Exercise 19 Page 1

a The addition of integers depends heavily on the numbers' signs and whether they are positive or negative. There are two situations we need to consider.

The signs are the same.
The signs are different.

In the situation where the signs are the same, we can add the absolute values of the numbers and keep the existing sign. Let's look at an example for both possible cases.

(+) + (+) = (+) \Rightarrow - 3 + 5 = 8 (-) + (-) = (-) \Rightarrow - 3 + (- 5) = - 8

When the signs are different, we subtract the absolute values of the numbers and keep the sign of the number whose absolute value is larger.

(+) + (-) = (+) \Rightarrow 5 + (- 3) = 2 (+) + (-) = (-) \Rightarrow 3 + (- 5) = - 2

b Subtraction can be thought of as adding the opposite. Therefore, the rules from Part A apply.

5 - 3 - 5 - 3 5 - (- 3) \Rightarrow - 5 + (- 3) = 2 \Rightarrow - 5 + (- 3) = - 8 \Rightarrow - 5 + 3 = 8

c When multiplying integers we have two possible situations, though the absolute value of the result does not change.

The signs are the same.
The signs are different.

In a situation where the signs are the same, the product will always be positive.

(+) \times (+) = (+) \Rightarrow (-) 5 \times 3 = 15 (-) \times (-) = (-) \Rightarrow (- 5) \times (- 3) = 15

Now, if the signs are opposite the product will always be negative.

(+) \times (-) = (-) \Rightarrow 5 \times (- 3) = - 15 (-) \times (+) = (-) \Rightarrow (- 5) \times 3 = - 15

d Division of integers follows the same rules as multiplication. If two integers have the same sign, their quotient is positive. Conversely, if their signs are opposite, the quotient is negative.

(+) \div (+) = (+) \Rightarrow 15 \div 3 = 5 (-) \div (-) = (+) \Rightarrow (- 15) \div (- 3) = 5 (+) \div (-) = (-) \Rightarrow 15 \div (- 3) = - 5 (-) \div (+) = (-) \Rightarrow (- 15) \div 3 = - 5

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Exercises p.1