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. SYSTE~ ND THEIR PROPERTIES. . 10
B IBLIOGf?JkFHY...... , • • . • • 22
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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. A letter shaped like oir letter t~~tI, one somewhat like our letter ~ and one somewhat like the Greek ~‘Pj, 71 “.
They often placed / or by each letter to show that it stood for a number. Some examples of Ionic Greek numbers are: A. e =
52i= ~A~,/ec-’iTJ= 92~
This number system used a base of ten and the principle of addition.
It provided a compact number system and makes unnecessary the use of
1E~es:~Hó~r4, An Introduction to the History of Mathematics (New York: Holt, Rinehart and Winston, 1961), p. 15. 2 Ibid., p.11. 7 repetition. The complete alphabet is shown in Figure 1~.
The ancient Hebrews had established a system of alphabetic numerals similar to the ones used by the Greeks. The alphabets i~ere exhausted when the symbol I~oo ~as reached, the letters for 1~o0 and 100 were combined by early writers to represent ~ooPfl and similarly up to 900Vfl7~ . Later scholars used the final form of the letters for
20, ILO, 50, 80 and 90 (that is, the form of the letter that would be used at the end of a wrd) to represent 500, 600 ..., 900. The Thou sands were represented by t~ same letters as the units. The scheme appears in Figure
The traditional Chinese—Japanese numeral system is a multiplicative grouping system to base 10. ~iting vertically the symbols of the two basic groups and of the number 5625 appear in Figure 7.
The Roman system of numerals maintained a strong position for nearly 2,000 years in c~nmerce, in scientific and theological litera— ture and belles lettres. It had the great advantage that for the mass
of users, it was only necessary to memorize four ]etters —— V, X, L and
C. The simplest of all operations was utilized, addition. Later the subtraction principle was used. The Romans often wrote four as 1111 and less often IV. Nine is written IX, but until the beginning of printing it appeared quite often as VflhI. Nineteen is written as but it has also appeared as ~ Ei~teen commonly appears as XVIII, but IIfl was also used. (1) was a favorite way to write 1,000, but later changed to M. Half of this symbol (1 or 1) led to the use of 0
1Encyclopedia Brittiania, XVI., 6]5. 8 for 500. Sometimes the Romans wrote a bar over a number. This
multiplies the value of the symbol by 1,000. For example, X = 10,000,
C l00,000,and II 22, 000. The Roman numerals continued in the use
in some schools until about the sixteenth century and were commonly used
in bookkeeping in European cpuntries until the eighteenth century.
Our ingenuous number system is that great instruments of scientific
progress which Bhaskara and his Hindu predecessors developed. Mathema
tics ~.ias functioning in India even before the time of Buddha, who was
born in 600 B. C.1
The Hindu-Arabic numeral system, named for the Hindus who in
vented it and the Arabs who transmitted it, is used today t~roughout the major portions of the world. The Hindu numbers were introduced into
Spain by the Moors in 711 A. D. From Spain they were transmitted to
Europe and then to America.
The number system which the Hindu gave us employs the base ten, in which there are ten and only ten symbols:
0, 1, 2, 3, L~, 5, 6, 7, 8, 9
With these ten symbols, all possible numbers can be written. The decimal system uses the idea of place value to represent the size of a group. For example, 555. The right hand five represents S units, the next five represents ten tiiaes as much as the first five. (5 10) and the third five ten times as much as the first five. The feature which distinguishes the decimal system from all other systems, i~ its place value idea and its symbol for zero. The early specimens did not have the idea of place value or a symbol for zero. The idea for zero
‘Veeriah V. Kota, “On the Hindu Numbering System.” Lecture de livered to Mathematics Seminar, Atlanta University, Atlanta, Georgia, July 2, l96)j. 9 may have come from the religious nature of the Hindus. The priests were the highest group in their social order, the rulers next, then the merchants and finally the ~orker. Above the priests ~as God.
The following diagram shows how the symbol may ha~e been invented :1 6od->o
.p~I~313 f~ui~ ~&S M ~ CI+4ilr~ Wo ~ when a person became. a priest he had gone as far as he could go and above that ~as God or Nir~i~ which meant ~mtIeteness~ Therefore, completeness ~inplied that they needed nothing more; hence, our symbol zero.
The symbol zero not only keeps other symbols in proper location, but also indicates that there are no groups of a certain size present in the number. Exam~1e 3OLj.O means there are three thousands, no hundreds, four tens and no ones.
Thus the Hindu-Arabic system has place value and face value.
These numerals are sho~n in Figure 7.
1Veeriah V. Kota, op. cit. CHAPTER III
SYSTEMS AND THEIR PROPERTIES
Pierre Simon LaPlace, a famous French mathematician, called the
decimal system one of the most useful inventions of the ~rld. The
names of the digits, the use of the piace value and the symbol for
zero make it the most convenient system used. It has been discovered
ti-at much can be learned about our base ten number system by consider
ing a system other than base ten. Following is a discussion on the
nature, properties and operations in the binary system and the duo—
decimal system as compared with the decimal system.
Although there are only 0, 1, 2, 3, L~, 5, 6, 7, 8, 9, in the
decimal system, we can construct ninbers as ]arge as we please by using
the symbols again and again. Starting with the first place on the right
which is the units place, each place to the left is ten times ~eater
than its predecessor. A general rule by thich an integer, N of base ten can be expressed is: Rule 1.1: N~ Qn (10)M * an—, (~o)”~ + ..~ -f-a (Io)-fQ0. and is represented by the symbol tlr On—a ... 0. Oa~
Example 1.1: If N~ 22,Z’17 then from rule 1.1 N Z27_’472(IO)’~
+ 2 (,o)~÷ 2 (j~)2 ~ ‘~ (,o)’ 1-7 ‘Io°.
Where the digits a0, a,, Q~., . . . , a,, are the successive re mainders when is divided repeatedly by ten and each digit is less than ten~.
Conversion to Non—decimal Systems
10 11
In the decimal system, the number ten is used as the base. Any
integer greater than one can serve as base. Thus, converting from the
decimal system to non—decimal systems is relevant and is discuss~d in
two methods.
Remainder method. —~ If we have a number expressedin base ten,
we may change it to base b as follows.: letting N be the number we
have to determine the integers O~, O~..g,.r.,C0in the expression
• * +~ b÷ a3, ithere
Dividing the above equation by b, we have R~I€ 1. 2 a2~÷a,÷ ~ That is, the remainder a0 of this division is the ).ast digit in the desired
representation. Dividing N’ by b, we obtain = Q~b~2÷a~_~ ~
-~ a. - W . * . + M~ Proceeding in this manner, we can obtain all the digits ~
This procedure is shown below:
Example 1.2: If 2~,2Lt7 is converted to base tweive, then and b 12 It follows that: ~ ______
~ 12[~ 12 T. 12L I L Ol The column on the right represents the successive remainders by applying rule 1.2. T~ence, the number in base twelve is 10 T ~ E, ~.hich is repre sented by rule 1.1, as f(l1’1).÷ a(12).+7~(I2?)÷-5CI2’)+ E(12°).
Therefore i2,247~~ JOTtTE12.
Example 1.3: If ~S~,10is converted to base two, then N=~ 5~ 8 and 2. 12
• it follows that:
2 32’? ~ 2 •I.6~i’ 1 2 ~a a 2 ‘ii o• 20 0 2 JQ_ 0 2 ~O
.~- -z •~T~ 2 I C 01
The column on the right represents the successive r~nainders obtained by applying rule 1.2.
Therefore,
Quotient method. —- The general rule for changing from base 10 to any other base b is to divide N by the highest po~er of ~ and contained in N. Then divide the remainder by the next 1o~r power of b and con tinuing until the divisor is the base itself. The final remainder in dicates the number of ones and the first quotient indicates the digit for’ the largest place value. That is if N is divided by br (r is the largest power of b contained in N), then the number in base b is:
Rule 1.3: C~ ~b *C.~—i,(h)’~’;+ ...ic,(b)+ Ce,, where c0 is the final remainder and C1, CZ,...C~are the successive quotient when N is divided by b. The representation of the number in symbols is Cii Ca—, CsP7—Z •.. Ci Co,. ~‘nere each digit is less than the base b.
Example 1.3: The conversion of ~ base two is: 3 .572)~ó7(I
2Y-•:2 ~-‘(0 2. —171 2~ Lt/ L2~L ~ 32 .2~q~: ~ ~3... ~r 1(0) igc:’i
2-‘—~2 7 3 4)~(D
) ~ (a remainder 0.
Therefore, (o6~~ IOO~iD~IO2 by the quotient method. Example ld~: The conversion of 22, 24’7~ to base twelve is: 12~ ~o,736 ~73~)22,2~7(/ 12g= 1,izS O373~4 12~ 1L14’ 17z8)JrJ/(O I2’~ 12 1 1’41/o jzJ 77(5 1
Therefore 22,2’/7,~ 1oY5E~
Similarly base ten numbers can be converted to any other system using these methods.
Computations. —— Numbers in bases two and twleve are added, sub tracted, multiplied and divided in the same manner as we perform the operations in base ten. For these computations, consider these addition and multiplication tables. iL~
Base Two I A I~ .±o1~ xQi 00i 0 Qo~ L110 .101 Base Twelve
Table 2-A
+0i234f56789T5
~2’ 23 4’5~ 7SqTE ~ 223qo67gq~ff/0// 334’ ~r~7i 9 re,on/z
44/ ~ ~, 7~’9 T ~/OJ/J2/3
£ 5 ~ 7 b’ 9 T ~ /o /1 /2 /3 /~‘ ~671qrE/~/,/~,3/y/~-
77 ~‘‘~ Tff/~ii/zi~/’,’/s/6 ~ 9 T if /0 /1 /2 /3 I’! /~/~ /7 q q r e /0 1/ /2 /3 /V /~/~ /7 /2
T T if /~ Ii /Z /3 /~ /~ /~ /7 /1 /9 E t-~/o// /2/3/~’/~/~/7/1/9/r ‘5
Base Twelve
Table 2—B XOI Z3~.567~9TC 0000000000000 /0/2 34t$~ 7197w 7 0 2 4’ ~ $~ 7 /0 /2 /4’ /6/, IT 3 0 3 4 q ic /3 M /92023 2~ 29 4 a 4 1 ‘0 /4/1 zo Z9Z~ 30 3~’~ ~ 0 5 713/2 2/ 2~ 26 ?~‘ 39 ~(Z ~‘7 6 0 6 lO ‘6 20 Z~ 3o 6 *~ ~ 7 07 ‘2 ‘9 ?N Z6 36 9/ /4’ ~ 5763 Y O I /4ZoZ~?~9o94’c~’~o68’74’ 9 0 ‘? /~ 23 3o 3q ~16 ~1 ;~ ~9 7613 ~ T/1z63orr4P76~4’9,.. 0e /rzqgg 9754~.s-7c’139zT/
Given the set of numbers S {a~, b1 c, . . .
~om our tables we e’an see that the product and sum of any ~ given numbers can~be found in the table. H~ice, the systems are closed under the operations of addition and multiplication.
For any two numbers, a and b, a 4- b = b -I- a and a.b = b.a.
The systems are therefore commutative under the operations of addition and multiplication. For any three numbers, a, b, and c -(b.i-c)=(~+b)~-c~tid a.(b.c,)= ca1b,~.c. Thus the associative law for addition and multiplication holds for any system. This can be verified from the table for each of the operations of addition and multiplication.
For any three numbers a, b, and c, the relation a(b+ c) ab+ ac. _____ IOI(O~
16
~ This indicates that multiplication is distributive over addition.
Example: Add 101102 and 110102 4+b 101101,
Hoo~0~
In the one’s column 0 ~- 0 = 0. In the two’s column 1+ 1 — 10.
0 is written and the 1 is regrouped with four’s column. The numbers
are added until all columns have been exhausted. Hence, the sum is
obtained for the two numbers.
101102 + 110102 110102 + 101102 The commutative property of addition for the binary number system.
Example: Add 3T0LIE12 and 51TT12 3T°’4~I-V 51T11V ~$T0LW~ Lt~3z~~7,~
From Table I-B. 3T0t~E12 ~fr 5flT~ = 51TT12 ÷ 3T0L~E12
The comuutative property of addition holds for the duodecimal system.
Example: Subtract 101112 from 110,1012 110, lOu.. 101II~.. !iHO~
Using Table 1-A we perform the subtraction in the same manner as we per
form subtraction in base ten. 17
Example: Subtract 19312 from T3212 T~21-..
The question is what we must add to 3 to get 12. From ~ table
base 12, 3 must be added to E to get 12. ~Jhat must we add to 9 to get
12? From thetable we add ~ to 9 to get 12, etc.
Example: Find the product of 10112 and 1012
/oi~ IC//ti
Using Table 1-B, multiplication is performed in the same manner as it is on base ten.
Example: Find the product of 10312 and L~212
2~6
‘fsô ~,a. Again, using Table 1—B, we multiply, get the partial products and add to get our total product.
Example: Divide 101002 by 1002 ______/~~~T7o ~ ~/00 /dO
Using Table 1-B, we divide in base two in the same manner as we divide 18
in base ten,
When 101002 is divided by 1002, the quotient is 1012. 1012 1002
101002.
Example: Divide L~306l2 by L,212 ______qz,2 P~3Qb~~. Io(~ ~Io’
When I~20612 is divided by U212, the quotient is 10312. 103 X L~2
1~306l2.
~ to the decimal system. -- In converting a number ~nto
base ten notation, we write the nimiber in expanded notation and add the results.
Example: Change ITE to base ten. /TE=(/Y,z~)+/o(12!) ~~I/(/zo)= /~i~/2Z~-// 277
Example: Change 1101012 to base ten, (~xz~’)-t(ix~ ~i~(OX23)~I.(,Xz2)* (oxz’)÷ (IXzo)
32+I6 +‘-~ -fri =5-3
Conversion from ñondecimal bases to nondecimal bases. ——
1. Express the desired base (base1) in terms of the base (base2)
of the given ni~nber.
2. Apply rule 1.2. 19
3. Proceeding from the last remainder obtained, these digits
represent the digits in the desired base (base1). If the
desired base (base1) is larger than the base of the given
ni~ither (base2).
Example: Change ~ to base two ~ithout using base ten. Apply ing rule 1.2.
z 5’i~~. 221o 2 g~ o 2~o 24~ 22o 210
Therefore ~~l2 = 10000002.
To prove ~~i2 10000002, simply change 10000002 to base12.
1. Express 12 in base2. 12 = 11002
Applying rule 1 • 2, we obtain //ool /OOc~oac~ /100~ o / / ~J—~ ~ i/oaf o /0/
Therefore, 10000002 = CONCLUSI ON
The binary system has only the symbols 1 and 0, the ~ and Toff’ of
the high speed computing machines. Children studying binary arithmetic would
find it easy to learn the number symbols. The addition rules reduce to 0 f 0
= 0, 0 ÷ 1 = 1, 1 + 1 10. The multiplication rules are these: OX 0 = 0,
.0 x l. 0, 1 X 1 = 1. It would be helpful to pupils in the first and second grades in learning number facts.
The base twelve number system may prove more useful than our decimal
system in some areas of our number system, such as expressing common fractions.
The Hindu~Arabic system from the East gave to the West a powerful tool with
~thich to develop modern mathematics. The fact that we can write a number as
large as we please and a number as small as we please, by u~ing scientific no
tation,has enabled scientists to write the distance between the stars to com
pute minute quantities necessary in a chemical formula, and to open up fields
of scientific endeavor which were undreamed of a few years ago.
It is inportant for teachers of young children to stop teaching by rote methods because it causes fear and a dislike for mathematics. Any successful
teacher ~o is going to contribute to future generations of mathematicians
has to investigate thoroughly the inner working of our numbering system. It
is amazing to note how few people do understand long division computations.
This may be due to the fact that the basic training at the elementary and
high school leveismight have neglected to give the basic operations of our
number system in a manner in which the pupil could successfully build his
mathematics into a powerful tool.
It is hoped that this Study of Our ~‘esent Numbering System with a
historical approach will maka teachers of mathema~cs, and people in general
20 21 realize that “Life is mathematics and mathematics is Life.” ______AandHistoryKarpinski,of Mathematics.L. C. The 2d.Hindu-Arabiced. New York:Numerals.The MacmillanBoston: Co.,Ginn
BIBLIOGRAPHY
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Bell, E. T. The Development of Mathematics. 2d ed. McGraw—Hill Book Co., Inc., 19145~.
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Struick, D. J. A Conoise History of Mathematics Tools. New York: Dover Publications, l9Lit3.
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Benton, William. Encyclopedia Brittanica. Chicago: London and Toronto, l9~8.
Lecture
Kota, Veeriah V. t~Q~ the Hindu Ni~nbering System.t’ Lecture delivered to Seminar in Mathematics, Atlanta University, Atlanta, Georgia,, July, 196L~.
Periodical
Young, Miriam H. II, No. ~. “Niwiber in the Western World,tt Arithmetic Teacher (May, l96I~), 337.
23