SCHOOL SCIENCEAND

MATHEMATICSVOL. XLV - DECEMBER, 19,45 WHOLE No. 398

^MATHEMATICIANS HAVE AGREED ..."CHARLES SALKIND

Samuel J. Tilden High School, Tilden Avenue and East 57th Street,Brooklyn, N: F.

In an elementary algebra textbook recently off the press, wefind the following:

Order of operations. One of the "catch" questions in arithmetics was:Does 24 -3 X2 equal 16 or 4?The meaning can be made clear by putting parentheses around the

work that should be done first. Thus: (24 -3) X2 == 16 24 -(3 X2) ==4If parentheses are not used the rule is: Working from left to right, do

the multiplications first, wherever found: then do the divisions: and, lastof’all, do the additions and subtractions.

There is definitely no difference of opinion about the hierarchyof operations: first, involution and evolution, then, multiplica-tion and division, finally, addition and subtraction. But someof us were reared on textbooks which caught that, when thechoice is between multiplication and division only, if paren-thesis are not used, the rule is: Working from left to right, dothe operations in the order in which they appear.An investigation of sixteen commonly used algebra texts

revealed a disconcerting lack of agreement about the priorityrating of these operations. Nine of the sixteen adopt the con-vention that multiplications and divisions are to be performedin the order in which they occur. Of these nine eight are explicitin their statement: the ninth implies this rule in giving theanswer 100 to the example 40�2-5. Four of the sixteen givethe rule quoted at the beginning of this article. In two of thesefour we find the statement, "Mathematicians have agreedupon the following order,n followed by the rule:

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786 SCHOOL SCIENCE AND MATHEMATICS

1. First perform all multiplications.2. Next perform all divisions from left to right.If this constitutes agreement, we shall have to find a new

definition for the word.We may conclude that, for want of a uniform convention

about the order of operations with respect to divisions andmultiplications, the only safe procedure is to use parentheses,or brackets and such.How important is this question? Occasionally, pupils are in a

position to make comparisons between textbooks. They areinclined to inflate a disagreement of this type beyond its im-portance. It would be desirable to have a single convention, butthe question is not too important.

Authors, titles and page references will be supplied uponrequest.

A NEW CIRCUIT FOR FREQUENCY MODULATIONA new radio circuit for frequency modulation (FM) receivers which

makes it possible for the first time to build a receiver that realizes the ad-vantages of FM at a cost comparable to that of standard band receivers,was described October 3 by Stuart Wm. Seeley, manager of the IndustryService Division of RCA Laboratories, in a paper delivered to the NewYork Section of the Institute of Radio Engineers.FM sets produced before the war, Mr. Seeley pointed out, required the

use of one or more tubes whose functions were solely that of noise suppres-sion. They contributed nothing to the volume of the receiver output.Furthermore, he said, to make these extra tubes fully effective, considera-ble amplification of the received signal was necessary. Although both ofthese requirements added noticeably to the cost of FM receivers, noise con-tinued to be present when the strength of a received signal fell below a cer-tain point called the threshold level.According to Mr. Seeley, the new RCA circuit, called a ratio detector, is

insensitive to electrical interference of all kinds, whether man-made byignition systems, oil burners and domestic appliances, or natural, such asatmospheric static.Mr. Seeley added that the new circuit is not only free of a critical

threshold signal level, operating equally effectively on strong and weakstations, but its incorporation in a receiver eliminates the need for addi-tional tubes and parts that formerly were considered necessary in fre-quency modulation receivers. It is this simplification, he said, that shouldreduce the manufacturing cost of FM receivers to a point comparable withthat of receivers covering the standard broadcast bands.

The national security of the United States demands that military sci-entists and industrial scientists continue their cooperation; for peace andsecurity rise and fall with science.

DAVID SARNOFF