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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 ^<b for X = 0 , 1, , m - 1. Proof: The proof will be carried out by repeated use of the division alogrithm. If a b we have the desired form at once with r^= a. and m = 0. If a S b. we may write (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.
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