To multiply integers with different signs, the product will always be negative. This stems from fundamental rules of integer multiplication.
Rules for Multiplying Integers
Here's a quick reference table summarizing the rules:
| Integer 1 | Integer 2 | Product | Rule Reference |
|---|---|---|---|
| Positive | Positive | Positive | RULE 2 |
| Positive | Negative | Negative | RULE 1 |
| Negative | Positive | Negative | RULE 1 |
| Negative | Negative | Positive | RULE 3 |
Explanation
When multiplying a positive integer by a negative integer (or vice versa), the result is always negative. The absolute values of the integers are multiplied as usual, but the sign of the product is negative. This is formally stated as RULE 1: The product of a positive integer and a negative integer is negative.
Examples
Here are some examples to illustrate the rule:
- *(+5) (-3) = -15**
- *(-7) (+2) = -14**
- *(10) (-4) = -40**
- *(-1) (1) = -1**
Why is the Product Negative?
Think of multiplication as repeated addition. For example, 3 * -2 means adding -2 three times: -2 + -2 + -2 = -6. This intuitively shows why a positive number multiplied by a negative number results in a negative product.
