The commutative property says that the order of the numbers does not change the answer. It applies to addition and multiplication, so 4 plus 7 gives the same result as 7 plus 4, and 3 times 5 equals 5 times 3. It does not apply to subtraction or division.
This guide explains the rule in plain language, shows where it works and where it fails, and separates it from the two properties it is most often confused with.
The Rule in Simple Terms
For any two numbers a and b:
- Addition: a + b = b + a
- Multiplication: a times b = b times a
The word comes from commute, meaning to move around. The numbers swap places and the total stays put. Britannica sets out the formal version in its entry on the commutative law.
Where It Works
Addition
12 + 38 = 50 and 38 + 12 = 50. This is why you can add a column of numbers from the bottom up and get the same total.
Multiplication
6 times 4 = 24 and 4 times 6 = 24. A rectangle of 6 rows by 4 columns holds the same number of squares as 4 rows by 6 columns. Turning the rectangle does not change the area.
Where It Fails
Subtraction
9 minus 4 = 5, but 4 minus 9 = minus 5. The order changes the answer, and often the sign.
Division
20 divided by 5 = 4, but 5 divided by 20 = 0.25. Again the order matters.
Some Advanced Operations
Matrix multiplication is not commutative, so A times B usually differs from B times A. Neither is function composition. Students often meet this for the first time in secondary school algebra, and it surprises them.
How It Differs From Associative and Distributive
- Commutative: changes the order. 2 + 3 = 3 + 2.
- Associative: changes the grouping. (2 + 3) + 4 = 2 + (3 + 4).
- Distributive: spreads multiplication over addition. 3 times (4 + 5) = 3 times 4 plus 3 times 5.
An easy memory hook: commute means move, associate means group, distribute means share out.
Why It Matters
The property looks obvious, yet it quietly supports a lot of everyday arithmetic.
- Mental maths. Reorder to make a friendly pair, such as 7 + 8 + 3 becoming 7 + 3 + 8.
- Checking work. Add a list in reverse order to test the total.
- Times tables. Learning 7 times 8 also gives you 8 times 7, halving the memorising.
- Algebra. Rearranging terms in an expression relies on it.
- Spreadsheets. A sum returns the same answer whatever order the rows sit in.
That last point shows up in ordinary record keeping. A column of maintenance costs totals the same whether it is sorted by date, by amount or by supplier, which is why clean numbers can be reordered safely when you review a property, as our guide on how property management companies measure real performance discusses.
Teaching It to Children
- Use counters in two groups and swap them over
- Draw an array, then rotate the paper a quarter turn
- Ask which order is easier to count, then agree both give the same total
- Show a subtraction example so the limits are clear from the start
Naming the property once the idea is understood works better than starting with the vocabulary.
Quick Practice
Decide whether each statement is true.
- 15 + 6 = 6 + 15
- 15 minus 6 = 6 minus 15
- 9 times 2 = 2 times 9
- 18 divided by 3 = 3 divided by 18
Only the first and third are true. Addition and multiplication commute. Subtraction and division do not.
Summary
The commutative property is a short rule with wide reach: order does not matter for addition and multiplication, and it does matter everywhere else. Recognising when it applies makes arithmetic faster and algebra far less mysterious.