CHAPTER II.

ADDITION AND SUBTRACTION.

INTEGRAL EXPRESSIONS.

24. If an algebraic expression contains only integral forms, that is, contains no letter in the denominator of any of its terms, it is called an integral expression. Thus, x3 +7cx2 — c3 −5cx and tax-bey are integral expres2c24b- C х is a fractional expression.

sions, but

a2- ab+b2

An integral expression may have for some values of the letters a fractional value, and a fractional expression an integral value. If, for instance, a stands for and b for, the integral expression 2a-56 stands for &- §; and the fractional expression stands 36

+

=

5 a

for ÷ 3 = 5. Integral and fractional expressions, therefore, are so named on account of the form of the expressions, and with no reference whatever to the numerical value of the expressions when definite numbers are put in place of the letters.

25. Definition of Addition. The process of finding the result when two or more numbers are taken together is called addition, and the result is called the sum.

26. Definition of Subtraction. The process of finding the result when one number is taken from another is called subtraction, and the result is called the difference or remainder.

The number taken away is called the subtrahend; the number from which the subtrahend is taken is called the minuend.

In practice the difference is found by discovering the number which must be added to the subtrahend to give the minuend. Hence the general definition of subtraction is

The operation of finding from two given numbers, called minuend and subtrahend, a third number, called difference, which added to the subtrahend will give the minuend.

An algebraic

27. Parentheses for Algebraic Numbers. number which is to be added or subtracted is often inclosed in a parenthesis, in order that the signs + and -, which are used to distinguish positive and negative numbers, may not be confounded with the + and signs that denote the operations of addition and subtraction. Thus + 4+(−3) · expresses the sum, and +4−(-3) expresses the difference, of the numbers +4 and — 3.

28. Addition of Algebraic Numbers. In order to add two algebraic numbers, we begin at the place in the series which the first number occupies and count, in the direction indicated by the sign of the second number, as many units as there are in the absolute value of the second number.

The sum of +4 + (+ 3) is found by counting from + 4 three units in the positive direction; that is, to the right, and is, therefore, +7. The sum of + 4 + (− 3) is found by counting from + 4 three units in the negative direction; that is, to the left, and is, therefore, + 1. 1 0 +1 +2 + 3 + 4 + 5 +6.....

5 -4 3 2

The sum of

1

4+ (+3) is found by counting from 4 three units

in the positive direction, and is, therefore, — 1.

The sum of - 4 + (− 3) is found by counting from - 4 three units in the negative direction, and is, therefore, — 7.

-

Hence to add two or more algebraic numbers, we have the following rules:

I. If the numbers have like signs. Find the sum of their absolute values, and prefix the common sign to the result.

II. If there are two numbers with unlike signs. Find the difference of their absolute values, and prefix to the result the sign of the greater number.

III. If there are more than two numbers with unlike signs. Combine the first two numbers and this result with the third number, and so on; or, find the sum of the positive numbers and the sum of the negative numbers, take the difference between the absolute values of these two sums, and prefix to the result the sign of the greater sum.

29. The result in each case is called the sum. It is often ⚫ called the algebraic sum, to distinguish it from the arithmetical sum, that is, the sum of the absolute values of the numbers.

30. Subtraction of Algebraic Numbers. In order to subtract one algebraic number from another, we begin at the place in the series which the minuend occupies and count, in the direction opposite to that indicated by the sign of the subtrahend, as many units as there are in the absolute value of the subtrahend.

Thus, the result of subtracting + 3 from + 4 is found by counting from 4 three units in the negative direction; that is, in the direction opposite to that indicated by the sign + before 3, and is, therefore, +1.

The result of subtracting - 3 from + 4 is found by counting from +4 three units in the positive direction; that is, in the direction opposite to that indicated by the sign – before 3, and is, therefore, + 7.

0 +1 +2 +3 +4 +5 +6.

The result of subtracting + 3 from

1

4 is found by counting from - 4 three units in the negative direction, and is, therefore, — 7. The result of subtracting - 3 from -4 is found by counting from - 4 three units in the positive direction, and is, therefore, — 1.

If we employ the general symbols a and b to represent the absolute values of any two algebraic numbers, we have

From (1) and (3), it is seen that subtracting a positive number is equivalent to adding an equal negative number. From (2) and (4), it is seen that subtracting a negative number is equivalent to adding an equal positive number.

Hence, to subtract one algebraic number from another:

Change the sign of the subtrahend, and add the subtrahend to the minuend.

This rule is consistent with the definition of subtraction given in § 26; for, if we have to subtract-4 from +3, we must add +4 to the subtrahend, -4, to cancel it, and then add 3 to obtain the minuend; that is, we must add +7 to the subtrahend to get the minuend, but +7 is obtained by changing the sign of the subtrahend, -4, making it +4, and adding it to +3, the minuend.

31. The Commutative Law of Addition. If we have a group of 3 things and another group of 4 things, we shall have a group of 7 things, whether we put the 3 things with the 4 things or the 4 things with the 3 things.

If now we have

3 to add to +4, we begin at + 4 in the series, count three units to the left, and arrive at +1; and if we have + 4 to add to 3, we begin at 3 in the series, count four units to the right, and arrive at +1.

That is,

+4+(-3)= −3+(+4).

Hence, if a and b stand for any two numbers whatever, we have

a+b=b+a.

This is called the commutative law of addition, and may be stated as follows:

Additions may be performed in any order.

32. The Associative Law of Addition. If we have several numbers to be added, the result will evidently be the same, whether we add the numbers in succession or arrange them groups and add the sums of these groups.

in

This is called the associative law of addition, and may be stated as follows:

The terms of an expression may be grouped in any

manner.