and the negative quantity in the second case is interpreted as a debt, that is, a sum of money opposite in character to the positive quantity, or gain, in the first case; in fact it may be said to possess a subtractive quality which would produce its effect on other transactions, or perhaps wholly counterbalance a sum gained.

(ii) Suppose a man starting from a given point were to walk along a straight road 100 yards forwards and then 70 yards backwards, his distance from the starting-point would be 30 yards. But if he first walks 70 yards forwards and then 100 yards backwards his distance from the starting-point would be 30 yards, but on the opposite side of it. As before we have

100 yards 70 yards=+30 yards,

70 yards 100 yards 30 yards.

In each of these cases the man's absolute distance from the starting point is the same; but by taking the positive and negative signs into account, we see that -30 is a distance from the starting point equal in magnitude but opposite in direction to the distance represented by +30. Thus the negative sign may here be taken as indicating a reversal of direction.

(iii) The freezing point of the Centigrade thermometer is marked zero, and a temperature of 15° C. means 15° above the freezing point, while a temperature 15° below the freezing point is indicated by -15° C.

19. Many other illustrations might be chosen; but it will be sufficient here to remind the student that a subtractive quantity is always opposite in character to an additive quantity of equal absolute value. In other words subtraction is the reverse of addition.

20. DEFINITION.

When terms do not differ, or when they differ only in their numerical coefficients, they are called like, otherwise they are called unlike. Thus 3a, 7a; 5a3b, 2a2b; 3a3b2, - 4a3b2 are pairs of like terms; and 4a, 3b; 7a2, 9a2b are pairs of unlike terms.

Addition of Like Terms.

Rule I. The sum of a number of like terms is a like term.
Rule II. If all the terms are positive, add the coefficients.

Example. Find the value of 8a+ 5a.

Here we have to increase S like things by 5 like things of the same kind, and the aggregate is 13 of such things;

for instance, Hence also, Similarly,

8 lbs. + 5 lbs. = 13 lbs.

Sa +5a= 13a.

8a+ 5a+a+2a + 6a = 22a.

Rule III. If all the terms are negative, add the coefficients numerically and prefix the minus sign to the sum.

Example. To find the sum of -3x, -5x, -7x, -X.

Here the word sum indicates the aggregate of 4 subtractive quantities of like character. In other words, we have to take away successively 3, 5, 7, 1 like things, and the result is the same as taking away 3+5+7+1 such things in the aggregate.

Thus the sum of 3x, -5x, -7x, -x is - 16x.

Rule IV. If the terms are not all of the same sign, add together separately the coefficients of all the positive terms and the coefficients of all the negative terms; the difference of these two results, preceded by the sign of the greater, will give the coefficient of the sum required.

Example 1. The sum of 17x and 8x is 9x, for the difference of 17 and 8 is 9, and the greater is positive.

Example 2. To find the sum of Sa, - 9a,

-a, 3a, 4a, -lla, a. The sum of the coefficients of the positive terms is 16.

The difference of these is 5, and the sign of the greater is negative; hence the required sum is -5a.

We need not however adhere strictly to this rule, for the terms may be added or subtracted in the order we find most convenient.

This process is called collecting terms.

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21. When quantities are connected by the signs + and the resulting expression is called their algebraical sum. Thus 11a-27a+13a= 3a states that the algebraical sum of 11a, -27a, 13a is equal to -3a.

22. The sum of two quantities numerically equal but with opposite signs is zero. Thus the sum of 5a and - 5a is 0.

EXAMPLES II.

CHAPTER III.

SIMPLE BRACKETS. ADDITION.

23. WHEN a number of arithmetical quantities are connected together by the signs and, the value of the result is the same in whatever order the terms are taken. This also holds in the case of algebraical quantities.

Thus ab+c is equivalent to a+c-b, for in the first of the two expressions b is taken from a, and c added to the result; in the second c is added to a, and b taken from the result. Similar reasoning applies to all algebraical expressions. Hence we may write the terms of an expression in any order we please. Thus it appears that the expression a-b may be written in the equivalent form - b+a.

To illustrate this we may suppose, as in Art. 18, that a represents a gain of a dollars, and -b a loss of b dollars: it is clearly immaterial whether the gain precedes the loss, or the loss precedes the gain.

24. Brackets ( ) are used to indicate that the terms enclosed within them are to be considered as one quantity. The full use of brackets will be considered in Chap. VII.; here we shall deal only with the simpler cases.

8+(13+5) means that 13 and 5 are to be added and their sum added to 8. It is clear that 13 and 5 may be added separately or together without altering the result.

Thus

8+(13+5)=8+13+5=26.

Similarly a+b+c) means that the sum of b and c is to be added to a.

8+(13-5) means that to 8 we are to add the excess of 13 over 5; now if we add 13 to 8 we have added 5 too much, and must therefore take 5 from the result.

Thus

8+(13-5)=8+13-5=16.

Similarly a +(b-c) means that to a we are to add b, diminished

by c.

Thus

a+(b-c)=a+b-c.

In like manner,

a+b-c+(d-e-f)=a+b-c+d-e-f.

By considering these results we are led to the following rule: Rule. When an expression within brackets is preceded by the sign, the brackets can be removed without making any change in the expression.

25. The expression a (b+c) means that from a we are to take the sum of b and c. The result will be the same whether b and c are subtracted separately or in one sum.

a− (b+c)=a-b-c.

Thus

Again, a− (b −c) means that from a we are to subtract the excess of b over c. If from a we take b we get a-b; but by so doing we shall have taken away c too much, and must therefore add c to a-b. Thus

In like manner,

a-(b−c)=a-b+c.

a − b − (c - d − e)=a−b-c+d+e.

Accordingly the following rule may be enunciated :

Rule. When an expression within brackets is preceded by the sign -, the brackets may be removed if the sign of every term within the brackets be changed.

Addition of Unlike Terms.

26. When two or more like terms are to be added together we have seen that they may be collected and the result expressed as a single like term. If, however, the terms are unlike they cannot be collected; thus in finding the sum of two unlike quantities a and b, all that can be done is to connect them by the sign of addition and leave the result in the form a+b.

27. We have now to consider the meaning of an expression like a+(-b). Here we have to find the result of taking a negative quantity -b together with a positive quantity a. Now -b implies a decrease, and to add it to a is the same in effect as to subtract b; thus

a+(-b)=a−b ;

- b.

that is, the algebraical sum of a and b is expressed by a –

28. It will be observed that in Algebra the word sum is used in a wider sense than in Arithmetic. Thus, in the language of Arithmetic, a-b signifies that b is to be subtracted from a,