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).