Counting begins historically with the fingers. Nearly
all groups of people, anciently as well as modernly, have been able to communicate
numbers at least as large as ten. In some languages, the names for
the first five or ten numbers are derived from the names of the respective
fingers. A certain tribe in New Guinea describes a number like 38,
“one man (20), both hands (10), a foot (5), and 3.” This
technique, unfortunately, soon reaches a limit as to the size of the number
that can be expressed. Many tribes, in fact, have no word for any number
greater than ten, and some have no word for any number beyond four, higher
numbers being represented by a single descriptor, such as “plenty”
or “a heap.”
Counting on the fingers is one technique illustrating a one-to-one correspondence, using one finger for each item being counted. The next generalization is to count by making marks on the ground, on a slate, or elsewhere. These two ideas give rise to the very common symbol for one, a vertical line, and its natural extension, two vertical lines for two, three for three, etc. Most early methods for writing numbers begin with these symbols. Later, other symbols to represent larger numbers were developed, symbols for five, ten, one hundred, and so on.
An obvious consequence of counting fingers, most early peoples used either a decimal system or some other base which is a multiple of five (the Aztecs, for example, apparently accustomed to using fingers and toes, used base 20). Only very few exceptions defied this rule, such as the Bolans of West Africa, who used base seven, and the Maories, who used base eleven.
The Egyptians used a decimal system and represented the first nine numbers by so many vertical lines. They used a to represent 10, a for 100, a lotus flower for 1000, a pointing finger for 10,000, a burbot for 100,000, a man in astonishment for 1,000,000, and for 10,000,000. Large number were represented by repetitions of these symbols. For example, 236 would be written
Certain trading practices necessitated the use of fractions, so the Egyptians used either a dot or an oval above a number to represent 1/n. They had a special symbol for 2/3, but other fractions had to be written with special care. For example, for 2/13, they would write 1/8+1/52+1/104, indicating the addition simply by juxtaposing the two numbers. This kind of conversion was certainly difficult and time-consuming, so the people would generally keep available a table of ways to write fractions of the form 2/n. In the Rhind papyrus, for example, written by a high priest named Ahmes, is found a table:
2/5 = 1/3 + 1/15
2/7 = 1/4 + 1/28
2/9 = 1/6 + 1/18
2/99 = 1/66 + 1/198
For larger fractions, like 5/11, they would first decompose this into 1/11+2/11+2/11, then reduce the last two fractions according to the table, combine like terms, and repeat. Thus Ahmes wrote that if nine loaves are divided among ten persons, the portion each receives is 2/3, 1/5, 1/30. He used the word “heap” to denote an unknown variable (which is interesting considering the aforementioned use of the same word to mean “many”) and solved algebra problems thus: “Heap: its two-thirds, its half, its one-seventh, its whole, make 33.” It other words,
2x/3+x/2+x/7+x=33He then gives the correct answer as 14, 1/4, 1/97, 1/56, 1/679, 1/776, 1/194, 1/388. It is interesting to note that the Greeks also used the same technique for writing fractions until the sixth century A.D.
Ahmes also uses certain algebraic symbols in the Rhind papyrus. For addition, he uses a pair of legs walking forward, for subtraction, legs walking backward. For equality, he uses a symbol that looks like an upside-down less than or equal to. In this way, Egypt might have been the forerunner of modern algebraic notations if the technique hadn’t been dismissed and forgotten.
While the Babylonians communicated with the Egyptians and were certainly familiar with their decimal numbers, they used a numerical system of their own invention. They had 60 numerals which were based on two symbols,
the first of which represents one, the second ten. A representative copy of a mathematical cuneiform tablet follows (CBS 8536 in the Museum of the University of Pennsylvania).
They wrote their numerals as follows:
Their use of these digits can by seen by one tablet which dates approximately from the period of Hammurabi (circa 2000 B.C.). A list of numbers begins 1, 4, 9, 16, 25, 36, 49. The next number is written 14, then 121, 140, 21, 224, and so on, until 581. Clearly, the first number represents sixties and the second number units. The Babylonians thus invented the vitally important idea of a place value for a digit. Unfortunately, they lacked one important ingredient: the zero. Having no way to fill a blank spot, the same number might represent 3, 180 (3×60), 10800 (3×60²), and so on. With such a wide difference as a factor of 60, this difficulty was, in practice, easily overcome by the context of the number. One theory (unconfirmed due to a lack of surviving clay tablets) holds that the Babylonians would leave a space between digits for internal zeros (as in 1030 for 3630, for example).
The Babylonians represented fractions in a way much more simple than the Egyptians, adding no complexity to their system. They would optionally use the rightmost digit as a fraction out of 60. Thus 1½ is represented 130 (exactly the same as 90).
The Babylonians might have been the forerunners of our modern numerical notation if their use of a place value had not been forgotten for thousands of years.
The Mayans were the first to fully utilize the principle of place value. They used a zero (which looks much like a half-closed eye) centuries before the Hindus did. They used dots and bars for the numerals from 1 to 19, with a bar representing 5 and a dot 1.
The numbers are written vertically, from top to bottom, with the higher positions representing larger quantities (just as the leftmost digits in our numbers today represent larger quantities than the digits to the right). They wrote in base 20, except in the third spot. The numbers were counted as kins, or days. Thus, the lowest digit gave the number of days. The second digit gave the number of uinals, or 20 days. The third digit gave the number of tuns, which were not 20 but 18 uinals, to make one year of 360 days. Successive digits have, once again, 20 times the value of a preceeding digit, so 20 tuns make a Katun, 20 Katuns make 1 cycle, 20 cycles make 1 great cycle. The largest number found in their codices is 12,489,781.
From the Dresden Codex, of the Maya, displaying numbers. The second column from the left, from above down, displays the numbers 9, 9, 16, 0, 0, which stand for 1,366,560. In the third column are the numbers 9, 9, 9, 16, 0, or 1,364,360. The original appears in black and red colors.
The Greeks, at first, used notation mirroring Roman numerals, with the usual vertical line, Ι, to represent one, Γor Π for five, Δ for ten, Η for one hundred, Χ for one thousand, etc. They soon gave up this notation for one of their own doing. They used the first nine letters of the alphabet to represent the numbers 1-9, the next nine letters to represent the multiples of 10 through 90, and, finally, another nine letters to represent the multiples of 100 through 900. They further used suffixes and indices to represent numbers up to 100,000,000. Unfortunately, there are only 24 letters in the Greek alphabet, while this system required 27. To supply the missing three letters, they re-introduced two letters, digamma and koppa, which were originally part of the Greek alphabet but had been discarded as being obsolete. For the last letter, they borrowed one from the Phoenician alphabet. They would also often write a horizontal line over long numbers to distinguish them from words.
A notable contribution to this system came from Diophantus. He made many contributions to arithmetic and algebra. He used a system of abbreviations to write his equations. He used one symbol, ϛ, to represent the unknown (i.e. the variable x). He denoted its square (or δύναμιϛ) by δύ, and its cube (or κύβοϛ) by κυ. Each was immediately followed by the coefficient of the variable or its power. To write a constant, he used μ to mean units (or μονάδεϛ). He used ι for equality but had no sign for addition beyond juxtaposition. Thus x³+2x²+5=x would be written κυαδύβμειϛ.
This notation, crude as it may seem, was ahead of its time, as it used symbols instead of words to represent unknown quantities. In this way, Diophantus might have been the forerunner of algebraic notation. Unfortunately, his writings were forgotten for hundreds of years until long after his ideas were obsolete.
The origins of Arabic numerals are obscure and much disputed, but they are doubtlessly the basis for the numerals in use around the world today. The numerals probably came from India and date from as early as 150 B.C. The zero came later, perhaps around A.D. 500, and it may have developed from a dot inserted to indicate a blank space. The progression that their numerals underwent can be seen in the table that follows.
Finally, the history of the development of numerals ends as the Arabic numerals take their shape, and by A.D. 1500, the numerals looked much the same as those we use today.BIBLIOGRAPHY
Ball, W. W. Rouse, A Short Account of the History of Mathematics, Dover Publications, Inc., New York, N.Y. (1908, reprinted 1960).
Cajori, Florian, A History of Mathematics, Fourth Edition, Chelsea Publishing Company, New York, N.Y. (1985).
—————, A History of Mathematical Notations, Volume I, The Open Court Publishing Company, La Salle, IL (1928).
Eves, Howard, Great Moments in Mathematics, The Mathematical Association of America (1980).
Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York (1972).
Scott, J. F., A History of Mathematics, From Antiquity to the Beginning of the Nineteenth Century, Taylor and Francis Ltd, London (1960).