any particular direction, +4 will stand for a distance of 4 units in that direction, and 4 will stand for a distance of 4 units in the opposite direction.

29. From the above examples it will be seen that the signs + and — will serve to distinguish between magnitudes of opposite kinds. Thus whatever + 4 may represent, 4 will represent an equal magnitude, but of the opposite kind. The signs and are therefore used in algebra with two entirely different meanings. In addition to their original meanings as signs of the operations of addition and subtraction respectively, they are also used as marks of distinction between magnitudes of opposite kinds.

30. A quantity to which the sign is prefixed is called a positive quantity, and a quantity to which the sign is prefixed is called a negative quantity.

The signs and - are called respectively the positive and negative signs.

31. The signs + and are often called signs of affection when they are used to indicate a quality of the quantities before whose symbols they are placed.

The sign, as a sign of affection, is frequently omitted; and when neither the nor the prefixed to a term, the + sign is to be understood.

sign is

32. The range of positive and negative algebraic numbers is obviously double that of the numbers which belong to arithmetic. Thus, the series of integers in arithmetic is

0 +1 +2 +3 +4

while the corresponding algebraic series is

...

4 -3 -2 -1 0 +1

+2 +3 +4

In comparing the terms of this double algebraic series, we adopt the convention that they are here arranged in the order of magnitude, so that

-3< −2<-1<0<1<2 <3,

and in general, a and b being positive numbers,

-ab, if a > b. [See Art. 47.]

33. The magnitude of a quantity considered independently of its quality, or of its sign, is called its absolute magnitude, or its absolute value.

Thus a rise of 4 feet and a fall of 4 feet are equal in absolute magnitude; so also + 4 and - 4 are equal in absolute magnitude, whatever the unit may be.

― ex

34. NOTE. Although there are many signs used in algebra, the name sign is often used to denote the two signs and clusively. When the sign of a quantity is spoken of, it means the + or sign which is prefixed to it; and when we are directed to change the signs of an expression, it means that we are to change the + or before every term into and +, respectively.

If a = 1, b = 2, and c = 3, find the values of

3. 213 9.

10. The reading of a thermometer was + 40°, and its mercury column then fell 53°. What was its reading after the change?

11. If a = 2, b = 3, and c=- 10, which is the greater, a × b or c?

12. Can the result of subtracting one positive number from another ever be greater than the minuend?

ADDITION.

CHAPTER III.

SUBTRACTION.

BRACKETS.

ADDITION.

35. The process of finding the result when two or more quantities are taken together is called addition, and the result is called the sum.

Since a positive quantity produces an increase, and a negative quantity produces a decrease, to add a positive quantity we must add its absolute magnitude, and to add a negative quantity we must subtract its absolute magnitude.

Thus, when we add + 4 to +6, we get +6+4= +10; and when we add 4 to 10, we get +10 - 4=+6. So also, when we add +b to a, we get a +b; and when we add b to a, we get a b.

Hence a +(+b) = a + b and a+ ( − b) = a b.

We therefore have the following rule for the addition of any term: To add any term, affix it to the expression to which it is to be added, with its sign unchanged.

Ex. A boy played two games; in the first game he won 6 points, and in the second he won 4 points (that is he lost 4). How many did he win altogether?

The total gain in the two games together is what is meant in algebra by the sum of the gains.

To obtain the total gain, we must add 4 to 6, and this operation is indicated by 6 +(− 4), which by the above is 6 4 = 2.

When numerical values are given to a and to b, the numerical values of a + b and a b can be found; but until we know what numbers a and b represent, we cannot take any further step, and the process is considered to be algebraically complete.

36. It should be noticed that when b is greater than a, the arithmetical operation denoted by a b is impossible; for we cannot take any number from a smaller number.

=

5, a b will be 3 But to subtract 5 is

-5, and we

Thus, if a = 3 and b cannot take 5 from 3. the same as to subtract 3 and 2 in succession, so that 3 5=3-3 − 2 = −2: we then consider that -2 is 2 which is to be subtracted from some other algebraical expression, or that 2 is two units of the kind opposite to that represented by 2. And if 2 is a final result, the latter is the only view that can be taken.

In some particular cases the quantities may be such that a negative result is without meaning; for instance, if we have to find the population of a town from certain given conditions; in this case the occurrence of a negative result would show that the given conditions could not be satisfied, but so also in this case would the occurrence of a fractional result.