THE ORDER OF OPERATIONS INELEMENTARY MATHEMATICS

MAURICE L. HARTUNGThe University of Chicago, Chicago, Illinois

Within the past year two articles on the order of operationsin computation have appeared in SCHOOL SCIENCE AND MATHE-MATICS. Both of the two authors mentioned that the questionis of minor importance, but nevertheless it is one which teachershave to deal with sooner or later. Actually, the situation pro-vides an excellent opportunity for helping students understandthe nature of mathematical symbolism.The problem of communication which is involved is perhaps

best illustrated by a numerical example like 24�3X2. This ex-pression may represent either 16 or 4, depending upon whetherthe operation of division or of multiplication is to take prece-dence. There is no logical basis for a decision on the order ofoperations in this situation.1 It is a matter which must besettled by agreement or convention. Some authors of element-ary texts ask the reader always to do multiplications first;others permit him to do the indicated operations in the orderin which they occur. As long as either of these conventions isadhered to, consistent results follow. The answer to the questionas to which convention is preferable must be sought in terms ofconvenience, and of consistency with a more extensive set ofoperational procedures.The parenthesis as a symbol of grouping is firmly established.

In expressions of the type a+bc the convention that multipli-cation is to be done first is also universally accepted. Thusa+bc is treated as a-\-(bc). The wide acceptance of this conven-tion makes it possible to dispense with the parenthesis in thisand similar situations, and thus illustrates the tendency ofmathematicians to choose notations in terms of simplicity andeconomy. There is no logical reason why the expression 2+4X’5should not be evaluated as 30. Doubtless merely the fact thatin a+bc, as ordinarily written, b and c have a closer spatialassociation than a and b has resulted in the dominance of themultiplication relation. If addition is given precedence, the dis-

1 The argument in the article by Mr. Neureiter in SCHOOL SCIENCE AND MATHEMATICS for June,1946, contains a logical fallacy. His "identity" (5a), namely (<z �b)’c =a �b’c, on which the argumentis based, ’^begs the question."

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tributive postulate of algebra could be written a+bc = (ac) + (be)This looks a bit queer, but a few numerical examples suffice toconvince students of the existence of the relation. Such consid-erations properly developed can help them come to see that thechoice of mathematical notations and conventions is the resultof historical circumstance and convenience, and not of logicalnecessity or inexorable "law." There is not one "right" answeror rule, excluding all others as "wrong."Turning back to the expression a�bXc, if one adopts the con-

vention that the indicated multiplication is to be done first,what does he gain? First, this convention is consistent with thecorresponding one in a+bXc. Second, this convention permitsthe commutation of b and c, but the alternative one does not.Then 24-3X2=24-2X3=4, whereas if division takes pre-cedence 24-3X2=16, but 24-2X3=36. The "order in whichthey occur" convention makes the result dependent upon spatialarrangement. It has the effect of making the closeness of opera-tive association of the numbers shift from multiplication to divi-sion depending upon the order in which they are written. Thusunder both conventions 24X3 �2 =36, and the 24X3 is treatedas more closely associated than the 3�2. In 24�3X2, however,the "order in which they occur" convention makes the divisionassociation stronger than the multiplication association. Theother convention maintains the supremacy of the multiplica-tion association in both cases, and also follows closely the cor-responding situation in 24+3X2 or 24+2X3.Many writers have suggested that one can escape the dilemma

by avoiding the symbol �, and using instead the parenthesis orfraction bar, which performs as a symbol of grouping. This isundoubtedly the best practice. To insist upon it without takingthe opportunity to discuss the problems of communication in-volved is, however, to overlook an opportunity for teachingabove the routine level. Moreover, now and then a studentwill solve a formula like i=prt in the form p=i�rt. The closespatial association of r and t, or past experience with the rela-tion, results in the omission of any symbol of grouping for rand t. If the student then substitutes numerical values, forexample ^’=54, r=.06, /=3, he may write ^=54-.06X3. Thespatial association here may be less close, and he is face to facewith the need for the convention under discussion. If the "orderin which they occur" convention is followed, the result is absurdin a practical sense. If the "multiplication first" convention is

754 SCHOOL SCIENCE AND MATHEMATICS

used, the result conforms to the realities of the underlying simpleinterest problem, and moreover, the principle ^multiplicationfirst^ conforms with the accepted convention when addition orsubtraction are also involved.On the other hand, if a student writes the Fahrenheit-Centi-

grade relation in the form F=9-5XC+32, orF==9/5C+32 asit is sometimes printed, the ^multiplication first^ conventionwould result in faulty communication. Similarly, forms likeC=5/9(F-32), or S==n/2(a+I) which appear in some books,are misleading, but C=5(F-32)/9 and S=n(a+l)/2 are not.The criterion of effective communication demands that, whatever the convention may be, the writer of mathematics mustavoid notations which permit an ambiguous interpretation.A word or two about the exponent notation may be in order

here. In the symbol ab71, the exponent n is closely associated withthe base b. The accepted convention calls for the numericalcomputation of 671 before multiplying by a. If, however, bothbases have the same exponent, as in a^y then the theorem ex-pressed by a^^^ab)71 states that if desired the multiplicationof the bases may precede the operation indicated by the exponent.Rare indeed is the textbook which states in full the conditionsunder which this theorem holds. Thus if a <0, b <0, and n == 1/m,where m is an even integer, the theorem fails. For example, ifthe usual conventions with respect to the treatment of complexnumbers are followed, (-4)l/2(-9)l/2^(-4. -9)172. Studentsoccasionally run into difficulty in checking the roots of a quad-ratic equation because of failure to observe these restrictions.A careful consideration of order of operations in arithmetic

and in algebra should be effective in reducing formalism and inavoiding the occurrence of errors. The ^canceling57 of terms inthe reduction of fractions, the replacement of (a2-}-^2)172 bya+b, and similar errors, are at least partially the result of aninadequate conception of the role of symbolism in the controlof the order of the indicated operations.

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