Ancient systems of numeration-stimulating, illuminating

Ancient systems of numeration-stimulating, illuminatingAuthor(s): IRVING M. COWLESource: The Arithmetic Teacher, Vol. 17, No. 5 (MAY 1970), pp. 413-416Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186225 .

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Ancient systems of numeration- stimulating, illuminating IRVING M. COWLE Fairleigh Dickinson University, Teaneck, New Jersey

Irving Cowle teaches mathematics education courses at the Teaneck Campus of Fairleigh Dickinson University. He has had extensive experience in public school teaching that included teaching grades 1-4 in a two-room rural school.

JJr lementary school children find the study of ancient systems of numeration a fascinating topic. Often an interest is kindled by this initial brush with these early civilizations that stimulates many a youngster to continue the study and investigate further, in depth, on his own. This child is richer for the experience and, properly guided, the entire group will benefit as well when the researcher reports his new discoveries to his classmates.

Highly appropriate in the intermediate grades, a topic of this type can be worked into the study of a major unit; take Egypt, for example. Certain students could investigate the system "of numeration - while others examined and reported on the art, music, industry, and other facets of that civilization. Even though such an approach might not be suitable or appropriate, the consideration of certain historical systems still has great value for both class and teacher. As initially mentioned, it is truly fascinating; children love the topic. Further, it traces the development of number systems from ancient times right up through the present day. With proper selection and emphasis, the youngsters can see the weak- nesses in past procedures and thereby develop a genuine appreciation of their own Hindu-Arabic system of numeration.

The elementary school classroom is a place for providing a reasonable historical background and hopefully, with many stu-

dents, kindling a desire to look further. The author selects the Egyptian, the Roman, and the Babylonian systems of numeration for study, in the order given. It might be pointed out that, though not necessarily sequential from the standpoint of the time of origin, there is a certain developmental basis for this order that may be apparent in the following development.

The earliest representation of numbers is generally attributed to the civilizations in Egypt and Mesopotamia (Babylonian) about 5,000 years ago. Both had evolved a method of chipping notches on wood to record the passing days. The priests of Egypt also worked out a system of mark- ing on papyrus, a writing material made from the pith of a reed (called sedge) that was sliced and pressed into a scroll. They used simple strokes, called a staff, for the numeral 1 , with combinations of these used all the way up to nine. It is interesting to note that the Egyptians, when writing, never put more than four symbols horizontally together. Thus, the numbers were represented as shown in figure 1. The reason for writing in this manner (doubtless because

i 1 и " m « ЯИ «i ||( Hl IIM llu !!! i 1 и " « m «i ЯИ II 111 111 lili j¡¡ 123456 7 8 9

Figure 1

Excellence in Mathematics Education - For All 413



the scroll had more room vertically than horizontally) can provoke a thoughtful, stimulating discussion in your classroom.

For ten, the Egyptians used a symbol that resembles what we now use for set intersection; this was known as a heelbone or yoke - and certainly a discussion of the origin of this name is highly appropriate in your classroom. The symbol for one hun-

,o-n dred will vary according to your historical source. It is sometimes called a scroll, and

9 9 Scroll Coil

at other times it is termed a coil The varia- tion poses no problem with your class; simply show both forms and then decide which way is to be used. The author leaves beyond 100 up to the individual student's own curiosity. The lotus flower

i '- i for 1,000 comes next and there are symbols for

10,000 ( the bent line, or fìnger { ) ,

100,000 (a fish Oí),

and

1,000,000 С man in astonishment Д ).

Children will note with interest that the Egyptian system is based on ten, just as in our own decimal system - but at that point the similarity ends. Theirs was a pure addition procedure, without place value. A per- son simply took the symbols, one by one, and totaled their values together. If many heelbones were written in a row, each individual symbol was worth no more than any other. Position meant nothing, nor was there any symbol for zero.

Two points worth emphasizing are the lengthy and cumbersome process necessary to write some numbers in the Egyptian system:

nnnin »-ППП III-

ODD III and the fact that, since position is mean- ingless,

|| П-» -* ПН- '2- It is then worthwhile and challenging for the students to change some of our own decimal system numerals into the Egyptian system and also to go from Egyptian to our numerals.

The Roman system of numeration, that evolved some three thousand years later, fits nicely into the sequence at this point. Though the Romans also had no place value or zero, their system had modifica- tions that represented real progress over the Egyptian system and was far simpler and less cumbersome to use. With but seven symbols, the Romans could represent any number they needed.

|=| io - X 100 - С 1,000 = M 5 = V 50 = L 500 = D

We may term the Roman's system a modi- fied-addition system since, with fours and nines, a subtraction principle is utilized. Emphasize to your class that, though we in our hindsight might extend this principle to write other numbers more easily, the Romans used but six combinations in subtraction. Thus, to be authentic in our us- age, we must abide by their procedure.

4 - IV 9 = IX 40 = XL 90 = XC

400 - CD 900 = CM

Let students see that 99 cannot be a tempt- ing 1С, but must be written as XCIX. By

414 The Arithmetic Teacher/ May 1970



illustrating with this particular number, the contrast with the Egyptian system is vividly evident, and one advantage of the Roman system becomes obvious.

Egyptian Roman

nnnni »»-ПОП III - *cix

nnnni The Romans had a procedure that was

particularly valuable when writing large numbers. By placing a line above the given symbol, it was then multiplied by 1,000. Thus XV = 1 5,000. Ask your class how they would write 99,000 in the Roman system (XCIX), and then in the Egyptian system (99 lotus flowers). At this point, practice in the reading and writing of Ro- man numerals is appropriate and valuable. Students should be able to readily compare the Egyptian and Roman systems, mention- ing the shortcomings of each, while point- ing out the obvious advantages the latter has over the Egyptian system.

The Babylonian system of numeration contained an element that represents a step beyond that of the Egyptians and Romans; they had place value. Though usually termed a base-sixty system, in actuality the place value really is evolved by taking first a product of ten (times the value of the initial symbol on the right; this gives the value of the second symbol), and then a product of six (times the second symbol; this gives the value of the third symbol) -

continuing in this manner. The important step forward was that no longer did two like symbols standing side by side necessarily have to represent the same value.

The Babylonians wrote with pointed sticks that made wedge-shaped signs on tablets of soft clay. As one moved from right to left, the value of each symbol in-

} - i - } 3,600 Each eOO Each 60 Each 10 Each I Each

creased. Unfortunately, there still was no symbol for zero. For this reason, one then finds alert students asking questions such as: "How do we know if

if is 61, 3660, or 3601?" My answer (somewhat facetiously) is "You have to be a Babylonian to know!" Because of this com- plication, it is a difficult number system to use. However, noting this obvious problem achieves in itself an important purpose -

creating a real awareness among the students of the value of having some sort of placeholder.

Thus, a better appreciation then devel- ops for our own Hindu-Arabic numeration system. Perhaps briefly, note its evolution beginning with the Hindus in 800 a.D., picked up by the Arabs in 900 A.D., then the Spanish in 1000 a.D., through the Ital- ians in 1400 a.D. Ultimately it became the system that is now in general use through- out the world; its advantages are now better appreciated, having noted the weak- nesses in earlier systems. Using ten symbols, and based on powers of ten, the Hindu- Arabic system has both place value and a symbol for zero.

A fascinating little historical note - re- portedly conceived over 1 ,000 years ago in Morocco - is the work of a mathematical genius of that day. It can make an interesting bulletin board display, with each of ten children taking a numeral and putting it on construction paper. This mathematician took the Hindu-Arabic numerals and at- tempted to draw them so that each contained the same number of angles as the number the symbol was to represent. This is how he did it.

Thus 0 contains no angles, 5 has five angles, and so forth. You will find your students checking out each of the numerals

Excellence in Mathematics Education - For All 415



for accuracy. Though the 1 and the 7 seem somewhat contrived, it makes an interesting project for any willing student to see if they may both have at one time been written in the manner shown.

A teacher who wishes to kindle a curiosity about ancient civilizations through a study of their numeration systems would find, in addition, that students are also de- veloping a new appreciation for our own decimal system. Hopefully, some of the children will be motivated to investigate the workings of other ancient numeration systems on their own; you might suggest that of the Mayas of Central America. Though completely cut off from the civilizations of the old world, the Mayans developed in-

dependently a base-twenty system in which they could write any number with the use of only three signs: a dot, a stroke, and a type of oval. There is ample documentation of their procedure that is readily accessible to the youthful researcher.

For an interesting and rewarding topic for study, try a brief look at the development of numeration systems by examining those of a few of the better-known ancient civilizations. Students find the unit stimulating and it gives them a tangible basis for a genuine appreciation of our own decimal system. With many youngsters this intro- duction stimulates a curiosity and a thirst that can only lead to further fruitful and worthwhile activity.

NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS

Joint meeting with the National Education Association

Thursday, July 2, 1970 San Francisco, California (Location to be announced) Host Organization: California Mathematics Council, Northern Section Chairman, Local Arrangements: Lyle Fisher, President, California

Mathematics Council, Northern Section Representative, NCTM Committee on Meetings: Kenneth С Skeen

Program 12:00 noon-12:45 p.m Showing of NCTM films 1 :00 p.m.-2:00 p.m First General Session

Presiding: Lyle Fisher, President, California Mathematics Council, Northern Section

Welcome: Lawrence D. Hawkinson, State President, California Mathematics Council

Speaker: Charles Allen, Los Angeles City Schools, Los Angeles, California

Subject: Creative Teaching for Low Achievers 2: 10 p.m.-3:00 p.m Second General Session

Presiding: William Leonard, Vice-President, California Mathematics Council, Northern Section

Speaker: Frank Allen, Elmhurst College, Elmhurst, Illinois Subject: Residual Values of School Mathematics

4 1 6 The A rithmetic Teacher /May 1 970



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Ancient systems of numeration-stimulating, illuminating