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476–c. 1185

The number symbols we use today called Hindu-Arabic numerals were originally developed in India and had their origins dating back to at least the time of King Asoka in the mid-third century B.C.E. By around the year 600, the Indians evidently dropped the symbols for numbers higher than 9 and began to use their symbols for 1 through 9 in our familiar place-value arrangement. Also at this time, a dot to represent zero started appearing. These ten basic symbols were then passed to the Arabs, where changes were made yielding the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that in turn were adopted by the Europeans. Of the ten symbols, the key symbol is the one for zero because it represented “no quantity,” which is necessary to represent certain amounts in a positional notation system. Therefore, the Indians were one of the first civilizations to adopt a positional notation, or place-value, system including a symbol for zero. This is one of the most profound mathematical discoveries ever made, in spite of its apparent simplicity.(1) It changed the most difficult calculations into simple ones and laid the foundation for the invention of arithmetic—the invention that released man’s mind once and for all from the grip of the abacus. And it gave us a number system that would stretch out to infinity in either direction.


Writing Numbers

Counting or reckoning was undoubtedly one of the earliest arts practiced by man. During 4,000 years from the earliest civilizations to the years 500–600, man developed a heritage of literature, art, philosophy, and religion. Many civilizations rose to great heights and fell before new conquerors. But over this entire span of time, man could only develop such a crude and complicated system of counting that experts alone could manage the simple calculations that we expect of children today. Their adding machine, the abacus, was not a convenience that was resorted to for the sake of speed and simplicity; it was an absolute necessity. During this entire period of time, not one single worthwhile improvement to the instrument or to the principles of calculation found its way into the minds of men. Then one day between 500 and 600, an event of world-shaking proportions took place—an unknown Hindu discovered zero!


The zero symbol is the soul of the positional notation system because without it, many quantities could not be represented. For example, the numbers ten and 1,000 require a zero symbol in a positional notation system with a base of ten. In a positional notation system, each successive digit starting on the right represents a different unit amount. This also means that each successive unit digit is a raised power of the base. When you write numbers, you are typically using base-10. In base-10, you have: 100 or 1, then 101 or 10, then 102 or 100, then 103 or 1,000. In base-10, you use the first nine symbols to count up to nine. To write ten, you use a combination of the symbol 1, to indicate one unit in the ten’s position, or ten units, and a zero symbol in the one’s position to represent zero units in the one’s position. Therefore, in base-10, the first digit, starting on the right, is the one’s position, the second digit is the ten’s unit, the third digit is the hundred’s unit, the fourth digit is the thousand’s unit, and so on.


In base-20, the first digit is the one’s position, the second digit is the twenty’s position, the third digit is the four hundred’s position, and the fourth digit is the eight thousand’s position. In base-20, you have: 200 or 1, then 201 or 20, then 202 or 400, then 203 or 8,000. In base-20, we use nineteen symbols up to nineteen, then twenty is written 10, i.e., one unit in the twenty’s position and zero units in the one’s position.


Today we take the base-10 counting system for granted, but it evolved over thousands of years. It is in use worldwide probably because it is the best combination of two elements–the number of symbols and the incorporation of positional notation. Because base-10 only has ten symbols, combined with the use of positional notation, it makes writing and reading both small and large quantities quite easy.


It is important to remember that base-10 is just one possible base system for representing quantities. The Babylonians used a base of 60 and the Mayans used a base of 20, or a vigesimal system. However, in both cases, they made some modifications. The Babylonians did not use 60 symbols, which would have been very cumbersome. Instead, they only used two symbols (one representing ten and the other, one) that would not normally be allowed using a base-60 system but made it much easier to write the whole range of numbers from small to large. In contrast, today’s base-10 system is followed strictly. Of significance today, the Babylonians’ use of a base-60, or sexagesimal, system for writing numbers resulted in our present division of the hour into 60 minutes and the circle into 360 degrees.


Here is a conversion problem: How do you express the quantity fifty in base-20? The answer is 2A–two units in the twenty’s position, or 2 x 20 = 40, plus ten units in the one’s position, or A (10) x 1 = 10. Brought together, 40 + 10 = 50. Note that you could substitute any symbol you wanted instead of the “A” (just so it is different than the other eighteen symbols).


In contrast, the system used by the Romans, which was eventually named Roman numerals, had only seven symbols but no zero symbol and did not employ the positional notation concept. Instead, they relied on repetition and combinations of seven symbols.



Still in use today, though infrequently, upper and lowercase Roman numbers appear in book introductions for pagination, in films for the year of production, on watch and clock faces, and other places.


(1) William C. Vergara, Mathematics in Everyday Things (New York, 1962), p. 235.

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