Commutative Property of Multiplication — Definition & Examples

Written by

Malcolm McKinsey

January 12, 2023

Fact-checked by

Paul Mazzola

Commutative property of multiplication definition

Commutative property of multiplication says that the order of factors in a multiplication sentence has no effect on the product. The commutative property of multiplication works on integers, fractions, decimals, exponents, and algebraic equations.

The commutative property of multiplication is one of the four main properties of multiplication. It is named after the ability of factors to commute, or move, in the number sentence without affecting the product.

The word “commutative” comes from a Latin root meaning “interchangeable”.

Switching the order of the multiplicand (the first factor) and the multiplier (the second factor) does not change the product.

• What is 4 × 5? The answer is 20.

• What is 5 × 4? The answer is also 20.

The order of the two factors, 4 and 5, did not affect the product, 20.

Commutative property of multiplication formula

The generic formula for the commutative property of multiplication is:

Any number of factors can be rearranged to yield the same product:

• 1 × 2 × 3 = 6

• 3 × 1 × 2 = 6

• 2 × 3 × 1 = 6

• 2 × 1 × 3 = 6

Often, when demonstrating the commutative property of multiplication, the product is shown in the middle of the multiple arrangements of the equation.

Commutative property of multiplication examples

Let's see the commutative property of multiplication in action with some examples:

In the first picture we can think of the set of five rubber ducks as the multiplicand, spread across from left to right. Beneath it, vertically, we have the multiplier, 4.

In the second picture we have one set of four rubber ducks arrayed left to right, the multiplicand. Then we have the multiplier, 5, vertically.

Whether we take a set of five rubber duckies and multiply them four times, as on the left, or we take a set of four rubber duckies and multiply them five times, as on the right, we still end up with 20 rubber duckies.

The commutative property of multiplication works on basic multiplication equations and algebraic equations. Here was see how to use commutative property of multiplication various multiplication sentences:

Integers:

• $6\times 7=42=7\times 6$

• $\mathrm{1,234}\times 0=0=0\times \mathrm{1,234}$

• $717\times 11=\mathrm{7,887}=11\times 717$

Exponents:

• ${6}^{2}\times {3}^{2}=\mathbf{324}={3}^{2}\times {6}^{2}$

• ${2}^{3}\times {4}^{3}=\mathbf{256}={4}^{3}\times {2}^{3}$

Fractions:

• $\frac{3}{4}\times \frac{7}{8}=\frac{\mathbf{21}}{\mathbf{32}}=\left(\frac{7}{8}\right)\left(\frac{3}{4}\right)$

• $\frac{9}{10}\times \frac{75}{100}=\frac{75}{100}\times \frac{9}{10}=\frac{\mathbf{675}}{\mathbf{1000}}\mathbf{=}\frac{\mathbf{27}}{\mathbf{40}}$ (simplified by dividing by $\frac{25}{25}$)

Decimals:

• $0.1234\times 0.987=0.1217958=0.987\times 0.1234$

• $411.52\times 0.3=123.456=0.3\times 411.52$

Variables:

• $4{x}^{2}\left(2\right)=\mathbf{32}=2\left(4{x}^{2}\right)$