It’s my understanding that our base-10 system of counting arose out of the evolutionary accident of two five-fingered hands. If this is true, is the decimal system considered to be universal out of pure primate solipsism? Or did we happen to just coincidentally luck into the right number of digits (double meaning intended)?
What if we had 12 fingers? Or an asymmetrical 11? Would pi still be pi? Would it still be irrational? If base-10 is not universal, what was the point of engraving prime numbers on Voyager’s gold LP? I mean, our prime numbers are only prime in a decimal system, right?
What other things that we think of as universal would be different? Or even impossible?
To elaborate: We could use the simplest numeration system imaginable:
| for 1,
|| for 2,
||| for 3,
…
||||||||||||||||| for 17,
etc.
||||||||||||||||| is still a prime number. There’s no way to arrange those marks into a rectangle (other than a 1x17 or 17x1 rectangle).
I’m not sure I understands the question about “universality” since we’ve historical evidence of a Babylonian counting systems with 60 as the base which gives us divisions of arc and time measures. Leaving that aside, base 10 is just a positional numbering system that’s neither good nor bad.
The base we use only affects what numbers look like when we write them down. It doesn’t affect their properties as numbers or the way they relate to one another. In base 11 or base 12 or base 8, 6 + 7 or 5 * 3 would still be the same number, but we’d write it differently. Pi would still be pi, but we wouldn’t write its value as 3.14159… It would still be irrational, though.
An irrational number is a real number that is not a rational number.
A rational number is a (number that can be written as a) ratio of two integers: a/b (b ≠ 0).
So, while pi would indeed be written as 10 in base pi, it would still be irrational. Irrationality is defined with respect to the integers, not with respect to how numbers are written in a particular base.
As already noted, there is evidence of the Babylonians using a base 60 system - our use of 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle, minutes and seconds as measure of arc can all be traced back to that.
Counting certain items by 12 is commonplace - hence the terms “dozen” and “gross” (a dozen dozens).
Several languages - Celtic influenced ones like Irish, Scottish, and French (back to the Gauls, which were a Celtic group) and others have traces of a base-20 system. We even have a bit of that in English when we speak of a score of something.
If I recall correctly, meso-Americans used a base-20 system as well.
So, while base 10 seems pretty common both historically and today, no, it’s not universal. Two, five fingered hands might have contributed to the commonness of base-10, but base-20 is “four hands” (perhaps an extension, in some cases, of units of five represented by a five-fingered hand, or maybe not).
The idea of a place-value number system might be universal. But the choice of base is totally arbitrary. We (at least, most of us nowadays) happened to choose base 10, because that’s our number of fingers. But any other choice of base would have been just as mathematically valid.
True, although there might be practical considerations that would make a base very much larger (or smaller) than 10 awkward to use.
For instance, a true base-60 system that works the way ours does would require learning 60 separate symbols for numbers, and a relatively huge multiplication table. But maybe that would be child’s play for some hypothetical alien race that used such a system.
Even human hands don’t require a base-10. You can have a base-12 system by counting the joints in your fingers. As you count, start touching each joint of your fingers with the tip of your thumb, starting from the tip of the index to the base of the pinky. Three joints per finger, 12 total. Use both hands and you have base-24.
(Edit–and written while jasg was posting the above.)
In fact, for everyday use, we would have been slightly better off if we had evolved with an extra finger on each hand. Base 12 would have made things slightly easier because of the factorability. The most commonly used fractions are 1/2, 1/3, 1/4. In base 10, 1/3 is a repeating decimal and 1/4 is two places. In base 12, those are all 1 place representations (I guess we can’t call them “decimal”). 1/5 becomes repeating, but it’s an equitable trade for not having an irritating repeating digit floating around every time you divide something other than an integer quantity with a factor of 3 into thirds. A larger multiplication table you would have to learn as a kid, but manageable.