Guatemala – a land of volcanoes and jungle, howler monkeys and the quetzal bird that no one seems to have ever seen except on the national flag. Pyramids soar above the jungle canopy, inducing vertigo in even the most hardened climbers. A proud indigenous population still wears their colourful dress despite the horrors such affiliations might elicit, following the civil war that raged here for 30 years, before peace was finally established in 1996.
But as I sit across from the volcanoes that surround the beautiful Lake Atitlán, I consider the mathematics that the Mayans created, and its importance alongside the other treasures in this country of wonders. And it’s mathematics that is still in evidence today, if you look carefully. I’ve got a 100 quetzales note in my wallet. It’s worth a little less than 10 pounds. In one corner of the note, 100 is written in the Hindu- Arabic numerals that are universally recognised across the world, but in the other corner is a strange symbol: a horizontal line with an ellipse below, struck through by another horizontal line and two little tear marks on one half of the ellipse.
This is how the ancient Mayans wrote the number 100 and the symbol has several fascinating features that illustrate the sophistication of the mathematics being done here in Guatemala, over 1,000 years ago. For a start, the Mayans didn’t count in units of 10, as we do when we use Hindu-fouArabic numerals, but in units of 20. We have symbols for the numbers from one to nine, but once we reach nine we then write ‘10’ to denote ‘one 10 and no units’. The Mayans on the other hand (or perhaps foot), kept counting with symbols to represent every number up to 19. Only when they reached 20 did they start a new column to denote the count of one lot of 20.
Choosing to count in units of tens, as we do, is nothing to do with any mathematical significance of the number 10, but relates exclusively to our anatomy. The Simpsons, who have eight fingers, presumably count in units of eight, rather than 10. The Mayans, who could see the toes on their feet, had 20 bits of their body to count on, which is the reason they count in units of 20. The symbols they use to count up to 20 are wonderfully economical, simply using dots and dashes to depict the numbers from one to 19. A dot is one. Two dots denotes two. Three dots is three. Four dots is four. But like a prisoner counting off the days till their release, instead of writing five dots the counter puts a horizontal line through four dots to represent the number five. So the horizontal line on my 100 quetzales note is the symbol for five. But it is the symbol below the line that is perhaps the most sophisticated Mayan invention. The ellipse is actually a graphic of a shell, and was the way the Mayans represented the number zero.
Very few cultures developed number systems incorporating a symbol for nothing. The Romans didn’t have one, which is why they needed new symbols each time they went from tens to hundreds to thousands. But the Mayans were a smart lot, and that shell indicates that my bank note consists of no units and five lots of 20 – making 100 quetzales in total.
One of the principle reasons that the Mayans developed such a sophisticated number system was to keep track of the astronomical passage of time. Their cycle of time is measured by the long count, which started on August 11 3114 BC. It uses five placeholders, and goes up to 20x20x20x18x20 days. That’s a total of 7890 years. Indeed the ‘end of the world’ that was due to strike humanity on December 21 2012 was simply the Mayan calendar ticking over, marking a new cycle of time. Like kids in the back of the car watching for the odometer to click over, we waited for the moment the Mayan date became 184.108.40.206.0. Fortunately the world didn’t end, leaving me time for my continuing mathematical adventures.
Marcus du Sautoy OBE is Professor of Mathematics at the University of Oxford