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I have been doing work in various bases and have been convincing myself of the fact that our base-10 number system is not special at all. Any other base ($2$, $16$, hell, even base $100$ could work if you can find enough characters to assign).

This got me thinking, would math work just as freely in non-integer bases?

Let me explain. A non-interger base would behave as integer bases do, with each digit being multiplied by a subsequent power of the base.

Example: $154$ in base-$\sqrt 2$ would represent $1\cdot (\sqrt{2})^2+5\cdot (\sqrt{2})^1+4\cdot (\sqrt{2})^0$.

This leads to some interesting questions about which digits to use, however.

In base-$10$, there are only $10$ digits. This idea translates easily to other integer bases, but the idea of a number of digits falls apart in non-integer bases, in fact, the idea of which digits to use seems vague too (why should $1$, $4$ and $5$, all integers, be digits in a base-$\sqrt 2$ system.)

However, the idea of only $10$ digits in base-$10$ is somewhat unnecessary. As weird as it is, you can use any digit (even non-integer digits) in base $10$.

Example: $(2)( \frac {1} {2})(50)(-3)_{10}=2\cdot 10^{3}+\frac {1} {2} \cdot 10^{2} + 50 \cdot 10^{1} + -3 \cdot 10^{0}=2547_{10}$

Some of you, myself included, may be cringing at the notation I've used for non-integer and non-positive digits, but when has bad notation ever stopped any of us?

I think the question quickly becomes whether we could, not whether we should. Obviously, we shouldn't, but again, the purists among us live life without worrying about math's "shoulds".

Basically my question is if any of you have seen any work, serious or not, on this type of base system. I would love to hear what different fields of math would be changed in different ways. Obviously certain numbers could be represented in easy ways (we could calculate ALL the digits of $\pi$ in base-$\pi$).

From first thought, the idea of primes seems difficult to grasp, but I'm sure many other things are build natually from integer bases, if not from a base-$10$ system.

Magna Wise
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  • I'd normally suggest that this question needs improvement with more thought/research, but I upvoted it because I'm curious to see what people come up with. Maybe someone should add a 'soft-question' tag? – JonathanZ Apr 26 '20 at 15:19
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    @Oliver Murfett: For an integer $b > 1$, the key benefit of the standard base-$b;$system is the uniqueness of representation. Once you lose that, the usefulness is not clear. – quasi Apr 26 '20 at 16:04
  • @quasi No positional number system can represent every number uniquely. For example, in base-10, $0.\overline{9} = 1$. However, in integer bases it is a lot easier to classify the numbers without unique representations. – Xander Henderson Apr 26 '20 at 18:27
  • Interestingly enough, this question seems to have come up on MSE in several forms in the past (see, for example, [1], [2], [3]). I'm not particularly satisfied with any of the answers, however... – Xander Henderson Apr 26 '20 at 18:31
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    Base 100 works fine. Suppose we agree to represent the 100 digits as 00, 01, … 99. Then the base-100 representation of the number $142857$ is 14 28 57. Easy-peasy! Note by the way that computers typically represent numbers in base 256, and this is exactly the method used by programmers to write such numbers on paper: the 256 possible digits are written 00, 01, … 09, 0a, 0b, … 0f, 10, … 1f, … ff, and the number $142857$, which in base 16 is represented as $22e09$, is written in base 256 as 02 2e 09. – MJD Apr 26 '20 at 19:39

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