Requirel'fflnts for ATLANTA, GEORGIA
ON FRACTIONS IN SOME NUMERATION SYSTEMS
A THESIS
SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREl'fflNTS FOR THE
DEGREE OF MASTER OF SCIENCE
BY
THEODORE ROOSEVELT NICHOLSON
DEPARTMENT OF MATHEMATICS
ATLANTA, GEORGIA
JANUARY 1965 TABLE OF CONTENTS
Page
CHAPTER I INTRODUCTION 1
A, Historical Background 1 B. Definitions 3
CHAPTER II FRACTIONS IN THE DECIMAL SYSTEl'I OF NUMERATION.... $
A. General Representation of Fractions 5 B. Conversion of Base Ten Fractions in Other Numeration Systems 5 C. Converting Any Decimal Fraction to Any Other System 8 D. Conversion of Fractions in Non-decimal Systems to Fractions in the Decimal System... 11
CHAPTER III ARITHMETICAL COMPUTATIONS 13
A. Addition of Fractions in Different Numeration Systems 13 B. Multiplication of Fractions in Different Numeration Systems 15
BIBLIOGRAPHY 18
ii \ «
CHAPTER I
INTRODUCTION
Much enphasis has been put on the expression of integers in base systems other than ten, but very little work has been done in the field of numeration systems regarding expression of fractions. Currently, many textbook writers are including this material in books for elemen- tary, junior and senior high schools. Many believe that the conversion of integers from one numeration system to another increases the stu¬ dent's computational ability. The author of this thesis through years of teaching experience has observed repeatedly that students experience great difficulty with computations involving fractions, and since stu¬ dents develop skills in arithmetical computations to a high degree by the use of integers and fractions, it is the purpose of this thesis to present methods of converting fractions from one system of numeration to another.
Historical Background
With the advent of the nuclear age a great deal of work was amassed in different numeration systems by the use of digital computers. Here the base two was used extensively, but in this thesis the author hopes to show ease of coirputation and conversion of fractions into base 7,
12, and 10,
The theorem on conversion of integers into different numeration systems will be extended to include the conversion of fractions in dif¬ ferent nximeration systems. It ns a well known and often used fact that any integer in base ten can be expressed uniquely by place value of the digits as being some power of the base ten. This being true, it follows
1 2 that fractions should be able to be expressed uniquely also.
Chapter II and Chapter III of this thesis will be entirely devoted to conversions and confutations. Chapter II will present conversions of fractions in different numeration systems and Chapter III will present computations in various numeration systems.
The history of numeration systems shows that base systems grew out of man's need to express quantities numerically. When man began to use counting more extensively, it became necessary to systematize the process.
A number b was selected as the base for counting, and names were assigned to the numbers 1, 2 b. Names for numbers larger than b were then essentially given by combinations of the number names previously selected. Research also indicates that man developed different base sys¬ tems by using his fingers on one hand, the spaces between his fingers on one hand, both hands, both hands and both feet, and many other aspects of his environment contributed to the development of different numeration systems.
Our own numeration system is an example of a place valued system with ten as a base. The symbols selected for use with this system are
0,1,2,3,U, ,b-l. Consequently, there are b basic symbols, which we cass digits. Therefore, for any integer, a, it can be written uniquely in the form
a = rj„b™ + r„i.itf"-l + + + Tq where CL = o, >,1,, ■'
Now this integer may be written by the sequence of symbols
■^n'^ *■ 1 •^2'^iAd
As a result of this, a basic symbol in any numeral simply represents a multiple of the base raised to a power. The above statement for the 3 unique expression of sin integer is to be extended in this thesis to in¬ clude the unique expression of a fraction or a mixed fraction.
The following definitions will be used:
1. Numeration system - a set of numbers along with one or more
operations.
2. Set - a collection of distinct objects.
3. Integers - a set of numbers consisting of the natursil numbers,
their negatives and zero.
U. Base - the number of units in a given digit place or decimal
place, which must be taken to denote one in the next higher
place.
5. Fraction - any number of the form a/b where a and b are natural
numbers.
6. Mixed fractions - a fraction and an integer together.
7. Algorithm - a step-by-step iterative process, completed in a
finite number of steps, for solving a given problem.
8. Numeral - a symbol used to denote a number.
9. Digit - a term applied to any of the integers occurring in a
number.
10. Place value - the value given to a digit by virtue of the place
it occupies in the number relative to the units place.
11. Quotient - the quantity resulting from the division of one
quantity by another.
12. Division - the inverse operation of multiplication,
13. Addition - the sura of two or more objects.
II4., Subtraction - the inverse operation of addition.
l5» Multiplication - the process of adding repeatedly,
16, Factor - one of the members of a product. h
17. Exponent - a number placed at the upper right of a symbol
which tells how mary times that symbol is to be used as a
factor.
18. Remainder - the divadent minus the product of the divisor
and the quotient. CHAPTER II
FRACTIONS IN THE DECIMAL SYSTEM OF NUMERATION
General Representation of Fractions. — A fraction is defined to be a number of the form where a and b are costing numbers.
The decimal expansion of a fraction between 0 and 1 has the follow¬ ing property:
After a finite number of terras, the expansion repeats, i. e., it takes the form
a. ... ^nr • • •■
For example r . /^ J ^ ^ ^ ^ '
where the sequence 63 repeats ad infitium. This shows that even though the fractions are non-terminating decimals, it is proper to indicate them in their decimal form as a terminated number with the repetition at the end of the decimal.
Conversion of base ten fractions to fractions in other numeration systems
Theorem. — Let b be a positive integer greater than 1. If a is any positive integer, there exist a non-negative integer m such that a can be expressed uniquely in the form
In (1) A — I* Td + p Td + .+ /H r, b + r^^ where 0-c r^b and Oi ^
(2) a = + 0 < r < b qOb r^ o
5 6
by the division algorithm, and clearly > 0.
If q < b, we set r = q and have the form with m = 1. On ^6 ^ I *0 (2)
the other hand, if b, we apply the division algorithm to q^ and b , and obtain
> 0, 0 r. ^ b.
Substituting in (2), we get
(3) a = q^b + r^ b + r^ . Now if q^-c b, we set r^ = q^ and have the desired expression (l)
with m = 2. If q > b, we write
q^= q^b + ^ 0 ^ r^ b, and it follows from (3) that
a = + r^ b^ + r,fc+ . If q^^ b, we set r^ = q^ and are finished. A repitition of pro¬
cess must finally yield the desired result, For
a >q^>q|>q^>
and if q >• b, then q , >0 and so there is a first one of these q's,
say q^_, > such that 0 •< then set r^^-s q^^ and a is expressed
in the form (l). The uniqueness of the form (1) follows the uniqueness
of the various remainders when the division algorithm is used.
Similar to the method of omitting the powers of ten in the decimal
system, we may omit the powers in any other base system and specify the
number a with reference to the base b by giving in order the "digits" r f
r , r, r.. If no base is indicated, it will be understood
that the base is ten. Otherwise the number a will be specified as given
in (2) by writing (r^ r„_, r^_^ r, r^ , For example;
^Neil H. McCoy, Introduction to Modern Algebra (Boston; Allyn and Bacon, Inc., I960), pp. 52-53, 7
(U23l)_j' simply means ^.5”^+ 2.^^+ 3.5 + ii.5 and it will be verfied that
{h23h)^ = li . 125 + 2.25 + 3.35 + U.l
= 5oo + 50 + 15 + i;
= 569.
In order to convert 569 to a niunber in base five, it is only neces¬ sary to use repeated division by the base five, 51^ 5[//J - /Ip - - 3
0
Here the remainders are written at the right and the division car¬ ried out until the last quotient is less than the base five. Reverse the order of the remainders to write the number in standard form. There¬ fore, 569 = (i;32U)5 .
Fractions in the decimal system of numeration may be changed to fractions in non-decimal systems of numerations by the same method that integers in the decimal system are converted to integers in the non¬ decimal system. However, certain characteristic changes result upon specific conversions. Some fractions can be represented exactly (i. e. they terminate). Other fractions cannot be represented exactly in the given system (as one-seventh cannot be represented exactly in the deci¬ mal system), but each fraction can be represented exactly in some other system (as one-seventh is exactly .2 in the base lU system). From this. it is obvious that any fraction can be exactly represented in some system. Similarly, if a particular fraction cannot be exactly represented 8
in a given system, then its digits when carried far enough to the right begin to repeat indefinitely in some sycle.
In conclusion, let us say that any fraction whatsoever can be pre¬
sented uniquely in some system as a terminating repeater, or as a non¬
terminating repeater.
Converting Any Decimal Fraction to Any
Other System
A decimal fraction here may be considered as a pure decimal frac¬
tion or a mixed decimal fraction. Regardless of which type of fraction
is presented in the decimal system, it may be converted to another sys¬
tem by the method of repeated division and repeated multiplication, or repeated multiplication. Notice that 133.U2 equals the decimal integer
133 plus the proper decimal fraction 0.U2. In converting this number
to the system with a given base b, we must convert its integral and
fractional parts separately. The method of converting integers has
previously been discussed (2) by the use of the division algorithm.
Now we convert the proper decimal fraction by a corresponding multipli¬
cation algorithm, i. e., multiply repeatedly by the base b and take
out the resulting integral parts in foward order. A more general state¬
ment of this method follows:
To convert any proper fraction to the system with base b, multiply
repeatedly by b and take out the resulting integral parts in forward
order. Hence, to convert 0.375 to the duodecimal system we perform the
following multiplications:
= 12 X .375 li.5 ... ii
= 12 X .5 6.0 ... 6
12 X .0 = 0.0 0 9
Therefore, .375 = .it6o
To convert .375 to the octal system we have
= X .. 8 .375 3.0 ... integral part is 3
= X .. 8 ,0 0.0 ... integral part is 0
Therefore .375 = .30
To convert .375 to the binary ss'^stem, we have
- 2 X .375 0.75 .. 00
= 2 X .75 1.5 .. 1
= 2 X .5 1.0 .. 1
2 X 0.0 =0.0 .. 0
Therefore .375 = To convert 221. 021875 to the binary system, we use both division auad multiplication algorithms as follows: / X 1/ / a . ■ • 0 » 1 1 7 . • • • / 1 . • • • ' ■ ■ ' ■ 0 ^ J ^ _x • 0 . Hence, 221 -(llOlllOl)^^ 10 For the proper decimal fraction, we have 2 X .021875 = 0 .Oii375 2 X .OU375 = 0 .0875 2 X .0875 a 0 .175 2 X .175 = 0 .35 2 X .70 1 .Uo 2 X .liO = 0 .80 2 X .80 = 1 .60 2 X ,60 1 .20 2 X .20 = 0 .UO which repeats from here on. Therefore, 221.021875 =(ll01101.000001011o}^ (The bar above 10110 signifies the repetition of 10110 as a repeating period). Now we can write the general expression for the conversion of any decimal fraction to a non-decimal fraction. That is, if we wish to con¬ vert any fraction between zero and one to any other system, we must ex¬ press the number in the form (p_^ p^ p,^ )^^ere the p's may or may not terminate. In order to do this, we must find the values of these p's such that I- a b -f Now multiplying the right member by b we obtain ->» -/ + — 11 Hence, p_^ equals the integral part resulting from the first multiplication. Multiplying the fractional part by b we get p^^plus a fractional part. Hence, equals the integral part resulting from the second multiplication and so forth. Conversion of Fractions in Non-decimal Systems To Fractions in the Decimal System According to the method previously discussed concerning place value of digits, the number 231,7382 in the decimal system equals i.oo -h 30 + I . 7 + .03 -t . ao f + . ooal^ ^ X5J.73S2 - .\--3 0o') + 1(10°) + !$ -ts (lo'*) Similarly, for the octal, teniary, binary, and duodecimal systems respectively, we have; 0 7 5. ay = y ^ 7-3'^-2-3° ^ l-J-' ^ 2-3'^ S + J±^ fo.j/oi^ = a'-v-a'V + r j2 4. y+yf/ 3* Z.^ 2^ ^3.S - / 37 -h f 0 - / 3 7.0 ^ f y /2* Thus, to convert the given number a in any system to the decimal system, merely expand in powers of the base and add the terms. The integral part (to the right of the point) is worked out separately, and then the two are added together. In general, for any b > 1, if afsany fraction, there exists an integer n such that a can be expressed uniquely in the form a ' i’"'fl, ,I i)^'' -t ■ ‘ P.,^ + ^ . -A f>_J> where o < ^ and O 'S ^ f ^ ~ 12 From the previous statement, the niimber a may now be noted as CHAPTER III ARITHMETICAL OPERATIONS Addition of fractions in different numeration systems. — Here we examine the basic arithmetical operations of addition and multiplication. We omit subtraction and division because they are inverse operations of addition and multiplication. Computations will be done in the following systems: binary, octal, and duodecimal. Binary Addition + 0 I 0 0 I I 10 13 lu Octal Addition A- 0 2 3 5' S 7 O 6 / :? i' ; 1 1 2 5^ 7 tn 1 n V $ 7 // 3 3 5 C 7 >6 // /i. H 5 7 // /:J- $ 5 L 7 // C, // /P- /f IS L 7 7 /? /r 7 // Binary Decimal I \ lO . oc I 7. 1^5' + ^ J 1 I . 0 I ( 3 - 1 0 i 6 1 . / fl 0 a 0.7 S’ c) Octal Duodecimal 7. / I / 5.5 •+ n (, XH'C. /f. 90 15 Multiplication of Fractions in Different Numeration Systems Decimal Multiplication X 0 / 3. 3 sr a 7 S 9 0 0 d c 0 c 6 0 6 0 S 1 0 / 1 3 7 c 7 9 /o 0 J2 C JH ^7 7C 7l /f Z! 3 D 3 c 9 /s /C ZO 0 7 Jz 3z 34 96 0 S' /6 // -to 3S 9sr 6 C Jl ti Zi n 49 C3 7 0 7 J7 3S' ^4 3Z 73- 52* 3 6 ? /£ l7z 3C 7s f 6 9 /f ^7 5-9 ^3 7X 9'/ Binary Multiplication X 0 1 & 6 6 1 Q 1 16 Octal Multiplication X c / ^ 3 9 ^ y o c 6 d 0 C 0 0 0 ^ 1 6 J a 3 ^ 5' 1 jy /C 1 6 7- 9 C /6 ^0 '%z ZS 3 6 3 c. H fV 9 /o /f 7i> 77 /I 3^ yj S' 6 S /; 3J 9y yz c c c /y az 3d 3ji, 02 (/ /Q 75 iy 93 7 6 7 ! uuoaecxmai Multiplication 1 ^ * 0- £ y s f C d U / 2 9 / 0 7 5 7 ? 9 £ Ul 0 % 9 9 /o /7 /y /r 9f /(f 3 c 9 /(J 3 0 /J 99 73 X4 X9 0 9 9 /o /y /? 20 29 7t 70 39 78 r 0 S /3 ft 7/ 2C 2e 39 39 72 99 /C 76 ZC () c lo 30 3C i6 y£> 50 55 it If uy ze. 3X. 7/ yr 53 5j( 5y 7 6 1 xt io 5o 57 6 /f ty yg 59 9f 73 5d ?7 74 53 C6 55 75 ? 6 f /£ IJ 7C ty y'L 56 5^ 58 74 S’ff d /r 52 37 ' 5C e 6 C- // 7? ^5 97 83 f2 17 Binary Decimal //./ //.7 X / 6.1 X r. 6 6T> 5^ ?-r S'?,S'0 / 0 OOAl Octal Duodecimal 2-3,i X C'l X :?y.r 3 5* 2. / ^ / 33-3Z ^ 8- ^.28 The methods of converting fractions from one numeration system to another, as discussed in this thesis, will enable students to perform arithmetical computations with fractions with greater ease, as well as with better understanding. BIBLIOGRAPHI Culbertson, James T., Mathematics and bogie Digital Computers. New York: D. Van Nostrand Co., Ltd., 195^. Denbow, Carl and Geedicke, V., Foundations of Mathematics. New York: Harper and Brothers, 1959. Eves, Howard, An Introduction to the Histoiy of Mathematics. New York: Holt, Rinehart and V7inston, 1961. Goffman, Casper, Real Functions . New York: Holt, Rinehart and Winstbn, 1961. James, Glenn and Brown, Robert C,, I-lathematics Dictionary. New York, 1959. Lang, Serge, A First Course in Calculus. Massachusetts: Addison-Wesley Publishing Co., Inc., I96I1. McCoy, Neil H., Introduction to Ifodem Algebra. Boston: Allyn and Bacon, Inc., i960. Newman, James R., The World of Mathematics. New York; Simon and Schuster, 1956. Olmsted, John M., Intennediate Analysis. New York; Appleton-Century-Crofts, Inc., 1956. Rose, Israel H., A Modern Introduction to College Mathematics. New York; John Wiley and Sons, Inc., i960. 18