# Mayan Number System Explained.

I have recently been studying the Mayans and have encountered their number system.

A dot represents 1

A line represents 5

A shell represents 0

The base of the number system is 20

During my research I understand how the numbers were written vertically each row ontop of each other was the new power of 20 starting at $20^0$ in row one.

I have attached a picture below I have little understanding of numbers over 20

Also 401 would be a dot over a shell over a dot.

Why is this true and how do the exponents and multiplying work in this system?

Further research led me to understand how 401 is a dot over a shell over a dot the dot would be multiplied by $20^2$ the shell (0) would be multiplied by 20^1 and the dot on the bottom would be multiplied by 20^0. These would all be added to get 401.

I feel this is not consist throughout the system. In this chart I have just found, the third row from the bottom is multiplied by 360 instead of $20^2$

Why is this? Is this chart correct.

See:

Another description of the Mayan numbering system, including important historical facts explaining how limited our knowledge is, is at the site http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Mayan_mathematics.html.

We have only a few surviving documents from the Mayan civilization. We have examples of large numbers written in a not-quite-base-$20$ system, in which the place value always increases by $20$ except once: the place value above $20$ is $360$ instead of $400$. But those examples are said to represent numbers of days in calendar calculations, and we think it would have been more convenient to deal with groups of $360$ days at a time and account in some other way for the extra $5.2422$ days that the Mayans knew the average solar year contains, rather than have groups of $400$ days, which would be $34.7578$ days too many per year.

We too have an irregularity in the way we tell time, in that it takes $60$ seconds to make one minute, $60$ minutes to make one hour, but only $24$ hours to make one day. Why not use factors of $60$ in all three places? This is much stranger than the Mayan system, which at least has the excuse that the day and the year are naturally-occurring units of time that we cannot redefine for our convenience; if we had all been born in a world in which the day was divided in $60$ equal parts instead of $24$, would we ever have noticed that there was anything wrong with our timekeeping?

Whether the Mayans used a true base-$20$ system when they were not writing calendars seems to be a matter of debate among historians. It seems that it is very hard to find good evidence supporting either side of that debate, due to the lack of surviving documents that would have used such numbers.

This is how base 10 works with the digits 0-9:

10^2 place    10^1 place     10^0 place
5             9              1        = 5x10^2 + 6x10^1 + 1x10^0 = 591


For base 20, we use the digits 0-9 and A-J (A is 10, B is 11, ..., J is 19). The same number, 561, in base 20 would be

20^2 place    20^1 place     20^0 place
1             9              B        = 1x20^2 + 9x20^1 + 11x20^0 = 591


Instead of writing the digits from left to right in order of significance, the Mayans wrote them tom to bottom. They also used different 'digits': 1 was a dot, 2 was two dots, 5 was a line, 17 was three lines and two dots, and a shell represented a zero. Note that these 'digits' too went from 0 to 19.

By 'exponents and multiplying' I guess you mean the $9 \times 20^1$ part of the number system. If you meant actual multiplication of two numbers (e.g. $143 \times 67$) or exponentiation (e.g. $12^{31}$ or something) I have no idea how they did it and whether this particular number system was good/bad for that.

Feel free to ask for clarification in the comments if needed.