(49.) Subtraction may be proved, as in arithmetic, by adding the remainder to the subtrahend. The sum should be equal to the minuend.

The term subtraction, it will be perceived, is used in a more general sense in algebra than in arithmetic. In arithmetic, where all quantities are regarded as positive, a number is always diminished by subtraction. But in algebra, the difference between two quantities may be numerically greater than either. Thus the difference between +a and —a is 2a.

(50.) The distinction between positive and negative quantities may be illustrated by the scale of a thermometer. The degrees above zero are considered positive, and those below zero negative. From five degrees above zero to five degrees below zero, the numbers stand thus:

+5, +4, +3, +2, +1, 0, −1, −2, −3, −4, −5.

The difference between five degrees above zero and five degrees below zero is ten degrees, which is numerically the sum of the two quantities.

QUEST.-How may subtraction be proved? What is the difference between arithmetical and algebraic subtraction? Illustrate the distinotion between positive and negative quantities.

(51.) In many cases the terms positive and negative are merely relative. They indicate some sort of opposition between two classes of quantities, such that if one class should be added, the other ought to be subtracted. Thus, if a ship sail alternately northward and southward, and the motion in one direction is called positive, the motion in the opposite direction. should be considered negative.

Suppose a ship, setting out from the equator, sails northward 50 miles, then southward 27 miles, then northward 15 miles, then southward again 22 miles, and we wish to know the last position of the ship. we call the northerly motion +, the whole may be expressed algebraically thus:

+50-27+15-22,

If

which reduces to +16. The positive sign of the result indicates that the ship was 16 miles north of the equator.

Suppose the same ship sails again 8 miles north, then 35 miles south, the whole may be expressed thus: +50-27+15-22+8-35,

which reduces to -11. The negative sign of the result indicates that the ship was now 11 miles south of the equator.

In this example we have considered the northerly motion + and the southerly motion - ; but we might, without impropriety, have considered the southerly motion and the northerly motion It is, however, indispensable that we adhere to the same system throughout, and retain the proper sign of the result,

QUEST.-Sometimes the terms positive and negative are merely relative. Illustrate this by the example of a ship

as this sign shows whether the ship was at any time north or south of the equator.

In the same manner, if we consider easterly motion +, westerly motion must be regarded as -, and vice versa. And, generally, when quantities which are estimated in different directions enter into the same algebraic expression, those which are measured in one direction being treated as +, those which are measured in the opposite direction must be regarded as -.

So, also, in estimating a man's property, gains and losses, being of an opposite character, must be affected with different signs. Suppose a man with a property of 1000 dollars loses 300 dollars, afterward gains 100, and then loses again 400 dollars, the whole may be expressed algebraically thus:

+1000-300+100-400, which reduces to +400. The sign of the result indicates that he has now 400 dollars remaining in his possession. Suppose he further gains 50 dollars and then loses 700 dollars. The whole may now be expressed thus:

+1000-300+100-400+50-700,

which reduces to -250. The sign of the result indicates that his losses exceed the sum of all his gains and the property originally in his possession; in other words, he owes 250 dollars more than he can pay, or, in common language, he is 250 dollars worse than nothing.

(52.) It is sometimes sufficient merely to indicate the subtraction of a polynomial without actually per

QUEST.-Illustrate the same principle by a case of gain and loss. How do we indicate the subtraction of a polynomial?

forming the operation. This is done by inclosing the polynomial in a parenthesis, and prefixing the sign —. Thus, 5a-3b-(3a-2b),

signifies that the entire quantity 3a-2b is to be subtracted from 5a-3b. The subtraction is here merely indicated. If we actually perform the operation, the expression becomes

According to this principle, polynomials may be written in a variety of forms. Thus,

Ex. 1. From 10ax+y take 3ax-y.

Ex. 2. From 17mx2+12 take 17mx2+12-b.

Ex. 3. From 12ab2x2-y' take 3ab2x2+y2.

Ans. 7ax+2y.

Ans. +b.

Ans. 9ab'x-2y1.

Ex. 4. From a+b take a−b.

Ans. 2b.

Ex. 5. From 5a2+b2+2c2-15 take 12+b2+5a2.

Ans.

Ex. 6. From 17x+4a-3b+25 take 12+2b-3a

+4.x.

Ans.

QUEST.-Give some of the different forms in which a polynomial may De written.

Ex. 7. From 3a+2b-7c+14y take 3y +4c-b+a.

Ans.

Ex. 8. From 28ax2-16a2x2+25a'x-13a* take

18ax+20a2x2-24a'x-7a".

Ans.

Ex. 9. From 8a'xy-5bx'y+17cxy'-9y' take a1xy

+3bx3y-13cxy2+20y3.

Ans.

Ex. 10. From 10ax2+13by'-17bc take 4bc-3by'

+6ax.

Ans.

Ex. 11. From 25x+32xy-6a take 2a+17-19xy

+3x.

Ans.

Ans.

Ex. 12. From 12x2 take 24x+3x-7x+16x.

Ex. 13. From 24a+176 take 20+3b-2a+7a-6b.

Ans.

Ex. 14. From 34-6ab+2y-5x take 2x + 3ab

-16+y.

Ex 15. From a' +abc-6 take 6+abc-a2.

Ans.

Ans.