Chapter 1.5, Problem 47E

To determine

To prove : The rules of adding integers.

Explanation of Solution

Given information :

The rules of adding integers:

1. Same sign: add the absolute values and use the common sign.
2. Different signs: subtract the lesser absolute value from the greater value and use the sign of the number with greater absolute value.
3. Opposites: the sum of a number and its opposite is 0.

Proof :

1. To prove the same sign rule:

Taking any two positive integers, for example;

2+4 =6.

On the number line from 2 moving 4 points along positive axis, the arrow ends at 6.



The sum has the positive sign as the two integers as we are moving on the positive axis of the number line.

Taking two negative integers, for example;

-2 +(-4) = -6.

On the number line from -2 moving 4 points along negative axis, the arrow ends at -6.



The sum has the same sign as the two integers.

2. To prove the different sign rule:

Taking two integers of different signs, for example;

2+(-6) =-4.

On the number line from 2 moving 6 points along negative axis, the arrow ends at -4.



The sum has the sign of -4 as the number 4 >2.

3. To prove the sum of a number and its opposite is 0.

Taking any integer, for example, 2

The opposite number of 2 is -2.

Thus 2+(-2)=0.