Commutative and Associative are largely used in Mathematics to solve questions or to prove some theorem. These properties help to solve the questions and determine properties.

It helps to calculate answers. Both have different meanings, but both of them are related to each other.

Both can be applied to multiplication.

## Key Takeaways

- The commutative property applies to both addition and multiplication, allowing numbers to be rearranged without affecting the result.
- The associative property also involves addition and multiplication but focuses on regrouping numbers without changing the outcome.
- Both properties are fundamental in mathematics, helping to simplify and solve equations more efficiently.

## Commutative vs Associative

The difference between Commutative and Associative is that Commutative arises from the word commute, whereas Associative comes from the word grouping. Commutative makes the numbers switch, but Associative makes the group of numbers switch with each other. The order of factors or addends does not change the answer.

A commutative operation is an operation that is independent of the order of its operands. The addition and multiplication of real numbers are commutative operations, since for any real number, “a” and “b”.

However, subtraction and division are not commutative operations. The exact definition depends on the type of algebra being used.

An associative operation (also called a commutative operation) is a mathematical operation that retains the order of the operands.

The numbers 3 and 4 are added together, followed by 4 and 3 being added together, which means that the order of addition doesn’t matter. The associative property also works for subtraction and multiplication.

## Comparison Table

Parameters of Comparison | Commutative | Associative |
---|---|---|

Originate | Commute | Group |

Meaning | Switch numbers | Numbers in a group |

Two numbers in addition | a+b = b+a | (a+b)+c = a+(b+c) |

Two numbers in multiplication | a*b = b*a | (a*b)*c = a*(b*c) |

Change | Order of addends | Grouping of addends |

Answer changes | The order of factors does not change the answer. | A group of factors does not change the answer. |

## What is Commutative?

While the commutative property of addition is relatively straightforward, the commutative property of multiplication is slightly more subtle.

Contrast the addition and multiplication of real numbers. In this case, we have not only a change in the order of the terms but also a change in the result!

This is something we do not see too. For example, if we consider why, then both 1+3 and 3+1 are equal to 4.

If we were to swap the order of these two terms, the answer would still be 4. In fact, every binary operation (including the empty operation) is commutative in a field or a ring.

A commutative operation is an operation in mathematics whose order does not matter. In other words, the result of any two operations with the same operands is always the same regardless of their order.

Commutative operations are very important for simplifying mathematical expressions and avoiding order of operations errors.

A commutative operation is defined as an operation that can be reversed.

For example, multiplying two numbers is commutative because multiplying the first number by the second number or vice versa will give the same result.

If we use the + operator on two numbers, the result may not always be the same.

## What is Associative?

Subtracting one number from another and then subtracting the second number from the first will give the same result as subtracting these two numbers in any order.

The associative property allows us to rewrite expressions in different ways without changing their value. For example, if we have two functions, f(x) and g(x).

An associative operation is a generalization of an operation defined between elements from a group that has a particular property.

Associative operations are common in many fields, such as mathematics, physics, philosophy, linguistics, and computer science.

The most familiar associative operation is an addition to the set of real numbers. That is, for any three real numbers, the sum is independent of the grouping of the operands: for example.

This remains true if one or more of the summands are zero. This property extends to all commutative operations involving real numbers.

The associative operation represents an arithmetic operation that has the same result regardless of the order in which the operands are evaluated.

Associative operation is an important property of the map that enables us to do things like vector addition:

The associative law for intersection states that the intersection of three sets can be computed by starting with the intersection of two sets and then applying the intersection to the third set.

## Main Differences Between Commutative and Associative

- Commutative comes from the commute, but associative comes from the group.
- Commutative can switch numbers, but Associative refers to making the numbers in a group.
- Commutative is a+b = b+a but Associative is a+(b+c) = (a+b)+c in addition.
- Commutative is a x b = b x a, but Associative is a x (b x c) = (a x b) x c in multiplication.
- Commutative can change the order of addends and ends, but Associative can change the grouping of addends.
- The change in the order of factors does not change the answer and changes in the order of a group of factors.

**References**

- https://www.sciencedirect.com/science/article/pii/S0732312312000351
- https://journals.sagepub.com/doi/abs/10.1177/2167702612455742

Last Updated : 11 June, 2023

Piyush Yadav has spent the past 25 years working as a physicist in the local community. He is a physicist passionate about making science more accessible to our readers. He holds a BSc in Natural Sciences and Post Graduate Diploma in Environmental Science. You can read more about him on his bio page.