III.

take the difference between the absolute values of these two sums, and prefix the sign of the greater sum to the result.

Since the order in which numbers are added is immaterial, we may add any two of the numbers, and then this sum to any third number, and so on.

The result is generally called the algebraic sum, in distinction from the arithmetical sum; that is, the sum of the absolute values of the numbers.

From the four cases of subtraction in § 64, we see that subtracting a positive number is equivalent to adding an equal negative number, and subtracting a negative number is equivalent to adding an equal positive number.

Subtract One Algebraic Number from Another,

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

68. Examples.

1. Find the sum of $3a$, $2a$, $a$, $5a$, $7a$.

The sum of the coefficients is $3+2+1+5+7=18$.

Hence the sum of the numbers is $18a$.

2. Find the sum of $-5c$, $-c$, $-3c$, $-4c$, $-2c$.

The sum of the coefficients is $-5-1-3-4-2=-15$.

Hence the sum of the numbers is $-15c$.

3. Find the sum of $8x$, $-9x$, $-x$, $3x$, $4x$, $-12x$, $x$.

The sum of the positive coefficients is $8+3+4+1=16$.

The sum of the negative coefficients is $-9-1-12=-22$.

The difference between $16$ and $22$ is $6$, and the sign of the greater is negative.

Hence the required sum is $-6x$.

Exercise 15.

Find the sum of:

1. $5c$, $23c$, $c$, $11c$.

2. $4a$, $3a$, $7a$, $10a$.

3. $7x$, $12x$, $11x$, $9x$.