Arithmetic
Commutative and Associative Properties
The commutative property says order does not matter: a + b = b + a and a × b = b × a. The associative property says grouping does not matter: (a + b) + c = a + (b + c). Both work for addition and multiplication, and both fail for subtraction and division. This guide shows each with examples, how to tell them apart, and how they speed up mental math.
The commutative and associative properties in plain words
The commutative property is about order: you can swap the two numbers in an addition or a multiplication and the answer stays the same. The associative property is about grouping: when you add or multiply three or more numbers, it does not matter which pair you work out first. Both properties hold for addition and multiplication. Neither holds for subtraction or division.
| Property | Addition | Multiplication | What it lets you do |
|---|---|---|---|
| Commutative | a + b = b + a | a × b = b × a | Change the order |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) | Change the grouping |
| Identity | a + 0 = a | a × 1 = a | Add 0 or multiply by 1 without changing a value |
A memory aid that works for most people: commute means to travel, and in the commutative property the numbers travel to new positions. Associate means to group with, and in the associative property the numbers change who they are grouped with while staying in the same order.
A third rule, the distributive property, connects multiplication with addition, as in 3 × (4 + 5) = 3 × 4 + 3 × 5. It has its own guide; this one covers order, grouping and identity.
The commutative property
The commutative property says that the order of two numbers does not change their sum or their product.
Example: commutative property of addition
Three apples and five apples make the same pile as five apples and three apples.
Example: commutative property of multiplication
Four rows of seven chairs and seven rows of four chairs are the same 28 chairs, just turned sideways.
This is why a multiplication chart is a mirror image of itself across its diagonal, and why you only need to learn 78 of its 144 facts.
The property works for every kind of number: whole numbers, fractions, decimals and negative integers. The fraction calculator will confirm the fraction case for any pair you try. 1/2 + 1/3 = 1/3 + 1/2 = 5/6, 0.5 × 8 = 8 × 0.5 = 4, and −3 + 8 = 8 + (−3) = 5.
Why it is true
Addition combines two collections, and a collection does not care which part arrived first. Multiplication counts a rectangle of objects, and turning the rectangle a quarter-turn swaps the number of rows and columns without adding or removing anything. Both pictures show the swap cannot change the total.
The associative property
The associative property says that when you add or multiply three or more numbers, you can group them in any way you like. The parentheses can move; the numbers stay in the same order.
Example: associative property of addition
Example: associative property of multiplication
Think of a box of chocolates 2 layers deep, 3 rows wide and 4 columns long. Count it layer by layer or row by row, and you still have 24 chocolates.
Because grouping does not matter, mathematicians write 2 + 3 + 4 and 2 × 3 × 4 with no parentheses at all. The answer is the same whichever pair you start with.
Commutative vs associative: how to tell them apart
The two properties are easy to mix up because they often appear together. The test is simple: look at what changed between the two sides.
- If the numbers changed places, it is the commutative property.
6 + 9 = 9 + 6. - If the numbers stayed in place but the parentheses moved, it is the associative property.
(6 + 9) + 1 = 6 + (9 + 1).
| Statement | What changed | Property |
|---|---|---|
7 × 3 = 3 × 7 | Order | Commutative (multiplication) |
(5 + 2) + 8 = 5 + (2 + 8) | Grouping | Associative (addition) |
x + 4 = 4 + x | Order | Commutative (addition) |
(4 × 5) × 6 = 4 × (5 × 6) | Grouping | Associative (multiplication) |
(3 + 9) + 1 = (9 + 3) + 1 | Order inside the parentheses | Commutative (addition) |
(2 × 9) × 5 = 2 × (5 × 9) | Both | Associative and commutative |
The last row is typical of real working: you regroup and reorder in the same move. Each step can be justified by one property or the other.
Using the properties for mental math
These properties are not just names to learn for a test. Together they say that a string of additions, or a string of multiplications, can be done in any order and any grouping. That is permission to pick the easiest route.
Example: 38 + 47 + 12 + 53
Look for pairs that make round numbers. 38 and 12 make 50; 47 and 53 make 100.
Example: 25 × 7 × 4
Multiplying 25 by 7 first gives 175, which is awkward. Swap the 7 and 4 so that 25 × 4 comes first.
The same idea works for 125 × 8 × 7 = 1,000 × 7 = 7,000 and 4 × 9 × 25 = 100 × 9 = 900. Pairs to look for: 2 × 5 = 10, 4 × 25 = 100, 8 × 125 = 1,000. For longer products the multiplication calculator shows each partial product.
In algebra
Collecting like terms is the commutative and associative properties at work. In 3x + 5 + 2x, you swap 5 and 2x, then group the x terms: 3x + 2x + 5 = 5x + 5. Writing x · 4 · y as 4xy is the same move for multiplication.
Why subtraction and division break both properties
Subtraction and division are not commutative and not associative. Swapping or regrouping changes the answer, so you cannot use the shortcuts above on them directly.
This is why the order of operations says to work subtraction and division from left to right. Without that rule, 8 − 3 + 2 could be read as 7 or as 3, and 6 ÷ 2 × 3 as 9 or as 1. Left to right gives 7 and 9, and the PEMDAS calculator shows each step in that order.
The way around it
Rewrite subtraction as adding a negative, and division as multiplying by a reciprocal. Then the properties apply again.
This trick depends on getting the signs right, which the negative number rules cover, and on the reciprocal idea explained in how to divide fractions.
Other operations that break them
Exponents are neither commutative nor associative. 2³ = 8 but 3² = 9. And (2³)² = 64 while 2^(3²) = 2⁹ = 512. The exponent rules say how powers do combine. Everyday actions are often non-commutative too: putting on socks then shoes is not the same as shoes then socks.
The identity property
The identity property names the number that leaves another unchanged. For addition that number is 0; for multiplication it is 1.
So 7 + 0 = 7 and 7 × 1 = 7. It sounds too obvious to need a name, but it powers a common technique: multiplying by a well-chosen form of 1. 2/2 equals 1, so 3/4 × 2/2 = 6/8 has the same value as 3/4. That is how equivalent fractions and common denominators are built.
Do not confuse the identity property of multiplication with the zero property: a × 0 = 0. Multiplying by 1 keeps a number; multiplying by 0 wipes it out.
Common mistakes
- Using the properties on subtraction or division.
9 − 5is not5 − 9. Convert to addition of a negative or multiplication by a reciprocal first. - Mixing up the two names. Order changed means commutative; parentheses moved means associative.
- Regrouping across a minus sign.
20 − 6 + 4is 18, not20 − 10 = 10. The minus sign belongs to the 6 only. - Mixing operations inside one regroup.
(2 + 3) × 4is 20, but2 + (3 × 4)is 14. The associative property only applies when every operation is the same one. - Calling a × 1 = a the zero property. That is the identity property; the zero property is a × 0 = 0.
Frequently asked questions
What is the commutative property?
The commutative property says that changing the order of two numbers does not change their sum or product: a + b = b + a and a × b = b × a. For example, 4 × 7 and 7 × 4 both equal 28. It holds for addition and multiplication, but not for subtraction or division.
What is the associative property?
The associative property says that when you add or multiply three or more numbers, the grouping does not change the result: (a + b) + c = a + (b + c). For example, (2 × 3) × 4 and 2 × (3 × 4) both equal 24. The numbers stay in the same order; only the parentheses move.
What is the difference between commutative and associative property?
The commutative property changes the order of numbers; the associative property changes their grouping. 6 + 9 = 9 + 6 is commutative because the numbers swapped places. (6 + 9) + 1 = 6 + (9 + 1) is associative because the numbers stayed put and only the parentheses moved.
Is subtraction commutative?
No. 10 − 4 = 6 but 4 − 10 = −6, so swapping the numbers changes the answer. Subtraction is not associative either. You can make it behave by rewriting it as addition of a negative: 10 − 4 = 10 + (−4), and that sum can be reordered as −4 + 10.
Is division associative?
No. (24 ÷ 4) ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 12. Because grouping changes the result, division is done from left to right in the order of operations. Division is also not commutative: 12 ÷ 3 = 4 while 3 ÷ 12 = 0.25.
What is an example of the commutative property of addition?
3 + 5 = 5 + 3, since both sides equal 8. The same rule works with any numbers, including fractions and negatives: 1/2 + 1/3 = 1/3 + 1/2 = 5/6 and −3 + 8 = 8 + (−3) = 5. In algebra it lets you write x + 4 as 4 + x.
What is the identity property?
The identity property says that adding 0 or multiplying by 1 leaves a number unchanged: a + 0 = a and a × 1 = a. Zero is the additive identity and one is the multiplicative identity. Multiplying by a form of 1, such as 2/2, is how equivalent fractions are made.
How do the associative and commutative properties help with mental math?
They let you reorder and regroup a sum or product to make friendly pairs. For 25 × 7 × 4, multiply 25 by 4 first to get 100, then 100 × 7 = 700. For 38 + 47 + 12 + 53, pair 38 with 12 and 47 with 53 to get 50 + 100 = 150.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/properties-of-operations/