# Commutative vs. Associative: What's the Difference?

By Janet White || Published on January 6, 2024
Commutative property refers to the ability to change the order of numbers without affecting the result, while associative property involves changing the grouping of numbers without changing the outcome.

## Key Differences

The commutative property, applicable in operations like addition and multiplication, states that the order of numbers does not affect the end result. For example, in addition, 3 + 5 is the same as 5 + 3. The associative property, on the other hand, deals with the grouping of numbers. It asserts that how numbers are grouped in an operation (addition or multiplication) does not change the sum or product. An example is (2 + 3) + 4 equals 2 + (3 + 4).
Commutative property simplifies calculations by allowing flexibility in the order of operations. This property is not applicable in subtraction and division. Associative property is helpful in making complex calculations easier by altering the grouping of numbers, especially in longer expressions. Like commutativity, it also does not apply to subtraction and division.
In the commutative property, the focus is on the sequence of the numbers or elements involved in the operation. This property is particularly useful in mental math and algorithmic calculations. The associative property emphasizes the way elements are associated or grouped together, which is crucial in algebraic expressions and solving equations.
The commutative property is represented by the equation a + b = b + a or a × b = b × a, showcasing the interchangeability of numbers. The associative property is symbolized by (a + b) + c = a + (b + c) or (a × b) × c = a × (b × c), indicating the same result despite different groupings.
Understanding the commutative property helps in recognizing patterns and relationships between numbers, enhancing problem-solving skills. The associative property aids in breaking down complex calculations into simpler parts, making it easier to compute mentally or with algorithms.
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## Comparison Chart

### Definition

Changing the order of numbers without affecting the outcome
Changing the grouping of numbers without affecting the outcome

### Applicable Operations

Addition and multiplication
Addition and multiplication

### Non-applicable Operations

Subtraction and division
Subtraction and division

### Mathematical Representation

A + b = b + a; a × b = b × a
(a + b) + c = a + (b + c); (a × b) × c = a × (b × c)

### Utility

Simplifies calculations by order flexibility
Makes complex calculations easier by altering groupings
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## Commutative and Associative Definitions

#### Commutative

Property where changing the order of numbers does not change the result.
In the commutative property, 7 + 9 is the same as 9 + 7.

#### Associative

A mathematical rule applied in addition and multiplication.
The associative property makes (7 × 4) × 2 the same as 7 × (4 × 2).

#### Commutative

A property where the order of terms does not affect the final answer.
The commutative property is evident when 15 × 2 equals 2 × 15.

#### Associative

A property where the grouping in equations does not affect the outcome.
The associative property applies to 1 + (2 + 3) being the same as (1 + 2) + 3.

#### Commutative

A fundamental principle in basic arithmetic operations.
The commutative property simplifies calculations like 8 + 12 being equal to 12 + 8.

#### Associative

Property where changing the grouping of numbers does not change the result.
The associative property shows (2 + 3) + 4 equals 2 + (3 + 4).

#### Commutative

The concept that sequence in certain operations is interchangeable.
Thanks to the commutative property, 3 + 6 + 9 equals 9 + 3 + 6.

#### Associative

The principle that the way numbers are grouped in an operation is flexible.
Thanks to the associative property, (5 + 6) + 7 equals 5 + (6 + 7).

#### Commutative

A mathematical principle applicable in addition and multiplication.
The commutative property allows 4 × 5 to equal 5 × 4.

#### Associative

Of, characterized by, resulting from, or causing association.

#### Commutative

Relating to, involving, or characterized by substitution, interchange, or exchange.

#### Associative

(Mathematics) Independent of the grouping of elements. For example, if a + (b + c) = (a + b) + c, the operation indicated by + is associative.

#### Commutative

Independent of order. Used of a logical or mathematical operation that combines objects or sets of objects two at a time. If a × b = b × a, the operation indicated by × is commutative.

#### Associative

Pertaining to, resulting from, or characterised by association; capable of associating; tending to associate or unite.

#### Commutative

Such that the order in which the operands are taken does not affect their image under the operation.
Addition on the real numbers is commutative because for any real numbers $s,t$, it is true that $s+t=t+s$.
Addition and multiplication are commutative operations but subtraction and division are not.

#### Associative

Such that, for any operands $a, b$ and $c$, $\left(a * b\right) * c = a * \left(b * c\right)$; (of a ring, etc.) whose multiplication operation is associative.

#### Commutative

Having a commutative operation.

#### Associative

(computing) Addressable by a key more complex than an integer index.
Associative memories were once given considerable attention.

#### Commutative

Such that any two sequences of morphisms with the same initial and final positions compose to the same morphism.

#### Associative

Having the quality of associating; tending or leading to association; as, the associative faculty.

#### Commutative

Relating to exchange; interchangeable.

#### Associative

Relating to or resulting from association;
Associative recall

#### Commutative

Relative to exchange; interchangeable; reciprocal.
Rich traders, from their success, are presumed . . . to have cultivated an habitual regard to commutative justice.

#### Associative

Characterized by or causing or resulting from association;
Associative learning

#### Commutative

Having the property of commutativity.

#### Associative

Fundamental in simplifying and solving complex arithmetic expressions.
The associative property aids in solving expressions like (3 × 5) × 4 equals 3 × (5 × 4).

#### Commutative

Of a binary operation; independent of order; as in e.g.
A x b = b x a

## FAQs

#### What is the associative property?

It's a principle where changing the grouping of numbers in an operation doesn't affect the outcome.

#### What does commutative mean in math?

It means the order of numbers in an operation (addition/multiplication) doesn't affect the result.

#### Can the commutative property be used in division?

No, it's not applicable in division and subtraction.

#### Is the associative property applicable in subtraction?

No, it only applies to addition and multiplication.

#### How does the commutative property help in calculations?

It allows flexibility in the order of operations, simplifying mental math.

#### Are commutative and associative properties the same?

No, commutative focuses on order, while associative focuses on grouping.

#### How do you teach the associative property?

By showing that different groupings in addition or multiplication give the same result.

#### How does the associative property work in addition?

For instance, (4 + 5) + 6 equals 4 + (5 + 6).

#### Why is the associative property important?

It helps break down complex expressions into simpler parts for easy computation.

#### What's a real-world example of the commutative property?

If you buy 3 apples and then 5 apples, it's the same as buying 5 apples and then 3 apples.

#### Can the associative property be visualized?

Yes, through grouping symbols in equations or with objects like blocks.

#### Can you give an example of the commutative property in multiplication?

Yes, 6 × 7 is the same as 7 × 6.

#### Does the commutative property apply to all numbers?

Yes, but only in addition and multiplication operations.

#### Are the commutative and associative properties used in algebra?

Yes, they are fundamental in simplifying and solving algebraic expressions.

#### Do computers use the commutative property in calculations?

Yes, it's used in algorithms for efficiency in computing.

#### Are there exceptions to these properties?

They don't apply to subtraction and division, and certain advanced mathematical operations.

#### Can the commutative property be applied to functions?

Only if the functions are commutative in nature, like addition or multiplication.

#### Do all mathematicians agree on these properties?

Yes, these are universally accepted principles in mathematics.

#### Is the commutative property intuitive?

For many people, yes, especially in basic arithmetic operations.

#### Is the associative property important in higher math?

Absolutely, it's essential in more complex equations and calculations.
About Author
Written by
Janet White
Janet White has been an esteemed writer and blogger for Difference Wiki. Holding a Master's degree in Science and Medical Journalism from the prestigious Boston University, she has consistently demonstrated her expertise and passion for her field. When she's not immersed in her work, Janet relishes her time exercising, delving into a good book, and cherishing moments with friends and family.