The Historical Roots of Elementary Mathematics [NOOK Book]


Exciting, hands-on approach to understanding fundamental underpinnings of modern arithmetic, algebra, geometry and number systems, by examining their origins in early Egyptian, Babylonian and Greek sources. Students can do division like the ancient Egyptians, solve quadratic equations like the Babylonians, and more.
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The Historical Roots of Elementary Mathematics

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Exciting, hands-on approach to understanding fundamental underpinnings of modern arithmetic, algebra, geometry and number systems, by examining their origins in early Egyptian, Babylonian and Greek sources. Students can do division like the ancient Egyptians, solve quadratic equations like the Babylonians, and more.
Read More Show Less

Product Details

  • ISBN-13: 9780486139685
  • Publisher: Dover Publications
  • Publication date: 11/13/2012
  • Series: Dover Books on Mathematics
  • Sold by: Barnes & Noble
  • Format: eBook
  • Pages: 336
  • Sales rank: 860,726
  • File size: 18 MB
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The Historical Roots of Elementary Mathematics

By Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient

Dover Publications, Inc.

Copyright © 1988 Harry Bunt, Lidy E. M. Van Welzenis, Phillip S. Jones and Jack D. Bedient
All rights reserved.
ISBN: 978-0-486-13968-5




The earliest written mathematics in existence today is engraved on the stone head of the ceremonial mace of the Egyptian king Menes, the founder of the first Pharaonic dynasty. He lived in about 3000 B.C. The hieroglyphics on the mace record the result of some of Menes' conquests. The inscriptions record a plunder of 400,000 oxen, 1,422,000 goats, and 120,000 prisoners. These numbers appear in Figure 1-1, together with the pictures of the ox, the goat, and the prisoner with his hands behind his back. Whether Menes exaggerated his conquests is interesting historically but does not matter mathematically. The point is that even at this early date, man was recording very large numbers. This suggests that some mathematics was used in the centuries before 3000 B.C., that is, before the invention of writing (the prehistoric period).

There are two ways of learning about the mind and culture of prehistoric humans. We have learned about them through the discovery of ancient artifacts, which were found and interpreted by archaeologists. We have also learned about prehistoric civilization by observing primitive cultures in the modern world and by making inferences as to how prehistoric thought and customs developed. In our study of the development of ideas and understandings, both approaches are useful.

One of the most exciting archaeological discoveries was reported in 1937 by Karl Absolom as a result of excavations in central Czechoslovakia. Absolom found a prehistoric wolf bone dating back 30,000 years. Several views are shown in Figure 1-2. Fifty-five notches, in groups of five, are cut into the bone. The first 25 are separated from the remaining notches by one of double length. Although we do not know how this bone was notched, the most plausible explanation is that some prehistoric man deliberately cut it. Perhaps he was recording the number of a collection, possibly of skins, of relatives, or of days since an event. It is reasonable to assume that he made a notch for each object in the collection that he was counting. If this interpretation is correct, then we can recognize in this prehistoric record rudimentary versions of two important mathematical concepts. One is the idea of a one-to-one correspondence between the elements of two different sets of objects, in this case between the set of notches on the bone and the set of whatever the prehistoric man was counting. The other is the idea of a base for a system of numeration. The arrangement of the notches in groups of 5 and of 25 indicates a rudimentary understanding of a base 5 system of numeration.

Anthropological studies reinforce our belief in the existence of prehistoric number ideas. A study of the western tribes of the Torres Straits, reported by A. C. Haddon in 1889, describes a tribe that had no written language which counted as follows: 1, urapun; 2, okosa; 3, okosaurapun; 4, okosa-okosa; 5, okosa-okosa-urapun; 6, okosaokosa-okosa. Everything greater than 6 they called ras. A student of modern mathematics would recognize in this system of counting the beginnings of a base 2 numeration system. If a Torres Strait native had recognized this idea, however, he would have used a different word for 4 and would not have said ras for numbers greater than 6.

A. Seidenberg has recently published a theory of the origin of counting (see reference 12 at the end of the chapter). He believes that counting was invented for use in early religious rituals. Many studies of primitive tribes as well as early Babylonian religious writings are cited which indicate that participants in religious rituals were called into the ritual in a definite order and that counting developed in connection with specifying this order. In his studies, Seidenberg found 2-counting to be the earliest counting that he could detect. This seems to indicate that the counting of the Torres Straits natives is consistent with a method of counting that was in use thousands of years earlier.

These two types of prehistoric number ideas, matching and counting, correspond to two different approaches to number which are common both in modern life and in modern education. One of these is the approach through the ideas of set and one-to-one correspondence between sets that have developed since the work of Georg Cantor (1845-1914) in the latter part of the nineteenth century. This treatment is sometimes referred to as a cardinal approach to number. At about the time that Cantor was developing the beginnings of modern set theory, Giuseppe Peano (1858-1932) was attempting to axiomatize the natural numbers and their arithmetic. To do this, he stated a set of five axioms. One of these axioms is that every natural number has a successor. Such a treatment is called an ordinal approach to number. It emphasizes the counting idea, in contrast to the matching idea stressed in the cardinal approach to number. These two approaches can be shown to be equivalent to each other, but our purpose here is merely to point out the antiquity of the underlying ideas which have recently been organized into important modern mathematical systems.

Evidences of other prehistoric mathematical ideas are not hard to find. One can read into primitive cave paintings some ideas of proportion and symmetry as skilled artists produced remarkably realistic drawings of animals and hunters. Ideas of number and one-to-one correspondence appear in connection with stickmen and four-legged animals. Elaborate geometric designs can be found on prehistoric pottery. Prehistoric drawings showing different views of a wagon and horses have been found in Europe. Sketches from the time of ancient Babylonia that might be plans of a building, perhaps a temple, have been unearthed. What appears to be a decimally divided ruler has been unearthed at Mohenjo-Daro in Pakistan. Interesting as these archeological findings are, from a mathematical point of view we shall find a study of the historic period to be more profitable. Let us therefore turn our attention to the earliest written mathematics, that of the Egyptians and the Babylonians.


Although monuments, inscriptions, and Menes' mace record the earliest written numbers, most of our knowledge of Egyptian mathematics comes from writings on papyrus. Papyrus is a paperlike substance made from the papyrus plant, which grows along the Nile River. From these writings we learn that mathematics was studied in Egypt as early as 2000 years before Christ. Why in Egypt?

Herodotus (about 450 B.C.) observed that the Egyptians were forced to reset the boundary markers of their fields after the spring flooding of the Nile destroyed them. For that purpose surveyors were needed with some practical knowledge of simple arithmetic and geometry. Many of their computations remain. However, it is typical of Egyptian mathematics that arithmetic processes and geometric relations are described without mention of the underlying general principles. Thus, we know how the Egyptians performed computations, but we can only guess at how they developed their methods. We look for the reasoning behind their methods by deciphering and studying the detailed solutions of many examples.

The Greek mathematician Democritus (about 460-370 B.C.) appreciated the mathematical knowledge of the Egyptians as highly as his own achievements in this field. He writes: "In the construction of lines with proofs I am surpassed by nobody, not even the so-called rope stretchers of Egypt." By rope stretchers he probably meant surveyors, whose main instrument was the stretched rope. Figure 1-3 shows a statue of a rope stretcher with his coil of rope.

The oldest known Egyptian mathematical texts contain mostly problems of a practical nature, such as computing the capacity of a granary, the number of bricks needed for the building of a store, or the stock of grain necessary for the preparation of a certain amount of bread or beer.

The Rhind papyrus is our best source of information about Egyptian arithmetic. It is named after an Englishman, A. Henry Rhind, who bought the text in Luxor in 1858 and sold it to the British Museum, where it is displayed. This papyrus, copied by a scribe, Ahmes, and sometimes called by his name, dates from about 1650 B.C., although, according to the writer, it had been taken from an older treatise written between 2000 and 1800 B.C. The text contains about 80 problems. Besides including solutions for many practical questions, some of which include geometric concepts, it contains a number of problems that are of no practical importance. We get the impression that the author posed himself problems and solved them for the fun of it.

There are four other, smaller Egyptian mathematical writings of some importance: the Moscow papyrus, the Kahun papyrus, the Berlin papyrus, and the leather roll. There are many small fragments and commercial papyri scattered around the world, but they furnish only slight information about Egyptian mathematics.

No definite place of discovery is known for the Moscow papyrus. It is named after the city where it is kept. A start was made on deciphering it in 1920. The complete document was published in 1930. The papyrus contains about 30 worked-out problems. Figure 1-9 (see page 38) contains a picture of a part of the papyrus.

About 1900 an Englishman discovered a papyrus in Kahun, hence its name. This papyrus contains applications of the arithmetic methods described in the Rhind papyrus, but it contains little more of importance.

Since in the course of years the leather roll had completely dried up and become hardened, it was extremely difficult to unfold it without destroying the text. Modern chemical processes have made it possible to soften and preserve it. The leather roll, which is displayed in the British Museum, will be discussed and shown in Section 1-9.


Egyptian numerical notation was very simple. It used symbols for 1, 10, 100, ..., 1,000,000. In hieroglyphics these symbols were:


The symbol for 1000 was a lotus flower, for 104 a finger with a bent tip, for 105 a tadpole, and for 106 a man with his arms uplifted. Look back at Menes' mace in Figure 1-1 for examples of these symbols.

The numbers 2 through 9 were represented by two, three, ..., nine vertical dashes, as follows:


Tens, hundreds, and so on, were treated likewise. For example,


These symbols were often combined to represent other numbers. For instance,


Here the hundreds are represented first, then the tens, and finally the units, just as in modern notation. Hieroglyphics were also written from right to left, in which case the symbols themselves were reversed. For example, 324 could also be written as


We further observe the following:

1. A symbol for zero was lacking. For instance, when writing 305, which we could not do without the zero, the Egyptian wrote


2. The numerals were written in base ten. One symbol replaced 10 symbols of the next smaller denomination.


1. See Figure 1-1. Determine the number of oxen, goats, and prisoners claimed by Menes on his mace. Compare your answers with those given in Section 1-1.

2. Write these numbers in hieroglyphics:

a. 53

b. 407

c. 2136

3. What numbers are represented by the following:


4. How many different types of symbols are needed to write the numbers 1 through 1,000,000 in hieroglyphics? How many in our own numeration system?

5. Add, in hieroglyphics,


How many number combinations did an Egyptian student need to memorize to be able to add? How many does a modern student need to memorize?

6. Multiply


Can you suggest a simple rule for multiplying by 10 using Egyptian numerals?


In hieroglyphic notation, addition did not cause any difficulty. It was even simpler than in our system. There were no combinations such as 7 + 5 = 12 to memorize. Since the Egyptians knew that 10 unit strokes could be replaced by [intersection], 10 symbols [intersection] by [??], and so on, they could proceed by counting symbols in the two numbers to be added. Thus, they would write the sum of


directly as


Having counted 10 vertical strokes, they wrote [intersection] and then marked down the remaining two strokes without having to know that 7 plus 5 is 12 and without having to think: "I'll write the 2 and carry 1 (or 10) in my mind." And so on.

Subtraction was performed as shown by the following example. If Egyptians wanted to compute 12 - 5, they thought: What will be needed to complete 5 to make 12? Such a completion was called skm (pronounced: saykam). We use a modern equivalent of this process in making change today. For instance, when $5.83 is paid with a $10 bill, the clerk counts the change from $5.83 up to $10.00; thus: $5.83 + $0.02 = $5.85; $5.85 +$0.05 = $5.90; $5.90 + $0.10 = $6.00; $6.00 + $4.00 = $10.00. The clerk does not say all of this as he counts the change into your hand, nor does he go back and add all the underscored numbers — 0.02, 0.05, 0.10, 4.00 — to find the total amount of your change and hence the difference between $10.00 and $5.83. This completion process is mathematically sound. In ordinary algebra, and even in more advanced mathematical systems, as well as in arithmetic, subtraction is always the inverse of addition. Every subtraction problem, such as 12 - 5 = ?, really gives the result (sum) of an addition and one of the addends and asks for the other addend. Thus, 12 - 5 = ? really means 12 = 5 + ?. Mathematically, addition is a fundamental operation. Subtraction is defined in terms of addition and cannot exist without it. This fact is recognized when children are taught to check subtraction by addition, and when the subtraction facts are taught along with the addition facts.

The Egyptian method of multiplication was quite different from ours. The Egyptians used two operations to multiply: doubling and adding. To compute 6 X 8, for instance, they reasoned as follows:

2 · 8 = 16

4 · 8 = 2 · (2 · 8) = 32.

Addition on the left gives: (2 + 4) • 8, or 6 X 8, and on the right: 16 + 32 = 48. Hence, 6 X 8 = 48.

Problem 32 of the Rhind papyrus shows the actual procedure used by the Egyptians to compute 12 X 12. It goes as follows (reading from right to left):


It corresponds to the following (reading from left to right):


From top to bottom we see the results of 1 X 12, 2 X 12, 4 X 12, and 8 X 12, which have been obtained by doubling. The sloping strokes next to the third and the fourth line indicate that only these lines are to be added to get the desired product. The symbol [??] in the bottom line of the calculation in hieroglyphics represents a papyrus roll and means "the result is the following."

By way of an exception, the Egyptian sometimes multiplied a number directly by 10 instead of adding twice the number and eight times the number. This was easily done in his notation; he just substituted [intersection] for I, [??] for [intersection], and so on.


Excerpted from The Historical Roots of Elementary Mathematics by Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient. Copyright © 1988 Harry Bunt, Lidy E. M. Van Welzenis, Phillip S. Jones and Jack D. Bedient. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

The Greek alphabet
1-1 Prehistoric mathematics
1-2 The earliest written mathematics
1-3 Numerical notation
1-4 Arithmetic operations
1-5 Multiplication
1-6 Fractions and division
1-7 The red auxiliary numbers
1-8 The 2 ÷ n table
1-9 The leather roll
1-10 Algebraic problems
1-11 Geometry
2-1 Some historical facts
2-2 Babylonian numerical notation
2-3 The fundamental operations
2-4 Extraction of roots
2-5 Babylonian algebra
2-6 A Babylonian text
2-7 Babylonian geometry
2-8 Approximations to p
2-9 Another problem and a farewell to the Babylonians
3-1 The earliest records
3-2 Greek numeration systems
3-3 Thales and his importance to mathematics
3-4 Pythagoras and the Pythagoreans
3-5 The Pythagoreans and music
3-6 Pythagorean arithmetica
3-7 Pythagorean numerology
3-8 Pythagorean astronomy
3-9 Pythagorean geometry
3-10 Incommensurable segments and irrational numbers
4-1 Introduction
4-2 Hippocrates of Chios and the quadrature of lunes
4-3 Other quadratures
4-4 Hippocrates' geometry
4-5 Duplication of the cube
4-6 The trisection problem
4-7 Hippias and squaring of the circle
4-8 The solutions of the Greek problems
5-1 Philosophy and philosophers
5-2 Plato
5-3 Aristotle and his theory of statements
5-4 Concepts and definitions
5-5 Special notations and undefined terms
6-1 Elements
6-2 The structure of the Elements of Euclid
6-3 The definitions
6-4 Postulates and common notions
6-5 The meaning of a construction
6-6 The purport of Postulate III
6-7 Congruence
6-8 Congruence (continued)
6-9 The theory of parallels
6-10 The comparison of areas
6-11 The theorem of Pythagoras
6-12 The difference between the Euclidean and the modern method of comparing areas
6-13 Geometric algebra and regular polygons
6-14 Number theory in the Elements
7-1 The span of Greek mathematics
7-2 Archimedes and Eratosthenes
7-3 Apollonius of Perga
7-4 Heron of Alexandria and Diophantus
7-5 Ptolemy and Pappus
7-6 Review of the Greek method
7-7 Objections to the Euclidean system
7-8 The meaning of deduction
7-9 Euclid's system is not purely deductive
7-10 How is geometry built up purely deductively?
7-11 A four-point system
8-1 Roman numerals
8-2 The abacus and tangible arithmetic
8-3 The Hindu-Arabic numerals
8-4 An early American place-value numeration system
8-5 Later developments in positional notation
8-6 Conversions between numeration systems
8-7 Addition and subtraction algorithms in nondecimal bases
8-8 Multiplication alogorithms in nondecimal bases
8-9 "Fractions, rational numbers, and place-value numeration"
8-10 Irrational numbers
8-11 Modern theoretical foundations of arithmetic
8-12 Modern numeration
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