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A base-10 number system requires 10 symbols. Any less, and you would not be able to represent every number. (Is this a correct assumption?) A base-16 system requires 16 symbols. How can I tell how many symbols a base-3.14 system takes. Or a base-9.9 system.

I'm defining a base-N system as a finite set of symbols $S$, and choosing a combination of these symbols $...s_3s_2s_1s_0.s_{-1}s_{-2}s_{-3}...$ where $s_x \in S$ results in the value $\sum\limits_{x=-\infty}^{\infty} s_xN^{x}$

Applying this to base-4.5 with the set of symbols ${0, 1, 2, 3, 4}$, we can take the number 1342.23 and evaluate it like this.

$$1*4.5^3 + 3*4.5^2 + 4*4.5 + 2 + 2*4.5^{-1} + 3*4.5^{-2} \approx 172.4676$$

My question is how small can I make $S$ for any given base-N system?

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    $\begingroup$ First one needs a definition of e.g. what a base-9.9 system is. $\endgroup$ – coffeemath Feb 10 at 12:01
  • $\begingroup$ The first statement is false. We only need two digits. The binary system can represent every number. In the case of a ( rather artificial ) base-system of a non-integer number, you have infinite many "digits". $\endgroup$ – Peter Feb 10 at 12:27
  • $\begingroup$ @coffeemath I saw an application of the $\phi$-base system here : math.stackexchange.com/questions/3103634/…. Does such a system actually have an application ? $\endgroup$ – Peter Feb 10 at 12:30
  • $\begingroup$ It is also problematic to make , lets say, a $\pi$-base represenation unique. $\endgroup$ – Peter Feb 10 at 12:33
  • $\begingroup$ @coffeemath I've defined what I mean (hopefully) more clearly. $\endgroup$ – Daffy Feb 10 at 23:57
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Without specifying exactly how numbers in the base-$n$ system are represented, we can't say that $n$ symbols are needed—since each "digit" might use more than one symbol. Here are two examples.

  1. We routinely represent time of day in base $60$, writing values like $15$:$18$:$32$. This uses only $11$ symbols ($10$ numerals and the colon).

  2. In computing or circuit design it's sometimes useful to represent numbers in binary-coded decimal (BCD) format. This represents each decimal digit by four binary digits (ie bits), thereby using only two symbols to represent a base-$10$ number. (Also the real symbols are voltages or similar in the machine, representing the $0$ and $1$ that we associate them with).

Of course, you might argue that the time example isn't base $60$ but really a sort of mixed system where going one place to the left multiplies by $6$ and $10$ alternately. Similarly you might argue that BCD isn't a base at all because a group of $4$ bits isn't allowed to go above $1001$.

Or you might argue that in the time example, each pair of digits counts as one symbol so there are really $60$ symbols after all.

So in the end it depends on how you're defining a base-$n$ system and a symbol.


P.S. you might want to look at the Babylonian number ssystem. They used base $60$, but achieved it by using one symbol to count units and another to count tens, assembling their base-$60$ digits out of those.

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