How is the base of a counting system defined

Question

I've just come across the mayan counting system, and I'm told it's base 20.

Here the mayans have their system as base 20, where they considered each of these as individual symbols despite the fact some symbols are clearly compounds of others. To me, the symbols remind me of a tally and thus I would consider using these symbols to represent base 5 to make more sense.

This got me thinking, what defines what base a counting system is in? What is it about the way the symbols are used/defined that determines what base is being used?

Answer 1

Conventional (Hindu-Arabic) numerals are a base-$10$ system because the value of a symbol depends on the position in which it is written, and each position multiplies the value of a symbol by $10$: the 3 in 371 is worth ten times as much as the 3 in 35, which is worth ten times as much as the 3 in 123.

Each position in the Mayan system is worth twenty times the previous position. There is no symbol which, written in one position, means five times as much as the same symbol in another position. You cannot write a dot in one place to mean $1$ and in different place to mean $5$; you can write a dot in a different place to mean $20$. A bar means $5$ or $5\cdot20$ or $5\cdot20\cdot 20$; it never means $5\cdot 5$ or $5\cdot 10$. Too write $10$ you need two bars, which, written in a different position, mean $10\cdot20$. So the system is pure base $20$.

There are hybrid systems. For example the same symbol in Babylonian numerals might mean $3$ or $3\times 6 = 18$ or $3\times 60 = 180$ depending on where it is written (or $3\div 10$ or $3\div 60$ for that matter). But the Mayan system isn't a hybrid in this way.

Answer 2

A number system is "base $N$" only when it has a place-value system with $N$ different values that can be inserted in each place.

The Mayan numerals $0$ through $19$ indeed do not show that they are "base anything," since in a base-$20$ system these numbers are all written as a single "digit" (in the "ones" place of the place-value system). Until you write a number that uses at least two places of the place-value system there is no indication of a "base" at all.

Here is a site where you can convert decimal numbers to Mayan notation to see how its place-value system works. Unfortunately this calculator seems to be able to write only three digits in Mayan notation, but that is enough to intuit that it is base-$20$ if you care to try it.