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The Mayan number system

Watch Erika and Eric Roldan's videos on Mayan arithmetic.
[music] The Mayan was a great Mesoamerican culture that had settled in Central America from 2000 BC to around 900 CE. They had a sophisticated number system. This allowed them to have precise astronomical measurements, and to develop accurate calendar calculations. They are also believed to be the first civilisation to have the concept of Zero. But how does the number system work? Well, to answer that question we can start by analyzing our current number system. Our number system is written using base 10. Because it is base 10, we need ten symbols to represent any real number that we desire to write down. We know perfectly well those symbols.
Those are the following: The order in which we arrange two or more of these symbols to represent a number is relevant, for example, it is not the same to be 19 years old than to be 91 years old. This happens because a position that a digit-symbol has on a number will give to it its different value for example, the digit 1 in the first position of a number stands for the number 1. But if the same digit goes to the second position, now the value is 10. And if we move again the digit to the third position, now its value will be 100, and so on…
As you can see, when you move a digit-symbol one position, you multiply by 10 the value the digit had in its previous position.
This is why, for instance, the number 7042, can be broken into this: Notice that we had summed the position value of each digit that formed the number 7042. Using power notation and setting that any number to the power of zero is one we have an alternative way to express this number. Now let us go back in space-time to analyze the Mayan system. The Mayan used a vigecimal number system this means that instead of representing the numbers in base 10 as we do, they had a number representation based on base 20. Remember how our number system has ten symbols because it is based in base 10.
Well, since the Mayan number system is based in base 20, they used 20 symbols to represent any number. It’s very useful to memorize the value in base 10, of each one of these Mayan symbols. What difference to you see between the symbols that we use in the base 10 numerical system, and Mayan number system symbols? That’s right! The symbols used in base 20, only use three different objects to form the whole set of twenty symbols. The dot which stands for a rock, the line which stands for a stick, and this little graphic that stands for a shell. We have now all the elements we need to learn to read Mayan numbers and to know their value in base 10.
What difference to you see between the arrangement of the symbols? Good answer! Now the symbols are in a column. When you move a symbol from bottom to top, the value of the symbol is multiplied by 20 with the correct position exponent. Now you know how to transform a number represented in Mayan number system, to a number with the same number represented in base 10. Let us analyze an example of how to transform a number represented in base 10, to a number represented in the Mayan numerical system. For instance, 1455, which in base 10 needs four digit positions to be represented. It turns out that it only takes three vertical digit positions to represent this number in the Mayan number system.
We have now all the tools needed to know how to do Mayan arithmetic. See you in the next session to learn how to sum, subtract, multiply and divide with Mayan numbers.

Erika Roldan is a mathematician from Mexico, who specialises in the history and philosophy of math and the popularisation of math in South America. I first met Erika at the Gathering for Gardner recreational math, magic and art conference in Atlanta.

In the this video and the following, with the help of her brother, Eric, they explain the Mayan base-20 number system and how to do Mayan arithmetic.

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