The Mayans devised a counting system that was able to represent very large
numbers by using *only* 3 symbols, a dot, a bar, and a symbol for zero,
or completion, usually a shell. The chart above shows the first complete cycle
of numbers. Like our numbering system, they used place values to expand this
system to allow the expression of very large values. Their system has two
significant differences from the system we use: 1) the place values are
arranged vertically, and 2) they use a base 20, or *vigesimal*, system.
This means that, instead of the number in the second postion having a value 10
times that of the numeral (as in 11 - 1 × 10 + 1 × 1), in the Mayan
system, the number in the second place has a value 20 times the value of the
numeral. The number in the third place has a value of (20)^{2}, or 400, times the value of the numeral.
This principle is illustrated in the chart below.

Sometimes this number will be expressed in the shorthand 3.10.6.13.17 in writings on the Mayan numeration system, especially when discussing dates that are recorded in stelae or monuments. Using this system for expressing numbers has 2 advantages: 1) large numbers can be easily expressed, so long time periods can be recorded; and 2) simple arithmetic can be easily accomplished, even without the need for literacy among the population. In the marketplace, sticks and pebbles, small bones and cacao beans, or other items readily at hand can be used to express the numbers in the same way that they are expressed on the monuments or in the books of the upper classes. Simple additions can be performed by simply combining 2 or more sets of symbols (within their same set). This is shown below.

For more complicated arithmetic, you must simply remember that you borrow or carry only when you reach 20, not 10, as shown below.

It is important to note that this number system was in use in Mesoamerica while the people of Europe were still struggling with the Roman numeral system. That system suffered from serious defects: there was no zero (0) in the system, and, as opposed to the Mayan system, the numbers were entirely symbolic, without direct connection to the number of items represented.

It is not known whether a system was developed for multiplication and division.

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