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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  
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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