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

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with fingers showed that the idea of numbers did not have to wait for spoken language to

develop (Smith 6).

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.

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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

picture-writing, 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 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

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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. 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

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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,

LXX50+20=70, and IV5-1=4, therefore 500+70+4=574. For the second number,

CD500-100=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 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 1128

i) 4 times XCIV

94

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-

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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 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, for 10, and for 100. A special symbol, П represents 5,

thus equals 5, so and 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

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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

304

k) 8 times ρκα

121

968 = ξη

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

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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

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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).

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

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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 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

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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 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

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574 475Egyptian Hieroglyphics

∩ ∩ ∩ ∩ ││ ∩ ∩ ∩ ││

∩ ∩ ∩ ∩ │││ ∩ ∩ ∩ ││

Roman Numerals DLXXIV CDLXXVAttic Greek

Babylonian Cuneiform

Chinese/Japanese

Alphabetic Greek φοδ υοεMayan ●

═══●●●●════

●───≡≡≡≡

Table 1: Written Numbers -- Problem 1.2 (a)-(g)

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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.