A STUDY OF OUR PRESENT NUMBERING SYSTEM: AN HISTO~t CAL A~RO&~H A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FUIJFILLMBNT OF THE REQUIREMBNTS FOR THE DE(~?.EE OF MASTER OF SCIENCE BY MAE FRANCES WIlSON DEPA1~rMBNT OF MATHEMATICS ATLANTA, GEORGIA AUGUST 196t~ ii ~‘- TABLE OF CONTENTS Page LIST OF FIGURES. • . • • • • Iii Chapter I. INTRODUCTION. .• . 1 II. NU_~4ERATION . 3 III. 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SN3± b~~9S-&~ZI qLL[LLLLCt~~ S~LINfl $IV~WflN ~ 9 e.in2t~ ~,1- co’ 4- 01 itt b 37d WYX~ 9 -ç 4.7 z I 5 7j1≥13(AJPAJ 3~3/IVd4’I— 3S3IVIfI) 741>13W fiN ~‘flQ ~I~HflN ~S~vdvf-~9~NIHD L ‘~T~!i •~7’~74,4J Qb’ FQ~/ ~v ~, Ob~ ~J9h-6 ~ZL ‘q~’ wi ~44OA/ -% A’~VW..t f7~fs~3,4/g4/ ~ b VA L C’? ~ ~4 I Q’b’ 914 ~9A 000/ P~ ~ $~TVV~flN ~IEWV-flG$EIR CHI&PrER I INTRODUCTION The meaningful approach to the teaching of arithmetic is widely accepted. A criterion test used to obtain a measurement of quantita tive understanding lies in the degree to which there exists an in si~t into the structure of the number system and number measurement. It i~s Pythagoras who stated that “Everything is number~ and he and his followers believed that although real integers were a part of the real world, tley were also endowed with quantitative characteristics.1 The author is interested in making a study of the quantitative characteristics of various number concepts as they appear and as they relate to the comprehension of place value. The major purpose here is to present the vie~oint that a study of the historical development of numbers and the properties of bases other than ten will help pupils achieve a better understanding of our adopted Hindu-Arabic system of place value. This treatise will include discussions of numeral sys tems and the ancient number systems. These are presented in Chapter II in as a brief a manner as the author deemed consistent with a thorough understanding. Chapter III deals with an analysis and interpretation of place value of the dec~iual system, binary system, and duodecimal system. For the most part emphasis is placed on operations and translating from one number system to the other. I wish to acknowledge my deep appreciation to Dr. Veeriah V. Kota Miriam H. Young, “Nirther in the Western World,” Arithmetic Teacher II, No. ~ (May, 19614), 337. 3. 2 for valuable guidance and advice in the successful completion of this thesis. CH&P~ER II NU~RATION The concept of number and the process of counting developed long before the time of recorded history. There is not a language ~ithout some numerals. Numerals are nb~äD for numbers. Numeration is the study of how symbols are written to represent numbers. Counting was an outgrowth of the need to keep records. Primitive people learned to keep records by numbers by employing the principle of one-to-one correspondence. The pebble, or the notch on a stick would represent a single sheep. Counts could be maintained by making coflections of pebbles or sticks, by making scratches in the dirt or on a stone, by cutting notches on a piece of wood or by tying knots in a string. As centuries passed, early people used sounds or names for numbers. Noon for one, and eyes, feet or hands for two. For larger numbers they used “lots,” “heap,” “school,” ~ ~more heap,” “many ~ When it becomes necessary to make more extensive counts, the counting process had to be systematized. This was done by arranging the numbers in basic groups. The pair system starts from parts of the human body that exist in pairs, like eyes, ears, hands, feet. The numerals in this system are one and two and form the following numerals by addition to the “pair”: i:2*i, ‘~2~-Z, 2~’-2”/, ~~2#2.~’-L, 7~ L#Z1-Z ,‘- /, .‘Z#-a* 2~+ 2, and so on. This system is found to be used by tribes in Australia, Africa and South lEncy~lopedia Brittanica,XVI, 6l~. 3 America The Quaternary system forms the numerals above four in this manner~ 5~ Ij7L./, 7~’*3, 6=~L,t.~/ o~ (2xt,L) A’~3 /6~ 4’~/~ The Thid Indians in California use this system because they use the spaces between the fingers instead of the fingers. The Quinary scale, the number system based on five, was the first scale.that was used extensively. In its pure form, this system is found in Africa where 10 = 2 hands, 25 — S hands. Anthropologists present strong evidence that the decimal system, base 10, is used simply because the human hands possess ten fingers. In this system we count to 10, then to ten tens, then to ten times ten and so on. All over the civilized world today we find men count ing by ten. Because the decimal system used the idea of place value, it is recognized as the most convenient system. The Duodecimal Society of I~merica has worked out a system of counting by twelves. Many mathematicians have expressed the opinion that the duodecimal system should serve as a better system in the representation of fractions as compared with our present decimal system because 12 has a greater number of factors than ten. There are evidences that twelve was often used as a group; for e~camp1e, 12 units equal one dozen, 12 inches equal one foot, 12 ounces equal one pound (old style), 12 pence equal one shilling, 12 lines equal one inch, 12 months in a year and 12 hours about the clock. ‘Op. cit., p. 615. S There were many number systems used throughout civilization. The most important among these are the following: the Egyptians, the Babylonians, the Greeks, Hebrew, Mayans, Chinese—Japanese, Ronans and Hindu-Arabic. These are discussed in the paragraphs which follow. One of the earliest systems of which we have definite historical record is that of the Egyptians. Because of the Egyptians well-developed government and business, they required. considerable use of numbers. Their picture numerals called hieroglyphics may be traced as far back as 3300 B. C. The Egyptian numerals are shown in Figure 1. Generally, they ~‘ote from right to left, as in the Semitic script. But the hiero glyphics were occasionally ~‘itten from left to right. In order to read larger numbers, the symbols were arranged in groups of three or four. These are shown in Figure 2. As an example in this system: z~szq III ~ 1) fl iii Five thousand years ago the Sumerians and Chaldeans had developed a high degree of civilization. They could read and ~ite and had a system of cuneiform (wedge shaped) numbers and numerals. About a thou sand years later, the Babylonians learned how to use these numerals and stamped them on clay tablets with a stylus and baked them in the sun. The Babylonian system was a mixed system with numbers less than 60 expressed by a simple grouping system to base 10 using the addi tion and subtraction principles. Numerals larger than 60 employed the principle of position with no zero symbol, for example~ ~2~5S5~ 2(6o~*2~~o)~+42(6o)*31 — ~ irv 4h ViAL Thus the reader had to carefuily study the context to determine what 6 the numeral meant. These numerals seemed to have continued in use for about three thousand years. These numerals are shown in Figire 3. The Greeks had several ways of writing their numerals, but the author in this thesis considered only two of them, the Herodianic (Attic) Greek and the Ionic Greek. The Attic or Herodjanic Greek numerals constituted a simple grouping system of base 10 formed from initial letters of number names. In addition to the symbols I,41H,X, coLI,Io,lO~ 1O~Io~ there is a special symbol for 5. This special symbol is an old form of fføp. P the initial letter of the Greek pente (five) and 4 is the initial letter of the Greek deks (ten).1 As an example in this number system, we have2$57= KXI1~’UUH1W P11 the symbol for five appears once alone and twice in combination with other symbols.2 The Ionic Greeks used the 2I~ letters of their alphabet together with three other symbols.