8. The commutative law likewise applies to the sum of three or more integers. That is:

The sum is the same for every order of adding the numbers.

9. Associative Law of Addition. If we have three rows of books, the number of books is the same whether those in the second row are first placed with the first row, and then those in the third row placed with these, or those in the third row placed with the second, and then all of these with the first row.

In symbols:

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

This fact is called the associative law of addition.

10. Graphical Representation. The associative law may be represented graphically thus:

The properties stated above are often used to abridge calculations. Thus, 7 +4+3 +6, are more easily added thus: (7+3) + (4 + 6).

11. Subtraction. It often happens that we wish to know how many objects are left when some of a set are taken away, or to know how much greater one number is than another. The process of finding this number is called subtraction. The number taken away is called the subtrahend, that from which it is taken, the minuend, and the result, the difference or the remainder.

12. The sign of subtraction is

13. Subtraction is the reverse of addition, and from every sum one or more differences can at once be read.

Thus, from 5+ 7 = 12 we read at once 12 – 5 = 7, 12-75.

And from 5+ 5 = 10, we read 10 – 5 = 5.

And from a + b = c, we read ca = b,

c-b=a.

Likewise, from a + b + c = d we read d — a = b + c,

d − (a + b) = c, etc.

14. There is no commutative law of subtraction.

For 7 - 4 is not the same as 4-7. In fact, the latter indicated difference has no meaning in arithmetic. We cannot take a larger number of objects from a smaller number.

15. In algebra, where numbers are often represented by letters, we may not know whether the minuend is larger than the subtrahend or not. For example, in ab, we do not know whether a is larger than b or not. But it is desirable that such expressions should have a meaning in all cases, and this is accomplished by the definition and use of relative numbers.

16. The First Extension of the Number System. Relative Numbers. Whenever quantities may be measured in one of two opposite senses such that a unit in one sense offsets a unit in the other sense, it is customary to call one of the senses the positive sense, and the other the negative sense, and numbers measuring changes in these senses are called positive and negative numbers respectively. (Examples, see Chap. IV, Pt. I.)

17. A number to be added is offset by an equal number to be subtracted; hence such numbers satisfy the above definition, and numbers to be added are called positive, and those to be subtracted are called negative. Consequently, positive and negative numbers are denoted by the signs + and respectively.

Thus, +5 means positive five, and denotes five units to be added or to be taken in the positive sense.

5 means negative five, and denotes five units to be subtracted, or to be taken in the negative sense.

18. Graphical Representation. Relative integers may be represented graphically thus:

It appears that the positive integers are represented by just the same set of points as the natural or absolute integers. For this and other reasons the absolute numbers are usually identified with positive numbers. Although it is usually convenient to do this, we have in fact the three classes of numbers: the absolute, the positive, and the negative. Thus, we may consider $5 without reference to its relation to an account, or we can consider it as $5 of assets, or we may consider it as $ 5 of debts.

19. According to Sec. 17, the signs +, denote the operations of addition or subtraction, or the positive or negative character of the numbers which these signs precede.

If it is necessary to distinguish a sign of character from a sign of operation, the former is put into a parenthesis with the number it affects.

Thus,

8(3), means: positive 8 minus negative 3.

When no sign of character is expressed, the sign plus is understood.

Similarly, 8a+ 9 a means: positive 8 a plus positive 9 a.

20. Absolute Value. The value of a relative number apart from its sign is called its absolute value.

ORAL EXERCISES

Read the following in full, according to Sec. 19:

1. The sum of positive 5 and positive 3.
2. The sum of positive a and negative b.
3. The difference of positive p and positive q.
4. The difference of negative 5 and positive 3.
5. The difference of negative x and positive y.
6. The sum of positive a and positive b.

7. The sum of negative ab and negative ab.

8. The sum of positive y and negative x.

9. The difference of positive xy and negative xy.

10. The difference of negative pq and positive mn.

21. Addition of Relative Numbers.

Just as 3 pounds

and

=

3 positive units+5 positive units 8 positive units, 3 negative units +5 negative units = 8 negative units.

To add units of opposite character, use is made of the defining property of relative numbers, that a unit in one sense offsets a unit in the other sense. Thus, to add 3 positive units and 7 negative units we notice that the 3 positive units offset

3 of the negative units and the result of adding the two will be 4 negative units.

That is, (+3)+(−7) = (+3) + ( − 3) + ( − 4) = — 4.

In general:

I. If two relative numbers have the same sign, the absolute value of the sum is the sum of the absolute values of the addends, and the sign of the sum is the common sign of the addends.

II. If two relative numbers have opposite signs, the absolute value of the sum is the difference of the absolute values of the addends, and the sign of the sum is the sign of the addend having the larger absolute value.

22. More than two numbers are added by repetition of the process just described. This may be done either :

(1) by adding the second number to the first; then the third number to the result, and so on; or

(2) by adding separately all the positive numbers and all the negative numbers, and then adding these two results.

23. It may be verified that the Commutative and the Associative Laws of Addition hold also for relative integers.

24. Subtraction of Relative Numbers. Since n units of one sense are offset by adding n units of the opposite sense, we may subtract n units of one sense by adding n units of the opposite sense.

25. Accordingly, subtraction may be regarded as the inverse of addition: To subtract a monomial, we add its opposite.

To subtract an algebraic expression consisting of more than one term, we subtract the terms one after another.

In general, to subtract any algebraic expression we may change the sign of each of its terms and add the result to the minuend.