
Fancied
Number
Symbols
Paper 1
History of Mathematics
September 19, 2001
1.2
Abstract:
The history of mathematics is hard to identify because it happened so
many years ago. Savages during the Prehistoric period had simple needs
and used their fingers to signal numbers rather than actual words. Egypt
developed along the Nile River shortly after the Prehistoric period. The
first known written mathematical work is credited to the Egyptians. The
Romans imitated other civilizations in philosophy, poetry, and art;
however, they had no desire to imitate mathematics. Their numeral
system is noteworthy because the subtraction principle is incorporated.
The Greeks developed two systems of writing: Herodianic or Attic
symbols and the Greek alphabet, beginning as early as the 6th century B.
C. This development is thought to be a decline in progression because
there were more symbols to memorize in the Greek alphabet. The
Babylonians wrote in cuneiform on clay tablets, which have been
recovered and are displayed around the world. They had two symbols for
writing in the sexagesimal base. The early Chinese used counting rods to
aid with calculations. The Chinese-Japanese system of writing is a
multiplicative system based on 10, written vertically, with symbols for 1-9
along with the powers of 10. The Mayans are credited for using a zero
and the principle of local value. A combination of dots and bars, written
vertically with the lowest order at the lowest position resembled the
Mayan numeral system. The process of mathematics evolved throughout
the ancient period, but the future provides for advancement.
Fancied Number Symbols
When attempting to outline the beginnings of mathematical history, the issue of defining
history and mathematics appears. Consider history to be a narration of recorded events. The
prehistoric period may be defined to be a “relation of incidents which probably happened even
before the advent of the human race”. There are many limitations when defining mathematics
(Smith 1). It is difficult to define mathematics without referring to science in some way,
however during the prehistoric era science had not yet evolved (2). So it is assumed that the
history of mathematics originated in that vague period which is called the beginning (1).
Primitive counting was a tedious process because the primitive man’s needs were very
simple. The only thing demanded by him was the size of his family, counting of enemies, or
similar use of small numbers. For a long time, savages used nouns to represent numbers of
multitude, such as heap, crowd, school, pack, and flock. Counting with fingers showed that the
idea of numbers did not have to wait for spoken language to develop (Smith 6).
1
Civilization in Egypt developed along great rivers, such as the Nile, which protected the
country and allowed for more uninterrupted development. The introduction of the Egyptian
calendar of twelve months of thirty days each, plus five feast days in the year 4241 B. C. is the
earliest dated event in human history. There is no authentic record of mathematical progress in
any other country dating farther back than the Egyptian calendar (Smith 42). During the reign of
Amenemhat III, about 1850 B. C., an extensive system of irrigation, knowledge of leveling,
surveying, and mensuration was carried out (45). The introduction of a sophisticated calendar
and gigantic pyramid structures are indications that Egypt had highly advanced technology and
sound mathematical foundations (Hofmann 8).
The earliest written mathematical work possessed consists of a papyrus written by an
Egyptian priest named Ahmes about 1000 B. C. It consists of a collection of problems in
arithmetic and geometry. Ahmes was familiar with ordinary processes of calculation, including
the use of fractions (Larrett 19). The mathematical ability of the early Egyptians was practical
and largely the result of acute observation. Their skill and courage in attempting such mighty
works with primitive tools and little theoretical training is admirable (20). The writing was done
with ink on papyrus, however papyrus dries up and crumbles so very few documents of ancient
Egypt have survived. There are two sizable papyri documents dated from 1700 B. C.: the
Moscow papyrus and the Rhind papyrus. The Rhind papyrus is also known as the Ahmes
papyrus after its author. There are 85 problems on the Rhind and 25 in the Moscow papyrus
intended as examples of typical problems and solutions (Kline 16).
The ancient Egyptians developed their own system of writing, hieroglyphics or picturewriting, as a form of recording an agreement, conveying a message, or making a calculation with
numbers (Gillings 4). Champollion, Young, and their successors gave insight into Egyptian
2
methods of numeration through the deciphering of hieroglyphics. The Egyptians used symbols
to represent higher powers of ten; “the symbol for 1 represents a vertical staff; that for 10,000 a
pointing finger; that for 100,000 a burbot; that for 1,000,000 a man in astonishment.” The
significance of the other symbols used is doubtful (Mathematics 11).
Writing in hieroglyphics was cumbersome because the unit symbol of each was repeated
as many times as there were units in that order, hence the additive principle was employed, thus
23 was written ∩∩
(Mathematics 11). Sometimes a large number of characters are required to
write a number and a smaller number usually requires more than a large number. The Egyptians
wrote from right to left like the Hebrews (5). Hieroglyphic writing is usually written vertically
downward. It is apparent that the objects, such as birds, reptiles, and human faces all face the
direction that the writing is coming. When translated into English the direction is reversed for
convenience (6).
When writing the numbers 574 and 475 in Egyptian hieroglyphics, three symbols are
involved. The
represents 100, ∩ is 10, and │equals 1. As shown on Table 1 (page 14), the
first number is written by using 5
‘s equal to 500, 7 ∩’s equal to 70, and 4 │’s to equal 4.
Reading from left to right this equals 574. The second number is written the same way but with
4
‘s and 5 │’s. The Egyptian way of writing numbers is simple because there are only a few
symbols to memorize.
The contrast between the Greek and Roman minds is distinctly shown in their attitude
toward the mathematical science. Romans were imitators of philosophy, poetry, and art;
however, they had no desire to imitate mathematics. What little mathematics they possessed
came not only from the Greeks but also from more ancient sources. It is probable that the
Roman notation came from old Etruscans who inhabited the district between the Arno and Tiber.
3
This system is noteworthy from the fact that a principle of subtraction is involved. “If a letter be
placed before another of greater value, its value is not to be added to, but subtracted from, that of
the greater”. A horizontal bar was placed over a letter to increase its value on thousand fold
(Mathematics 63).
The Romans engaged in three different kinds of arithmetic calculations: reckoning on the
fingers, upon the abacus, and by tables prepared for a purpose. Finger-symbolism was known as
early as the time of King Numa (Mathematics 63). The second mode of calculation, the abacus,
was elementary instruction in Rome. The most commonly used abacus was covered with dust
and then divided into columns, with each column supplied with pebbles, which served for
calculation. Arithmetic tables were used to aid in multiplying large numbers. These tables were
prepared by Victorious of Aquitania and contained a peculiar notation for fractions. Victorious
is also known for finding the correct date for Easter, which was published in 457 A. D. (65).
Written Roman numbers are more difficult to write because of the subtraction principle.
In this language I=1, V=5, X=10, L=50, C=100, and D=500. The I can only come before V or X
to equal either 4 or 9 respectively. It is also true that X can precede L and C to represent 40 and
90 respectively. Finally in this example the C can come before D to equal 400 (Eves 16).
Referring to table 1, the Romans wrote 574 as DLXXIV and 475 as CDLXXV. Let’s dissect
both numbers. For the first, the D=500, LXX50+20=70, and IV5-1=4, therefore
500+70+4=574. For the second number, CD500-100=400, LXX is the same (70), and V=5,
hence 400+70+5=475.
Finding ¼ of MCXXVIII and 4 times XCIV involved some translating from Roman into
English and then back into Roman numerals. MCXXVIII equals 1128 and when divided by 4 it
4
equals 282. This number can be written in Roman numerals as CCLXXXII. In English, XCIV
equals 94 and 4 times that is equal to 376, which is CCCLXXVI in Roman numerals.
Roman Numerals:
h)
¼ of MCXXVIII
i)
4 times XCIV
94
1128
94
282
4 1128  282  CCLXXXII
4
376  CCCLXXVI
The earliest mathematical accomplishments of the Greeks are almost completely
unknown (Hofmann 13). The oldest known Grecian numerical symbols were the Herodianic
signs, which date from the 6th to 1st centuries B. C. These signs occur frequently in Athenian
inscriptions and are generally called Attic. These symbols were replaced by the alphabetic
numerals. This change was for the worse because old Attic numerals were less burdensome on
the memory because they contained fewer symbols (Mathematics 52). From the 5th century B.
C. on, Milesian letter-numbers became demonstrable to write the numbers from 1 to 9, tens from
10-90, and hundreds from 100-900 represented by twenty-four standard letters of the Greek
alphabet and three older letters (stigma, koppa, sampi). The thousands were indicated by a low
stroke before the letter-numeral, unit fractions by an accent mark after the denominator, and
individual symbols were used for ½ and 2/3 (Hofmann 13).
The representation of numbers by letters dates back to Greek antiquity. Aristotle was
known to use single capital letters or two letters for designation of magnitude or number
(Mathematical 1). The small Greek letters were used to represent numbers and the Greek
capitals represented general numbers. Cicero used letters for the designation of quantities, such
5
as τû for known quantities and γâ and other symbols for unknown quantities (2). While the
Greeks used Greek letters for the representation of magnitude, the use of Latin letters became
common during the Middle Ages (5). Charles Babbage once suggested the rule that all letters
that denote quantity should be printed in italics, but those indicating operations should be printed
in roman characters (6).
Like the Egyptians the Attic Greek numerals are based on 10. The numbers are
represented by for 1,
equals 5, so
and
for 10, and
for 100. A special symbol, П represents 5, thus
represent 50 and 500 respectively (Eves 16). Refer to Table 1 to see
how 574 and 475 are written in Attic Greek. The symbols are read from left to right and are a
combination of the Egyptian’s repetition technique along with simple addition.
The Greek alphabet was created around 450 B. C., consisting of 24 alphabetic Greek
letters and three symbols for the obsolete (Eves 18). The alphabetic Greek numbers 574 (phi
omicron delta) and 475 (upsilon omicron epsilon) are shown on Table 1.
Again, writing 1/8 of
τδ and 8 times ρκα is a translation problem. The Greek number, τδ, is equal to 304, so when
divided by 8 results in 38. This can be written in alphabetic Greek as lambda eta, or λη. The
English number 121 equals the alphabetic Greek ρκα. When multiplied by eight the outcome is
968, which is written as
ξη, sampi xi eta
Alphabetic Greek:
j)
1/8 of τδ
k)
8 times ρκα
304
121
38
8 304  38  
121
 8  968 =
ξη
968
6
Early Babylonia endured from about 3100 to about 2100 B. C. The first great ruler was
Sargon. His remarkable career flourished about 2750 B. C. beginning in Akkad. It was partly
due to the proximity of territory that the people of Akkad and Babylonia adopted business
methods, astronomy, calendar, measurements, and the numerals of the more highly cultivated
Sumerians (Smith 37). During the reign of Hammurabi, the world’s first great code of laws was
written and the calendar was reformed. Part of our knowledge of Babylonian arithmetic was
found on tablets used by pupils in the oldest known schoolhouse, which was discovered by the
French in 1894. The early Babylonians developed knowledge of computation, mensuration, and
commercial practice in spite of an awkward numeral system (38).
The early Babylonians developed a numeral system during the 28th century B. C. They
also used a round and pointed stick resulting in a circular, semicircular, or wedge-shaped
(cuneiform) character on the surface of clay tablets (Smith 36). The clay tablets were baked by
the sun or fire for preservation. There are thousands of them available for study in various
museums. These tablets show that the Babylonians were familiar with bills, receipts, notes,
accounts, and systems of measurement nearly 3000 years before Christ. There is no other part of
the world that has evidence of commercial mathematics at this early date. There is also evidence
of a scientific calendar where probably the first use of a scale of 60 in counting is present (37).
The use of number 60, finally suggested the development of sexagesimal fractions that are still
used in the division of degrees, hours, and minutes. The early Babylonians also believed the
circle of the year consisted of 360 days. Therefore 60 is thought of as a mystical number (41).
Most of the clay tablets inscribed with text of mathematical content in cuneiform script
originated from Tigris and Euphrates basin over a period extending from the second millennium
to the second century B. C. The number system was based on wedge-shaped ones (
) and
hook-shaped tens (
), which were used to write the numbers from 1-59 (Hofmann 5).
Grotefend believes the character used for 10 to originally have been the picture of two hands
held in prayer. Numbers below 100 were expressed by symbols whose values had to be added.
The symbols of higher order always appear to the left of those of lower order. Sumerian
inscriptions disclose not only the decimal system, but also a sexagesimal one constructed for
weights and measures (Mathematics 4).
Written numbers in Babylonian cuneiform are difficult because it uses the base 60. In
order to write 574, it is necessary to change the number into base 60. The quotient of 574/60 is 9
R34. Therefore, the answer written in cuneiform has nine wedge-shaped ones to represent the 9,
and three hook-shaped tens (30) plus four wedge-shaped ones to equal the remainder, 34. The
second number is written the same way, by dividing 475 by 60 to equal 7 R55. There are seven
wedge-shaped ones along with five hook-shaped tens plus five wedge-shaped ones to equal
seven remainder 55.
There is little knowledge of early Chinese literature. The historical period begins around
the 8th century BC with the reign of Wu Wang, the Martial Prince (Smith 22). During the reign
of Huang-ti in 2704 B. C., the Chia-tsŭ, or sexagesimal system, was established by Ta-nao. The
emperor was also interested in mathematics and astronomy; during his reign an eclipse of the sun
was observed. The decimal system of counting is also assigned to this period (24). The oldest
Chinese work of mathematical interest is an anonymous publication, called Chou-pei. The
Chou-pei is believed to reveal the state of mathematics and astronomy in China as early as 1100
B. C. The Pythagorean theorem of the right triangle appears to be known at that early date. The
most celebrated Chinese Text on arithmetic is called the Chiu-chang (Mathematics 71).
8
In ancient times the legend of Lì Shŏu creating arithmetic circulated widely. It is only
possible that the history of number evolved gradually as the practical requirements of human
activity progressed (Crossley 1). There are two other legends involved in the beginning of
ancient mathematics, regarding the quipu knots, and the gnomon and compass. People tied knots
to remind themselves of particular matters, so it is reasonable to assume knots were also used to
record numbers. According to legend Chuí invented the gnomon and compass (2).
Although it is important to explore events from legends and myths, clues found in ancient
cultural artifacts, which have been excavated, are more important (Crossley 3). The Chinese
used counting rods to aid with mathematical calculations. These counting rods, called Chóu, are
small bamboo rods. Ancient Chinese mathematicians arranged the rods into different
configurations to represent numbers and then perform calculations using these rods. The
manipulation of these Chóu to perform calculations is known as ‘chóu suàn (6).
The traditional Chinese-Japanese numeral system is also a multiplicative system based on
10. The symbols are written vertically from top to bottom (Eves 17). This system of writing,
unlike those previously mentioned, used symbols for numbers 1-9 along with the powers of 10.
When writing the number 574, several symbols are needed. Referring to table 1 and reading the
symbols from top to bottom, the first represents 5, the second 102, then 7, next 10, and finally 4.
When read from top to bottom the number is said word for word as five hundred seven tens
(seventy) four. This way of writing numbers resembles the English language when read. The
number 475 is similar to the previous number, however the 4 and 5 switched positions.
Practically nothing is known about the Mayan system of writing numbers (Hofmann 11).
Mayan number systems and chronology are remarkable for the extent of early development. The
Mayans in the flatlands of Central America evolved a vigesimal number system employing a
9
zero and the principle of local value (Mathematics 69). Numbers below 20 were written by a
combination of dots (ones) and bars (fives). The Mayan system was positional with a symbol for
0 (Hofmann 11). The zero is represented by a symbol that looks like a closed half-closed eye.
The numbers are written vertically with the lowest order being assigned the lowest position
(Mathematics 69).
The ratio of increase of successive units was 20 in all positions except the third
(Mathematics 69). The gradations were 1, 20, 18X20=360, 360X20=7200, etc (Hofmann 11).
This system was related to dividing the year into eighteen 20-day months and five days regarded
as days of ill luck. This calendar system also groups days into 13-day weeks. Exact data about
the arithmetic of the Mayans is still missing (12).
The different gradations of the Mayan numeral system make it complicated to translate
Mayan into English numbers. The first step in converting 574 from Mayan numerals into
English, is dividing 574 by 20 to find 28 R14. Next the number 28 is divided by 18 to obtain an
answer of 1 R10. The number is written using dashes and dots starting at the top with 1
representing the second answer, then 10 the remainder, and finally 14 standing for the remainder
of the first quotient. Dividing 475 by 20 to obtain 23 R15, and then dividing 23 by 18 to get 1
R5 aids in translating 475 into Mayan numerals. Again from top to bottom the number reads 1
the answer to the second quotient, 5 being the remainder, and then the first quotient’s remainder
15.
Throughout the Ancient period several forms of written numbers evolved along with
elementary arithmetic. Many western civilizations, such as Egyptians, Romans, Greeks,
Babylonians, Chinese-Japanese, and Mayan, all had different languages and ways of counting.
Each individual group used their own number base system making them very distinct. The
10
English number system resembles a combination of the primitive counting systems. The events
in each country during the Ancient period reflected upon the use and development of
mathematics in that area. The process of mathematics throughout the ancient period was
progressive; however, the future allows for several great strides.
Achievement is but another milestone along the highway of progress—the end of the journey lies
ever beyond. --Anonymous
11
574
Egyptian Hieroglyphics
Roman Numerals
∩ ∩ ∩ ∩ ││
∩∩∩
││
475
∩ ∩ ∩ ∩ │││
∩∩∩
││
DLXXIV
CDLXXV
Alphabetic Greek
φοδ
υοε
Mayan
●
═══
●●●●
════
●
───
≡≡≡≡
Attic Greek
Babylonian Cuneiform
Chinese/Japanese
Table 1: Written Numbers -- Problem 1.2 (a)-(g)
12
Works Cited
Cajori, Florian. A History of Mathematical Notations. Illinois, Open Court, 1952.
Cajori, Florian. A History of Mathematics. New York, Macmillan, 1919.
Crossley, John N. Chinese Mathematics: A Concise History. New York, Oxford University
Press, 1987.
Eves, Howard. An Introduction to the History of Mathematics. New York, Harcourt College,
1992.
Gillings, Richard J. Mathematics in the Time of the Pharoahs. New York, Dover Publications,
1982.
Hofmann, Joseph Ehrenfried. The History of Mathematics. New York, Philosophical Library,
1957.
Kline, Morris. Mathematical Thought from Ancient Too Modern Times. New York, Oxford
University Press, 1972.
Larrett, Denham. The Story of Mathematics. New York, Greenberg, 1926.
Smith, David Eugene. History of Mathematics: General Survey of the History of Elementary
Mathematics. New York, Ginn, 1923.
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