Commutative Property

The commutative property (alternatively called the commutative properties of addition and multiplication or simply the commutative laws) states that the order in which you add or multiply two real numbers does not affect the result. This is an extremely important concept across multiple levels of mathematics, so this article will provide several examples to help you understand the full implications of the commutative property. Let's get started.

The commutative property of addition

The commutative property of addition states that it doesn't matter which real number you add first: The resulting sum will always be the same. This can be expressed mathematically as:

$a+b=b+a$

For example, you know that $3+5=8$ and that $5+3=8$ . It doesn't matter which number is written first as the answer is still 8.

The commutative property of addition also holds when negative numbers are involved. For instance, $20+\left(-3\right)=17$ and $\left(-3\right)+20=17$ . Likewise, fractions and decimals follow the commutative property of addition. Put another way, $0.3+0.4=0.4+0.3$ .

However, the operation of subtraction is not commutative because $3-5=-2$ but $5-3=2$ . The order of the terms changes the result, so subtraction is sometimes called noncommutative or anticommutative.

The commutative property of multiplication

The commutative property of multiplication states that it doesn't matter which real number you multiply first: the resulting product will always be the same. This can be expressed mathematically as:

$a\left(b\right)=b\left(a\right)$

For instance, we know that $4\times 5=20$ and that $5\times 4=20$ . Switching the positions of 5 and 4 doesn't change the answer, thus proving the validity of the commutative property of multiplication.

Similarly, the commutative property of multiplication holds when negative numbers are involved. For example, $\left(-2\right)\left(8\right)=-16$ and $\left(8\right)\left(-2\right)=-16$ . Again, fractions, decimals, and mixed numbers also follow the commutative property of multiplication: $2\frac{1}{2}\times 3\frac{1}{4}=3\frac{1}{4}\times 2\frac{1}{2}$

You may already be thinking that division is not commutative, and you would be right. Remember that fractions are another way of expressing division. The fraction $\frac{1}{2}=0.5$ , but the fraction $\frac{2}{1}=2$ . The quotients are different, which means the order of the terms matters. The same is true for problems in the format a divided by b does not equal b divided by a or a long division sign.

How the commutative property relates to the associative property

While the commutative property states that the order in which you add or multiply real numbers doesn't change the result, the associative property states that the grouping of three or more numbers doesn't affect the results. For example, $(a+b)+c=a+(b+c)$ . While distinct properties, the two concepts are closely related. Be careful not to confuse them!

Practice problems on the commutative property

a. If $x+y=27$ and both x and y are real numbers, what must $y+x$ equal?

$27$

b. If $x\left(y\right)=18$ and both x and y are real numbers, what must $y\left(x\right)$ equal?

$18$

c. Write a mathematical statement that illustrates how the commutative property of addition works.

$1+2=2+1$

d. Write a mathematical statement that illustrates how the commutative property of multiplication works.

$6\times 5=5\times 6$

e. In your own words, explain the difference between the commutative property and the associative property.

Commutative Property: The order of terms doesn't matter.

Associative Property: Grouping of terms doesn't matter.

Topics related to the Commutative Property

Associative Property

Distributive Property

Properties of Multiplication

Flashcards covering the Commutative Property

3rd Grade Math Flashcards

Basic Arithmetic Flashcards

Practice tests covering the Commutative Property

Common Core: 3rd Grade Math Diagnostic Tests

3rd Grade Math Practice Tests

Varsity Tutors can help students understand the commutative property

The commutative property is essential to the study of algebra, calculus, mathematical proofs, working with matrices, and more. As such, students who fail to grasp it now will be behind their peers for years to come. Fortunately, an experienced 1-on-1 math tutor can present the commutative property and other topics in fresh ways to deepen a pupil's understanding of the material. If you have any questions regarding private math instruction or are ready to sign up right now, the friendly Educational Directors at Varsity Tutors are standing by to assist you!