# Minimum number of symbols required for a base-N positional number system

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?

• First one needs a definition of e.g. what a base-9.9 system is. – coffeemath Feb 10 at 12:01
• 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". – Peter Feb 10 at 12:27
• @coffeemath I saw an application of the $\phi$-base system here : math.stackexchange.com/questions/3103634/…. Does such a system actually have an application ? – Peter Feb 10 at 12:30
• It is also problematic to make , lets say, a $\pi$-base represenation unique. – Peter Feb 10 at 12:33
• @coffeemath I've defined what I mean (hopefully) more clearly. – Daffy Feb 10 at 23:57

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.