Multiplying Integers: Videos & Practice Problems

When multiplying two integers that have the same sign, the product is always positive. This means if you multiply a positive integer by another positive integer, the result is positive. Similarly, if you multiply a negative integer by another negative integer, the product is also positive. For example, $3\times 6=18$ and $-3\times -6=18$. This rule is important because it helps us understand how the signs affect the final product when multiplying integers.

When multiplying more than two integers, the sign of the product depends on the number of negative factors. If there is an even number of negative integers, the product is positive. If there is an odd number of negative integers, the product is negative. For example, multiplying $3\times -2\times -7$ results in a positive product because there are two negative factors (an even number). Conversely, multiplying $-6\times -5\times -2$ results in a negative product because there are three negative factors (an odd number). This pattern helps simplify the process of determining the sign without multiplying each step individually.

Multiplying a negative integer by a positive integer results in a negative product because the signs are different. The rule states that when the signs of the two factors are opposite, the product is negative. For example, $3\times -6=-18$. This happens because multiplication can be thought of as repeated addition, and adding a negative number multiple times results in a negative sum. This rule is consistent regardless of the order of multiplication due to the commutative property, so $-6\times 3=-18$ as well.

Multiplication can be viewed as repeated addition when dealing with positive integers, where multiplying $3\times 6$ means adding 6 three times: $6+6+6=18$. When negative integers are involved, this idea still applies if the multiplier is positive. For example, $3\times -6$ can be seen as adding $-6$ three times: $-6+-6+-6=-18$. However, when the multiplier is negative, repeated addition is less intuitive, and patterns or sign rules are used instead to determine the product.

When you multiply any integer by zero, the product is always zero. This is because zero times any number means you have zero groups of that number, which results in nothing. For example, $0\times -6=0$ and $-3\times 0=0$. This rule is fundamental in multiplication and applies regardless of the signs of the integers involved.