Simple calculations can be made using your fingers with the Maya vertical vigesimal system without grids, beans, or bars.
If we were conducting trade or simply needed to do some simple and quick math, gathering slaves to help count using their fingers and toes is not very convenient. Plus, what if we were a lower status merchant or priest and didn’t have any slaves to help us count.
Making a quick “grid system” on the ground would work, but we’d still have to gather sticks and stones to do the counting and that’s no good in a hurry.
So we must reason that the Maya would have simplified their system of basic counting so they could count on their fingers, like we do with our decimal system of 10’s. Simply using all our fingers and toes to count to 20 is NOT convenient, especially if you have your shoes on. The Maya simply had to have an easier way to count through their base system of 20 with just their hands. Especially when conducting trade.
The Maya didn’t count linear in 20’s like our current modern system does with 10’s. Additionally, the Maya didn’t have specific characters to designate specific numbers, like the Arabic system that uses specific symbols for 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The Maya used a system of dashes (lines) and dots to represent a value.
They used a tiered stack of lines and dots to represent number values. They didn’t use a specific symbol to represent a specific number. In fact, the Maya method of writing numbers never counts past four dots or three bars. The symbols always change and goes up to the next value.
A count past 4 dots is exchanged for a bar (value=5) and 4 bars (4×5) is exchanged for a dot on the next tier valued at 20.
<see : Ancient Maya Arithmetic for more information>
This is how they could have huge, long count numbers with relatively few symbols to represent the value amount.
The Maya were able to count and represent values using their fingers, but they did it a different way than we do today. They didn’t count using all ten fingers to reach the count of 10, they only used two fingers to represent 10. Let me explain:
Assuming the Maya counted in the same manner as we do, counting each finger with a value of 1. They would count up and stop with using four fingers up to the valuer of 4. At the next value (5), they would use a separate finger to represent the value of 5. Whereas, when we use our fingers to count, we use all five fingers on our hand to display the value of 5. The Maya only needed to use their fingers on one hand to count up to four dots, then count a bar value of 5 on a separate hand. They would then combine bars with dots to continue their count. The value of 6 was one bar and one dot, or one finger from each hand, representing a bar and a dot.
Okay, let’s break this down visually.
To count using the Maya vigesimal system, on one hand each of your four fingers represents a value of 1 (1 dot each). On the other hand, each one of your fingers represents a value of 5 (1 bar each). When you count, you count on your hand with the value of 1 in sequence of 1, 2, 3, 4.
On the first hand, count: 1, 2, 3, 4.
When you reach 5, close the fingers on the first hand and you raise 1 finger on the other in the value of 5.
Remember, one finger on the other hand represents five fingers on the first hand.
On the first hand, you continue the count: 6, 7, 8, 9.
Each finger on the left hand = 5, and each finger on the right hand = 1.
When you reach 10, all the fingers on the first hand go down and the second hand now displays two fingers representing the value of 10 (5 + 5).
The value of 10 is represented by two fingers on the second hand (5 + 5 = 10).
On the first hand, with two fingers on the second hand out, you continue the count: 11,12,13,14.
Then at the count of 15, raise an additional finger, totaling to three fingers which represents the value of 15 (5 + 5 + 5).
Thus, with three fingers out on the second hand, we continue the count on the first hand with: 16,17,18,19.
At the count of 20, you close the fingers on the first hand and display all four fingers (or alternatively use your thumb) on the second hand to the value of 20 ( 5 + 5 + 5 + 5 ( or with the thumb representing 1 value on the second tier, which equals 20 )).
Wow, that is amazing, I think we just unlocked the mystery to a Maya simplified counting system. We can easily count to 20 with just our two hands and use the same dot and dash system to represent counts of 1 for dots and dashes as 5’s.
The count is not in linear values counting to 20, it is in the value of ones and fives and their relative position of counting up in segments of 4 and 5, with the sum of 4 or 5 being the next value, represented by the next symbol and position. Simply when four of the the “fives” are used up in their sequence, they go to the next level or tier.
The value of one and five also changes as it goes up each level. This is why there are no number symbols in Maya counting, only dots for values of 1 and bars that represent the value of 5 dots. There is no “20,” there is only one dot on a second tier, which merely represents the value of four bars on the previous lower tier.
Using the ‘finger method,’ you can even alter the system slightly and give each thumb the value of 20 (one dot each on tier two in Maya thinking). With both thumbs having the value of 20 each, equaling 40 ( 2 x 20 ). The value of ‘one’ for each finger on the first hand ( 4 x 1 ), and the four fingers on the other hand with each valued at 5 ( 4 x 5 ). We can easily count to 64 ( 20 x 2 + 4 x 1 + 4 x 5 ) with just two hands. Whereas, previously we could only count to 10 using the same fingers with the decimal system.
Interesting as well, that we can count to 64 with our hands using the Maya system. It appears to be a number significant to a computer’s binary method of counting using on and off switches that we call bytes. There are 8 bits to a byte. A row of 8 bytes is 64 bits. Of course this multiplies up to 128, 256, 512, etc., which I am sure you recognize as numbers used in computers as: gigabytes (GB), megabytes (MB) and Kilobytes (KB).
A computer or calculator counts using a binary system of off and on switches, each with a value of 0 or 1. This is broken down into a sequence of 1, 2, 4, 8, 16, 32, 64, 128 , 256, 512 typically.
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binarycoded decimal numeral of the traditional sexagesimal time.
Without going into greater detail about binary counting and how to calculate using the binary system, we can see how the Maya counting system is similar to the binary system computers use. This means the Maya could use their system to perform complex calculations needed to accurately plot astronomy, calendars, and the mathematics necessary for the engineering feats they had achieved.
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Written by Njord Kane © 2013, 2016 Spangenhelm Publishing
 Read: <Ancient Maya Arithmetic>
 Read: <The Ancient Maya understood Value of Zero>
 Read: <The ‘Four Slave’ Example How the Maya count to 8000>
 Read: <Maya Math – The Grid System>
 Read: <Counting the Maya Way – The Finger Method>
 Read: <The Maya Abacus>
 Read: <The Ancient Maya Concept of Fractions>
Article source:

This article is an excerpt from the book: Kane, Njord. “Chapter 12 – Ancient Maya Arithmetic.” The Maya : The Story of a People. 2nd ed. Yukon: Spangenhelm, 2016. ISBN: 9781943066032 Used by permission from the author and publisher exclusively for use on readicon.com only.
Sources:
 Kane, Njord. The Maya : The Story of a People. 2nd ed. Yukon: Spangenhelm, 2016. ISBN: 9781943066032
 Kane, Njord. Maya Math Simplified. 2nd ed. Yukon: Spangenhelm, 2016. ISBN: 9781943066087
 Georges Ifrah, “From One to Zero, a Universal History of Numbers”, Penguin Books, 1987.
 G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
 J B Lambert, B OwnbeyMcLaughlin, and C D McLaughlin, Maya arithmetic, Amer. Sci. 68 (3) (1980), 249255.
 Lounsbury, Floyd G. Maya Numeration, Computation, and Calendrical Astronomy. In Dictionary Of Scientific Biography. New York, New York. Charles Scribner’s Sons. Volume 15, Supplement 1. 1978. P. 759818.
 McAnany, A. Patricia (1998). “Ancestors and the Classic Maya Built Environment.”
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written by Njord Kane © 2013, 2016 Spangenhelm Publishing
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