Properties of Multiplication

In this section, we review the properties of multiplication. We will also review the basic operation of multiplication with whole numbers. When we multiply with whole numbers, we are using a shortcut for repeated addition. Suppose we have the following problem:
7 + 7 + 7 + 7 + 7
We could take the time and add all those 7’s, but it’s easier to use a multiplication table and find that 5 x 7 = 35. Generally speaking, we all learn our multiplication table (times table) up to 12 x 12 and use these facts to multiply small and large numbers.
Next, we will identify the parts of a multiplication problem. The numbers being multiplied together are known as factors. The result of the multiplication is known as the product.
Example 1: Identify the parts of 6 x 5 = 30
6 is a factor
5 is a factor
30 is the product or result of multiplying 6 and 5
We also discuss the various properties of multiplication: the commutative property, the associative property, the distributive property, the identity property of 1, and the multiplication property of zero.

The Commutative Property of Multiplication

states that we can place factors in any order when multiplying. Basically, we can multiply in any order, without changing the result.

Example 2: Rewrite 4 x 3 using the commutative property of multiplication
For this problem, we switch the order of the factors
3 x 4
4 x 3 = 12
3 x 4 = 12
In either order, the result of multiplying the factors 3 and 4 gives us 12

The Associative Property of Multiplication

states that we can group our multiplication of three factors or more in any order, without changing the result.

Example 3: Rewrite (7 x 2) x 3
For this problem, we switch the grouping of the factors
7 x (2 x 3)
(7 x 2) x 3 = 14 x 3 = 42
7 x (2 x 3) = 7 x 6 = 42
Either way the multiplication is grouped, the answer is 42

The Distributive Property of Multiplication

states that multiplication is distributive over addition/subtraction.

Example 4: Use the distributive property to simplify 4(3 + 2)
4(3 + 2) = 4 x 3 + 4 x 2 = 12 + 8 = 20
If we sum 3 and 2 first, the result is the same:
4(3 + 2) = 4 (5) = 20

The Identity Property of 1

states that we can multiply any number by 1, and it will remain unchanged.

Example 5: What is the product of 342 x 1?
342 x 1 = 342, any number multiplied by 1 remains unchanged

The Multiplication Property of Zero

states when we multiply by zero, the result is always zero.

Example 6: What is the product of 1,322,511 x 0?
1,322,511 x 0 = 0, the result of any number multiplied by zero is zero.