All About Division With Rational Numbers�Variation on a Theme

Herbert HannonWestern Michigan University, KalamazoOf Michigan. 49001

Much has been written on the topic of division with rational num-bers. It is the purpose of this article to summarize and also to extendthe treatment of division to include some aspects usually omittedfrom such presentations.The "divides relation" for the system of whole numbers selects

ordered pairs (a) 6), o^O, and a whole number n such that aXw==&.The topics of factors and multiples relate directly to the above wherea and n are called the factors of 6, and b is called the nth multiple of a.The Division Algorism extends division to include all non-zero

whole numbers as possible divisors of each whole number. The re-lationship is expressed as

(divisor X Quotient) + remainder === Dividend

or Dividend � divisor == Quotient fi remainder

The ordered pair (Q, r) relates the Quotient and the remainder where0<r<d, The division process is accomplished by repeated subtrac-tion of sets the size of the divisor from the Dividend set until aremainder set is obtained which is either zero or is less than thedivisor set.When the system of rational numbers is introduced, it is often

assumed that the Division Algorism is no longer applicable since forall a, b which are rational numbers, there is a rational number c suchthat aX6==^. Then c�a==b, where o^O.The following presentation will assume that a is a positive rational

and that b and c are positive rationals or zero. When using theDivision Algorism as applied to a rational field [1] it will be assumedalso that Q will be a positive integer or zero.

Consider the following problem:Mary has 2^ yards of ribbon which she wishes to divide into equal-sized pieceseach 2/3 of a yard long. How many such pieces can she get and how muchribbon will be left over?

The "division" can be accomplished in a physical sense by actuallycutting the ribbon into the required lengths, counting the number ofpieces obtained and finally measuring the length of ribbon left over.A physical model or diagram may be used to illustrate the procedure:

501

502 School Science and Mathematics

\����remainder = ^0|1^21 2^

1��§�-h�j��1��j��f��J-

Mary will have 3 pieces of ribbon each 2/3 of a yard long and therewill be a piece of ribbon left over which is ^ yard long.

By directly subtracting By changing the thirds and halves

2^ � | == 1| to sixths

If - t - 1 3(g)j

U - i - i t) ¥-or ^_^

2i - (3 X f) = i i - i = r 0 < 1 < iThe Division Algorism has been

used to obtain the desired solution

21 - i == 3 fi icheck:

(i X 3) + i = 21The writer strongly recommends the introduction of the DivisionAlgorism as a means of making division of rationals meaningful whenthe division situation in the practical problem is one of measurementand a remainder expressed in the same units as the dividend is re-

quired.When division of rationals is defined as the inverse operation of

multiplication of rationals, many different but mathematicallyequivalent methods of solution may be introduced. Since the divisionoperation is closed with respect to the rationals, the division processmay be considered as that of finding a missing factor when the prod-uct and one of its factors are known. Division will "undo" a multi-plication.

LEAST COMMON DENOMINATOR METHOD

Perhaps this is the most difficult method to explain mathematicallyHowever, it is found in textbooks as early as 1850 and has been usedextensively. The example is solved as follows:

2i - j = -I - t - -V- -- i - 15 - 4 = 3i

Equivalent fractions are introduced in which the thirds and halvesare expressed as sixths. The numerator numbers of these equivalent

Division with Rational Numbers 503

fractions are then directly compared by division. The division is

accomplished by dividing the whole numbers 15 and 4, and the re-

mainder is expressed as a fractional part of the whole number whichis the divisor.

3i4) 15

-12

3

A physical diagram or model may be used to illustrate the ^division.7’

The Fractional Units are Expressed as Sixths

0 1 2 2^ j. 1 2 3 4 5 6,7 8 9 10 11 121 13 14 15, \1�1���h�f���M�������I | I I | j

! j iis|of^

’ 1 ’2’3’4’ 1’2’3’41 1’2’3’ 4J 1’ 2’ 3’ 4JL.-.,._.__J

The remainder is composed of 3 of the 4 parts into which the divisormust be divided, i.e., |- is |- of ^.

It will be noted that we have solved a problem mathematicallyequivalent to the original, since the ratio of 2^ to | is the same as thatof 15 to 4. In a physical sense we may interpret the solution as findinghow many complete sets of ^ there are in a set of i^-, and also thefractional part of a set that is left over.The writer has usually called the two methods referred to above

,5^1-mathematical Methods since they require a physical model or

graphical representation for their interpretation. Appropriate math-ematics which relates to the model is then used to accomplish thedivision without constructing the model for each example to besolved.

In the remainder of this article the processes used do not appeal togeometric representation. Solutions are obtained directly throughdefinitions, the properties of the rational numbers, and the propertiesof equivalence relations. The basic interpretation will be:

factor X factor == product <==> product ~ factor = factor

504 School Science and Mathematics

THE LEAST COMMON MULTIPLE METHOD

Steps Reason1) 2^- � | == n Given: Find the missing factor.

2^ Definition of a rational number as an indicated2) "y ,. . .

^ division, wm � n == �

n

3) 2^ X 6 A rational number is an equivalence class.

i X 6 m m X d

n n X d

4) -y- Multiplication of rationals

5) 3^ == D Mixed numeral form of the rational

°) 2^ � | == Q == 3i Transitive property of equality

Step 3 can be expanded to show that

2i X 6 (2 + i) X 6 (2 X 6) + (i X f) 12+3� ^2t� ^4,

X 6J X6(| X f) 6

Since the computation in the above steps can usually be donementally, the method is usually quite easy to apply. It is essentiallya method of obtaining an equivalent ratio in which the numbers areexpressed as relatively prime whole numbers, (step 4 above)

MULTIPLY BY THE RECIPROCAL METHODThis method depends upon the definition of reciprocals which are

defined as pairs of rationals whose product is always 1.

f a b \a/-°i-^\b a~ )’���= 1 ),where �^ 0.

< b a /b

StepsReasons

1) 2^ � § == QGiven: Find the missing factor

2|-Definition of a rational number as an indi-2)

3)

|cated division

2|r XtArational number is an equivalence class.

| Xfi and | are reciprocals. Also

21 X ^4) ���� Definition of reciprocals, (i X f = 1)

Division with Rational Numbers 505

5) 2^ X t = D 1 is the identity element for division

a � 1 = a

6) 2^ � | = D = 2^ X | Transitive property of equality

7) 2^ X i = 3i = D by multiplication of rationals

The Transitive Property of Equality makes it unnecessary to takesteps 2 through 5 once the validity of this chain of steps has been estab-lished. We may quickly generalize that

a c a d

T’~T""TX7’for all rationals when

c

7"»-It is undoubtedly because of this simple generalization and ease ofcomputation that this method is used in general practice almostexclusively.

INVERSE OF MULTIPLICATION METHOD

In the number sentence c�a==b it is assumed that ^n is the prod-uct of the factors a and 5. Since there are many names for eachrational number (i.e., A rational number is an equivalence class), itmay be assumed that "c^ must be renamed since it usually names theproduct as a fraction expressed in lowest terms.The following interpretation is needed to apply and explain the

details of the process used.

m p m X p Definition of multiplication of

n q n X q rationals. (See step 4 below)Steps Reasons

1) 2^�|==D Given: Find the missing factor.

2) i�1 Using the improper fraction representa-

tion of the rational number

(5X6) 2 30 2 A rational number is an equivalence class.3)

(2X6)*

3 ^ 12’3Note: 6 is the L.C.M. of 2 and 3

4)^(15X2)-2 15X(2-r-2) Division will "undo" multiplication, i.e.,

(4X3)�3 4X(3�3) Division is the inverse of multiplication.

506 School Science and Mathematics

_15X1 15

"4X1 ""T30-2 15

or ���==� by division directly.12-3 4 ’

5) -y-=3i== D Improper fraction form changed to mixed

numeral form of the rational number.

The Transitive Property of Equality relates Step 5 to Step 1.Perhaps this method which explicitly shows division as the inverse

of multiplication has not found wide acceptance because of the extrasteps needed as well as the extra computation which results by usingthe L.C.M. in Step 3. It should be noted that the check as shown be-low would quickly verify the reasonableness of Step 3.

^ i .3 ^2 ls

^2 (5 X 3) X 2 5 X (3 X 2) 5X6

Check: 3jX � = � X � = ������� = ������� == ���3 4 3 (2X2)X3 2X(2X3) 2X6

30 5== � == � =2^ (See step 3 above)

i. Zt JL

USING EQUATIONS TO SOLVE A DIVISION SENTENCE

Steps Reasons

1) 2^�| =n Given: Find the simplest name for Q

2) § X 0^== 2^ An equivalentform of the division equation.

3) I X (j X D) ^ U X j Multiplication property of equality

4) (I X t) X D == 2^ X I- Associative property of multiplication

5) 1 X D ^ U X | Definition of reciprocals: (| X j) = 1

6) Q == 2^ X | 1 is the identity element for multiplication

7) So 2^ � | = 2i X | Transitive property of equality

Again we may relate Step 1 to Step 7 by the Transitive Propertyof Equality and conclude that:To divide by a rational number, multiply by its reciprocal.

a c a c c�-�==� x � . �^0.b d b d d

SUMMARY

Several variations of the above solution patterns may be found incurrent textbooks and in the literature, but in the opinion of the

Division with Rational Numbers 507

writer they do not represent any different mathematical approach.Detail may be introduced or deleted which may alter the appearanceof the solution however.Whenever a concept of measurement would seem to be the purpose

of the division in a practical sense and where a remainder is desiredexpressed in the original units of measurement, the Division Algorismapproach would seem to be the best explanation of the division. Ofcourse the answer obtained by other methods may be directly re-lated to the problem situation by interpreting the fractional portionof the answer numeral as a portion of the divisor. In our example,i °f i^’i- This was the remainder obtained by using the DivisionAlgorism. The "finding the missing factor" approach used in theother methods of division presented in this article is directly relatedto "finding a missing dimension.^ For example:

If the area of a rectangle is 2^ sq. ft. and its width is § ft., what is the length?

So the length is 3^ ft.

Certainly no one will ever need all of the methods developed in thispaper to accomplish a given division. Such methods do lend a com-pleteness to the study of division of rationals and they provide forenrichment of the curriculum content as well as for a more completeunderstanding of the topic of fractions and rational numbers ingeneral.

BIBLIOGRAPHY1. BIRKHOFF AND McLANE: A Survey of Modern Algebra, Macmillan Co., 1953,

p. 69.

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