The **commutative property** states that changing the order in which numbers are added or multiplied does not affect the outcome. While the commutative property applies to addition and multiplication, it does not work for subtraction and division. It applies to most numbers, but there are some sets that are excluded.

### Commutative Property Definition and General Form

#### For Addition:

The commutative property of addition states that for all real numbers *a* and *b*, *a*+*b* = *b*+*a*

#### For Multiplication:

The commutative property of multiplication states that for all real numbers *a* and *b*, *a*×*b* = *b*×*a*

### Applicability to Various Types of Numbers

The commutative property applies to:

- Whole Numbers
- Integers
- Fractions
- Real Numbers
- Complex Numbers
- Irrational Numbers

It does **not** apply to matrix multiplication, vector cross products, or function composition, which are non-commutative operations.

### Examples of Commutative Property

#### For Addition:

- 3 + 4 = 4 + 3 = 7
- 7 + (−7) = (−7) + 7 = 0
- ½ + ⅓ = ⅓ + ½

#### For Multiplication:

- 5 × 6 = 6 × 5 = 30
- ⅔ × ¾ = ¾ × ⅔
- (2+
*i*) × (3+2*i*) = (3+2*i*) × (2+*i*)

Note that you can’t combine the property for expressions that include both addition and multiplication because of the order of operations. For example:

1 + 2 x 3 **≠** 2 + 1 x 3

7 **≠** 5

### Non-Applicability to Subtraction and Division

The commutative property does not apply to subtraction and division. In other words, subtraction and division are **non-commutative**.

#### For Subtraction:

*a*−*b*** ≠*** b*−*a*

Example: 5−2 ≠ 2−5

#### For Division:

*b/a* **≠*** a/b*

Example: 4/2 ≠ 2/4

### Comparing Commutative and Associative Properties

Students often confuse the commutative and associative properties. While the commutative property deals with the order in which numbers are added or multiplied, the associative property deals with the grouping of numbers. The associative property states:

Addition: (*a*+*b*)+*c* = *a*+(*b*+*c*)

Multiplication: (*a*×*b*)×*c* = *a*×(*b*×*c*)

Both the commutative and associative properties apply to addition and multiplication, but not to subtraction and division.

### Commutative Property Examples for Different Types of Numbers

Here are examples of the commutative property for various types of numbers:

#### Whole Numbers:

- Addition: 4+5 = 5+4
- Multiplication: 6×7 = 7×6

#### Integers:

- Addition: −3+4 = 4+(−3)
- Multiplication: −2×3 = 3×(−2)

#### Fractions:

- Addition: ⅔+¼ = ¼+⅔
- Multiplication: ⅔×½ = ½×⅔

#### Variables:

- Addition:
*a*+*b*=*b*+*a* - Multiplication:
*ac*=*ca*

#### Real Numbers:

- Addition:
*π*+*e*=*e*+*π* - Multiplication:
*π*×*e*=*e*×*π*

#### Complex Numbers:

- Addition: (2+3
*i*) + (4+5*i*) = (4+5*i*) + (2+3*i*) - Multiplication: (2+3
*i*) × (4+5*i*) = (4+5*i*) × (2+3*i*)

#### Irrational Numbers:

- Addition:
*√*2 +*√*3 =*√*3 +*√*2 - Multiplication:
*√*2 ×*√*3 =*√*3 ×*√*2

### Commutative Property Practice Problems

#### Identifying the Commutative Property

Which of the following shows the commutative property for multiplication?

- 2 + 4 = 4 + 2
- 1 x 3 x 5 = 5 x 3 x 1
- (1 x 3) x 5 = 1 x (3 x 5)
- 2 + 4 x 1
**≠**1 x 4 + 2

The correct answer is 1 x 3 x 5 = 5 x 3 x 1

The first expression involves addition and not multiplication. The third expression illustrates the associative property. The last equation did not illustrate the property for multiplication.

#### Grocery Shopping With the Commutative Property

Alice and Bob went grocery shopping together. Alice bought 2 apples and 4 bananas, while Bob bought 5 apples and 3 bananas. Would it have made a difference if they had bought their groceries in a different order? Using the commutative property, prove whether the total number of each type of fruit they bought would be different if they switched the order.

Alice’s Purchase

- Alice bought 2 apples and 4 bananas.

Bob’s Purchase

- Bob bought 5 apples and 3 bananas.

Apply the commutative property to the word problem:

For apples:

- Alice + Bob = 2 + 5 = 7
- Bob + Alice = 5 + 2 = 7

Both scenarios yield 7 apples.

For bananas:

- Alice + Bob = 4 + 3 = 7
- Bob + Alice = 3 + 4 = 7

Both scenarios yield 7 bananas. Switching the order of purchasing has no effect on the total number of fruit.

### References

- Copi, Irving M.; Cohen, Carl; McMahon, Kenneth (2014).
*Introduction to Logic*(14th ed.). Essex: Pearson Education. ISBN 9781292024820. - Durbin, John R. (1992).
*Modern Algebra: An Introduction*(3rd ed.). New York: Wiley. ISBN 978-0-471-51001-7. - Gregory, D. F. (1840). “On the real nature of symbolical algebra“.
*Transactions of the Royal Society of Edinburgh*. 14: 208–216. - Hungerford, Thomas W. (1974).
*Algebra*(1st ed.). Springer. ISBN 978-0387905181.