When numbers were dotty: Marcus du Sautoy on Mayan mathematics and the way we count

2012; is it going to be the end of time, or the beginning of a new time??? Marcus Du Sautoy explains the Mayan number system and the relation it has with our time. Read on!!

Doom-mongers across the world are already warming up as we approach another of those dates that harbinger the end of the world. Fuelled by the Hollywood blockbuster 2012, December 21 2012 is now down in the diary as the date time will end. The reason for the choice of dates is thanks to a bit of mathematical counting by the Mayans.

The Mayan number system

The Mesoamerican culture of the Maya was at its height from AD200 to 900 and extended from southern Mexico through Guatemala to El Salvador. According to the creation account documented in the Mayan text the Popul Vuh, the universe containing humans began on August 11 3114BC. The prediction of the end of this universe is based on the way the Mayans count.

To write down their numbers, they had a very simple system. They used a dot for 1, two dots for 2, three dots for 3. Just like a prisoner chalking off the days on the prison wall, once they got to five, instead of writing five dots, they would simply put a line through the four dots. So a line corresponds to 5.

It is interesting that the system works on the principle that our brains can quickly distinguish small quantities – we can tell the difference between one, two, three and four things – but beyond that, it gets progressively harder. In the Hindu-Arabic numerals that we use today, we have symbols for all the numbers up to 9 and then instead of creating a new symbol for 10 (like the Romans did) we write 10, where the positioning of the one indicates “one lot of 10”.

The Mayans, however, used their dots and dashes to count all the way to 19 (written as three lines with four dots on top) before they started a new column which was used to keep track of the number of 20s.

The fact that we work in 10s as opposed to any other number is purely a consequence of our anatomy. We use our 10 fingers to count and this has led to a number system that has 10 symbols. It is believed that the reason the Mayans count all the way to 20 before starting a new column is because the Mayans used both fingers and toes to count. If we had a different anatomy we would probably have developed a different system of counting. For example, The Simpsons have eight fingers so they would have developed a counting system that went 1, 2, 3, 4, 5, 6, 7 and then went 10 to record the number 8.

The Mayans weren’t the first to develop an alternative to our base 10 system. Around 2000BC, the Babylonians became one of the first cultures to use the idea of a place-value number system. But instead of using powers of 10, the Babylonians developed a number system that worked in base 60. They had different combinations of symbols for all the numbers from 1 to 59, and when they reached 60 they started a new “60s” column to the left and recorded one lot of 60, in the same way that in the decimal system we place a 1 in the “10s” column when the units column passes 9.

The choice of base 60 is in some ways much more mathematically justified than the decimal system or the Mayan system based on 20. It is a highly divisible number which makes it very powerful for doing calculations. For example, if I have 60 beans, I can divide them in a multitude of ways (30x2, 20x3, 15x4, 12x5, 10x6). We see many hangovers of the Babylonian base 60 today. There are 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle.

In our number system, when we write 100, the third column denotes the number of hundreds or 10x10s. So in the Babylonian system, the third column indicates the number of 60x60s or 3,600. According to this principle, the third column in the Mayan system should have denoted the number of 400s (20x20), but bizarrely it represents how many 360s (20x18) there are. This strange choice is connected with the cycles of the Mayan calendar. One cycle consists of 18 months of 20 days. (That’s only 360 days. To make up the year to 365 days, they added an extra month of five “bad days”, which were seen as unlucky.)

According to Mayan culture, one cycle of time comes to an end when the number of days since the beginning of creation clicks over to record a 13 in the fifth column and 0s in the other four columns. Like kids in the back of the car waiting for the milometer to click over, this is the reason the doom-mongers have picked out December 21 next year as the end of time. Most Mayans, however, are far less apocalyptic. They are quite content to start afresh and set off counting from the beginning again. -The Telegraph