is that it can be difficult for human readers to grasp numbers at a glance. They could be used for arithmetic internally in computers, but they still make life complex for the programmer, and in the majority of cases this outweighs the few theoretical nice points they have.

Systems with fractional or irrational bases can be imagined, but there we quickly begin to lose nice properties. The number of different digits we work with is by necessity always a natural number -- and since it cannot match an irrational base we either end up in a situation where there are significant gaps in the number line that we cannot represent (because there are too few digits), or we end up with numbers that have many different representations.

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.

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