Arithmetic on a Different Base: Not All Counting Systems Add Up To 10

By Mr. Ghaz, December 27, 2009

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Arithmetic on a Different Base: Not All Counting Systems Add Up To 10

The numbering systems of many societies and cultures around the world today are based an either 5 or 10, the number of fingers on one or both hands. Among the exceptions: the systems used by the Bolans of West Africa, who were said to have counted in 7’s, and by the Maoris of New Zealand, who traditionally used a numbering system based on 11.

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But other numbering systems of the past also featured numbers other than 5 or 10. For example, the ancient Babylonians worked with a system based on 60. They did not have the Arabic numbering system in use today, but instead used the end of a stick to draw a cuneiform, a vertical wedge-shaped symbol, on a soft clay tablet. This symbol represented one unit. A horizontal wedge represented 10 units.

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The wedge shapes were then grouped together in columns. Wedges were placed in the first column until they added up to 59; a single wedge in an adjacent column to the left symbolized the numeral 60. A wedge in the third column to the left represented 60 times 60 (3,600), and a wedge in the fourth column 60 times 60 times 60 (216,000).

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The Babylonians are not the only people who employed a different system. The Mayas, an ancient people of Central America, used a number system based on 20. The first column contained symbols that totaled 20; in the column to the left a single symbol represented 20. In the third column to the left a symbol represented 18 times 20 (360). The number was again multiplied by 20 in each of the columns to the left. The Mayas may have given 360 this special place in their system because it was close to the number of days in the year and therefore made calculations for their remarkably advanced calendar easier.

One-Two Counting

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The familiar base-10 system in use throughout the world today has been superseded. An overwhelming number of calculations are now carried out by computers, in which every number is symbolized by sequences of just two numbers: 0 and 1.

Called the binary system, the value of a 1 doubles each time that it is moved one place to the left. Therefore, 1 = 1, 1 and 0 = 2, 1 and 0 and 0 = 4, and 1 and 0 and 0 and 0 = 8. The binary system is essential for computers; the 0’s and 1’s are represented by the opening and closing (“on” or “off”) of the switches in an electronic circuit: 0 when the switch is closed, 1 when it is open.

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However, many computer experts find it difficult to move back and forth between the decimal and the binary systems, and use still another system as an intermediate step between the two: the hexadecimal, or “hex,” which is based on 16. Hex uses the numerals 0 through 9-together with the letters A through F to represent the numbers from 10 to 15; 16 are represented as 10.

While the hexadecimal system may baffle adults accustomed to more traditional ways of counting, many young computer buffs around the world consider it merely child’s play.