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Prealgebra: Operations

Properties of Multiplication

Problems

Problems

In the last section, we learned how to add long strings of numbers using the properties of addition. Similarly, it is sometimes necessary to multiply long strings of numbers without a calculator; this task is made easier by learning some of the properties of multiplication.

Multiplication and addition have some similar properties. Like addition, multiplication has a Commutative Property and an Associative Property.

Commutative Property

The commutative property for multiplication states that for any numbers a and b , the following is always true:

a×b = b×a

For example, 3×4 = 4×3 . We can see that this is true because 3×4 = 12 and 4×3 = 12 , so 3×4 = 4×3 . Just as in addition, we can multiply a long string of numbers in any order. This can make multiplication without a calculator easier. For example:

4×6×5 = 4×5×6 = 20×6 = 120

Associative Property

The associative property for multiplication states that for any numbers a , b , and c , the following is always true:

(a×bc = a×(b×c)

For example, (2×5)×6 = 2×(5×6) . We can see that this is true because (2×5)×6 = 10×6 = 60 , and 2×(5×6) = 2×30 = 60 . Thus, (2×5)×6 = 2×(5×6)

Identity Property

Multiplication also has its own Identity Property. This property states that when any number is multiplied by 1, it does not change its identity. For any number a, the following is always true:


a×1 = a  
a = a  

For example, 45×1 = 45 . 1×123 = 123 .

We can remember these three properties of multiplication just as we can remember the corresponding properties of addition. With the Commutative Property of Multiplication, when only multiplication is involved, numbers can move ("commute") to anywhere in the expression. With the Associative Property of Multiplication, any numbers that are being multiplied together can "associate" with each other. Also, multiplying by 1 does not change the Identity of a number.

Zero Product Property

Multiplication has two additional properties. The first is the Zero Product Property. This says that any number multiplied by 0 is equal to 0. For any number a, the following are always true:


a× 0 = 0  
a = 0  

For example, 3×0 = 0 . 4, 567, 892, 435×0 = 0 .
Because multiplication commutes, if you are multiplying a long string of numbers that contains 0, you can move 0 to the beginning of the expression:

4×234×7×9×16×0×54 = 0×4×234×7×9×16×54

Because multiplication associates, this expression is equal to:

0×(4×234×7×9×16×54) = 0.

Thus, when multiplying any string of numbers, if 0 is one of the numbers, then the answer is always 0.

Distributive Property of Multiplication over Addition

The final property of multiplication is the Distributive Property of Multiplication over Addition. This property says that for any numbers a , b , and c , the following is always true:

a×(b + c) = (a×b) + (a×c).

For example, 3×(5 + 1) = (3×5) + (3×1) . We can see that this is true because 3×(5 + 1) = 3×6 = 18 and (3×5) + (3×1) = 15 + 3 = 18 .

Examples

Just like the properties of addition, these properties of multiplication can be used in any order. Here are some examples to make the properties more familiar:

Example 1. 2×13×5 = ?
Commutative Property: 2×13×5 = 2×5×13
2×5×13 = 10×13 = 130

Example 2. 8×(5×9) = ?
Associative Property: 8×(5×9) = (8×5)×9
(8×5)×9 = 40×9 = 360

Example 3. 43×9×0×7 = ?
Zero Product Property: 43×9×0×7 = 0

Example 4. 1×591 = ?
Identity Property: 1×591 = 591

Example 5. 6×(2 + 20)
Distributive Property: 6×(2 + 20) = (6×2) + (6×20)
(6×2) + (6×20) = 12 + 120 = 132

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