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Commute — Definition, Formula & Examples

Two numbers or operations commute when you can swap their order and still get the same result. For example, 3 + 5 and 5 + 3 both equal 8, so addition commutes.

Two elements aa and bb commute under an operation ∗* if a∗b=b∗aa * b = b * a. When every pair of elements in a set commutes under a given operation, that operation is said to satisfy the commutative property.

Key Formula

a∗b=b∗aa * b = b * a
Where:
  • aa = First element
  • bb = Second element
  • ∗* = A binary operation (such as addition or multiplication)

How It Works

To check whether two values commute under an operation, perform the operation in both orders and compare the results. If you get the same answer both ways, they commute. Addition and multiplication of real numbers always commute: a+b=b+aa + b = b + a and a×b=b×aa \times b = b \times a. Subtraction and division generally do not commute: 7−3≠3−77 - 3 \neq 3 - 7.

Worked Example

Problem: Do 4 and 9 commute under subtraction?
Compute in original order: Subtract 9 from 4.
4−9=−54 - 9 = -5
Compute in reversed order: Subtract 4 from 9.
9−4=59 - 4 = 5
Compare results: Since −5≠5-5 \neq 5, the two results are different.
Answer: No, 4 and 9 do not commute under subtraction because 4−9≠9−44 - 9 \neq 9 - 4.

Why It Matters

Knowing which operations commute lets you rearrange terms to simplify expressions and solve equations more efficiently. In algebra, you rely on the fact that addition and multiplication commute every time you reorder terms or factor an expression.

Common Mistakes

Mistake: Assuming all operations commute because addition and multiplication do.
Correction: Subtraction and division do not commute. Always verify: a−ba - b does not generally equal b−ab - a, and a÷ba \div b does not generally equal b÷ab \div a.