 # Is the base of a number counting system irrelevant to mathematics?

Let’s say humans had 6, or 8, or 12 fingers, and used a different number base for their mathematics than 10. Would this effect anything in terms of the ways mathematics formulas and geometric paradigms are constructed, or is base essentially irrelevant to the fundamental nature of math theory.

The base of the number system is utterly irrelevant to mathematics. There are a number of practical reasons why a base 8 or 16 is preferable, but this has nothing to do with mathematics.

To head off the inevitable next question, the base doesn’t even have to be an integer. It can be any number, rational or irrational. Nothing in math changes, except for the physical expression of the results.

You can even use non-basis represntations (e.g., the Cantor expansion).

How does one count with non integer bases? 1, 2, 3, 3.1415… ?

Okay, take pi as your base.

1 is pi^0 or 1
10 is pi^1 or pi which is about 3.141
100 is pi^2 which is a little more than 9
etc.

0.1 is pi^-1 or 1/pi, which is somewhere around 0.3

Rational numbers will have an infinite number of digits and (some) irrationals will have a finite number of digits.

Digits is the wrong word because it implies base ten, but you know what I mean.

The trick for multiplying by nine wouldn’t work in any other base, but there would (I think) be a similar trick for whatever number was one less than the base (assuming you use an integer)

That still does not answer my question. In say base 8 you have 8 symbols 0,1,2,3,4,5,6,7. What symbols do you have in base pi? If pi is to weird because it is irrational use 3.5. 0,1,2 then what?

I wish I had 3.14159265 fingers.

Tripler
Mmmm hmm.

Yea…ummmm…thanks for that. I guess I just won’t sleep for a few days.

It’s also possible to work out how to use a negative base for the number system and how arithmetic could be done in such a system.

Well, base zero might present a few difficulties.

And, base one isn’t much better.

1. In any base n, the symbols are integers in the range [0, n).

arithmetic is particularly simple in base 2 compared to base 10.

Division becomes shift and subtract.

These techniques will work in other bases, but each stage requires a variable number of iterations, where in binary, either a step is done or it is skipped.

As an example, when dividing in base 10 you have to estimate how many (0-9) times the denominator “goesinta” the numerator, make a trial multiplication,compare, and either subtract or adjust the estimate and iterate.

In binary it either goesinta or it doesn’t, which can be determined by inspection.

You use arbitrary symbols, just as in any other base. 1, 2, 3, 4, and 5 are utterly arbitrary symbols.

In hexadecimal, the “digits” are 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

ö, ÷, ø, ù, þ, and ÿ can be the first six representations of an irrational base, defined to mean what the equivalents are in that base.

Expression is completely arbitrary. We’re just conditioned to think in terms of base 10, but that’s cultural, not mathematical.

There was an interesting book that I read a long time ago, an alternate history where early man competed with advanced reptiles for supremacy, and the reptiles were very culturally superior. Anyway, one of the things that you just figure out from context is that the reptiles have, I think, eight fingers and therefore their math is base 8.

I understand all that. I am just confused about how many symbols are needed for base 3.5. For base 2, 2 symbols are needed for base 10, 10 symbols are needed what about for base 3.5? Do you need 3 symbols or 4 symbols?

As TJDude said above, you’d need 3 symbols: 0, 1, and pi. Pi is only 3.1415… in base 10. In base pi, pi would simply be pi.

How would you represent 2 given those symbols?

What? Where is TJDude saying that. There is something I am missing here. Why would there be a symbol for pi? In base 10 there is no symbol for 10. It is made with two symbols?

There is one area where the number base has a practical relevance: the conversion of fractions to “decimal” numbers. In base 10 the only fractions that can be converted to a precise decimal number are those whose denominators are whole number multiples of 2 and/or five - the factors of 10. Thus in base 10 the fraction 1/3 converts to the imprecise 0.333… . But in base 3, 1/3 is 0.1, the first positon to the right of the decimal, er, trimal point representing thirds (in the same base, 1/2 is 0.111… [one third plus one ninth plus one twenty-seventy &c. &c.).

I wouldn’t dare try to restate 1/3 in base pi. . . .