PLUS AND MINUS IN ALGEBRA 435

PLUS AND MINUS SIGNS IN ALGEBRA.

BY Jos. A. NYBEBG,Hyde Park High School^ Chicago.

The beginner in algebra is always confronted with the necessityof learning the double significance, of the + and � signs. Theyare signs of the operations of addition and subtraction, and alsosigns of quality, positive and negative. In the language ofgrammar they are both verbs and adjectives. In reconcilingthese two views, we either must make elaborate explanations,or else attempt to dodge the trouble by omitting any referenceto it. The present paper explains a mode of treatment of thisdifficulty.

After the pupil has been shown the existence in nature ofboth positive and negative numbers, the signs + and � beingused as signs of quality or adjectives, we consider the questionof addition. In adding a column of three or more numbers theteacher calls attention to the fact that the list of numbers mayas well be written in a line instead of in a column, the numbersbeing separated by commas, as +17, �6, �13, +5, �18, justas we find similar problems stated in arithmetic. However,inasmuch as the signs separate the numbers so that 17 and 6cannot be interpreted as 176, the commas may be omitted.This omission was impossible in arithmetic unless we in someway spaced the numbers. In other words, algebra needs nosign for addition. Addition is to be understood as long as noother operation is indicated. When handling polynomials alsowe may eliminate the need of a sign for addition by adopting anew attitude toward parentheses.

Consider the expression 3(2a�5). Instead of saying that theparenthesis is a symbol for grouping of terms, we say that thepresence of a parenthesis should always suggest multiplication,i. e., a quantity in a parenthesis should always be multipliedby the quantity preceding it. Hence �36(2a�56) is a problemin multiplication, the multiplier being �3&. And 3a(5a�36)�3&(2a�5&) is a problem in two multiplications, followed by ’theproblem of addition. The pupil should not think of this as aproblem in multiplication followed by a substraction, for the�sign is an adjective qualifying 3& and is not a verb. The addi-tion follows the multiplication because no other operation isindicated.When a quantity like �S(2x�y)+(�3x+2y) is met, I ex-

plain to the class that if no multiplier is present before the paren-

436 SCHOOL SCIENCE AND MATHEMATICS

thesis, the number 1 is understood and the problem means�3(2o;�2/), +1(�3^+2?/). This is consistent with previousexperience for we write a letter x when we mean Ix, and write�a;

meaning �Ix. Thus 4:(2x�3y)� (Qx�y) means that 6x�yis to be multiplied by � 1 and added to the result of the othermultiplication.

Let us see what changes in the order of the exercises this newattitude toward a parenthesis will involve. Regarding theparenthesis in the old way as a symbol for grouping, the pupilfirst learns to remove parenthesis from such an expression as(3x-y)-(Qx-8y) and is later taught that ^{x-2y) -2(x+Gy)means 2x+12y is to be subtracted from ^x�6y, and the teachermust avoid single expressions like �6(2^�3^) because thepupil will want to know from what he is to subtract the 12x�18y.Note how many different explanations are necessary. Butusing the new definition of a parenthesis, all these problems areof one kind, problems in multiplication. The teacher shouldreverse the order illustrated and consider first, quantities like-6(2x-3y); second, �6(i2x-3y)–2(x-7y) as this uses addi-tion after the multiplications; third, .quantities like (3x�2y)–(6x+8y) where the invisible number 1 is understood.Having eliminated the + sign for addition, the � sign for

subtraction can also be dispensed with. We begin as usual byexplaining that subtraction in algebra does not mean what isleft after something is taken away, but involves finding whatmust be added to the subtrahend to obtain the minuend. Thewriter has always found that this can be done easiest by usingequations, a subject which is usually begun fairly early in thecourse, and should be available when subtraction is reached.Thus, let x= what must be added to �6 to obtain 10. Arrangeseveral problems on the blackboard in such a scheme as:

Subtrahend �6 +5 �7Minuend +10 -3 -2

-G+x == 10 5+x = -3 -7+x = -2+6 =4-6-5 == -5 +7 = +7

x = 16 x == -8 a; = 5

Then ask: "What was added to each member of the equation?What was added to the minuend in each problem?" This analy-sis is easier to understand than why the loss of a debt is the sameas the gain of an asset, etc. The pupil sees that subtractioninvolves two steps, the multiplication by �1 followed by an

addition. Hence no sign for subtraction is ever necessary.

THE DISMAL SWAMP 437

Wishing to subtract 6x�7 from Sx�9 we write 3x�9�(6x�7)meaning ^x �� 9, � 1 (6a; � 7) the comma and the one being omittedaccording to custom.Thus the verbs + and � are eliminated from our grammar;

only the adjectives + and � remain.The usefulness of this attitude toward the + and � signs and

parentheses can be seen in simplifying such an expression as3(a-26)(4a-56)-5(2a-~&)(a+4&). This would be done asfollows: it equals

3(4a2-13a&-^-1052), -5(2a24-7a&-4&2)= 12a2-39a6+30&2-10a2-35a5+2062= -W^^ab+QOb2.

Again in later work, the pupil would be taught that theequation (3;r-7)/6 - (2x+6)/3 + 2(x-6)/5 == 0 should beregarded as l/^x-7), -l/3(2a;+6), +2/^x-Q) == 0 andthen as 5(3a;-7)~10(2a;+6)+12(a;-6)=0.

THE DISMAL SWAMP OF VIRGINIA AND NORTH CAROLINA.Few regions in America are more adorned by nature or more interesting

to the tourist and scientist than the Dismal Swamp of Virginia and NorthCarolina. Though the entire region, may present a dismal appearancein winter and some parts of it in all seasons, the swamp is annuallyvisited by many pleasure seekers and has long been a place of studyand absorbing interest to the geologist, the botanist, and the zoologist.It lies in the Coastal Plain of Southeastern Virginia and NortheasternNorth Carolina. Most of the surface consists of recently formed peat,the residuum resulting from the arrested decomposition of vegetation,but the underlying rocks are older and record events .that occured thous-sands of years ago, in the pleistocene epoch. The peat ranges in depthfrom one to twenty feet. Contrary to popular belief this peat has anti-septic and preservative properites, and consequen ly much of the surfacewater is pure. Though no remains of primitive man or of extinct animalslike those uncovered in the bogs of Ireland have been found in the DismalSwamp, the peat contains many well-preserved trunks of cypress treesthat lived long before America was settled by our ancestors.The region may be readily reached from ^ orfolk by canals, whose banks,

shaded by stately trees and graceful vines, afford an ever-changing scenefrom the deck of the little stearcer that daily plies their waters. Whenthe swamp was young it was entirely covered by water, but much of thewater has drained off through these canals, and large areas are now dry.In the center is a picturesque body of water called Lake Drummond, theorigin of which is a mooted question. According to the most plausiblehypothesis that has been advanced, it is the remnant of a large body ofdeep water which once covered the entire region. The water in this lake,because of its remarkable keeping property, was used in earlier years fordrinking on trans-Atlantic voyages. It is amber-colored and is knownlocally as "juniper water." As the name implies, the peculiar color hasbeen ascribed to the bark of the white cedar (juniper), which abounds inthe swamp. It seems more likely however, that this color is given to thewater by its finely divided vegetal content or by the dye extracted fromthe brown peat.� U. S. Geol. Survey.