Multiplying and Dividing Integers

Multiplying and dividing integers is straightforward once you memorize a few simple rules about positive and negative signs. The great news is that the rules for multiplication are exactly the same as the rules for division!

The Basic Sign Rules

When multiplying or dividing two integers, look at their signs:

1. Same Signs Give a Positive Result: If both numbers are positive or both are negative, the answer is always positive.

   • Positive $\times$ Positive = Positive
   • Negative $\times$ Negative = Positive
   • Negative $\div$ Negative = Positive

2. Different Signs Give a Negative Result: If one number is positive and the other is negative, the answer is always negative.

   • Positive $\times$ Negative = Negative
   • Negative $\div$ Positive = Negative

Example Problems

Let's apply these rules to some basic examples:

Example 1: Calculate $(-3) \times 4$

• We are multiplying a negative number by a positive number (different signs).
• First, multiply the numbers normally: $3 \times 4 = 12$.
• Apply the sign rule: Different signs mean the answer is negative.
• Answer: $-12$

Example 2: Calculate $(-12) \div (-3)$

• We are dividing a negative number by a negative number (same signs).
• First, divide normally: $12 \div 3 = 4$.
• Apply the sign rule: Same signs mean the answer is positive.
• Answer: $4$

Multiplying Several Integers

When you need to multiply more than two integers together, simply count how many negative numbers are in the problem:

• Even count of negatives: The final product is positive.
• Odd count of negatives: The final product is negative.

Example 3: Calculate $(-2) \times (-5) \times (-1)$

• Count the negative numbers: There are $3$ negative numbers. Since $3$ is an odd number, our final answer will be negative.
• Multiply the numbers normally: $2 \times 5 \times 1 = 10$.
• Apply the sign rule: Odd count means the answer is negative.
• Answer: $-10$