III.

Thus 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, $+ 1$ .

The sum of $- 4 + (+ 3)$ 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, $- 7$ .

63. Subtraction.

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, and is, therefore, $+ 7$ .

The result of subtracting $+ 3$ from $- 4$ 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, $- 1$ .

Collecting the results obtained in addition and subtraction, we have: $c > c \textsc A d d i t i o n . & \textsc S u b t r a c t i o n . + 4 + (- 3) = + 4 - 3 = + 1 . & + 4 - (+ 3) = + 4 - 3 = + 1 . + 4 + (+ 3) = + 4 + 3 = + 7 . & + 4 - (- 3) = + 4 + 3 = + 7 . - 4 + (- 3) = - 4 - 3 = - 7 . & - 4 - (+ 3) = - 4 - 3 = - 7 . - 4 + (+ 3) = - 4 + 3 = - 1 . & - 4 - (- 3) = - 4 + 3 = - 1 .$

From these four cases of addition, therefore,

Add Algebraic Numbers,

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

If the numbers have unlike signs, find the difference of their absolute values, and prefix the sign of the greater number to the result.

If there are more than two numbers, find the sum of the positive numbers and the sum of the negative numbers,