Commutative
Operation
Any operation ⊕ for which a⊕b = b⊕a for
all values of a and b. Addition and
multiplication are both commutative. Subtraction, division,
and composition of functions are
not. For example,
5 + 6 = 6 + 5 but 5 – 6 ≠ 6 – 5.
More: Commutativity isn't just a property of
an operation alone. It's actually a property of an operation over
a particular set. For
example, when
we say
addition
is commutative
over the set
of real numbers, we mean that a + b = b + a for
all real numbers a and b. Subtraction is not
commutative over real numbers since we can't say that a – b
= b – a for
all real numbers a and b. Even though a – b
= b – a whenever
a and b are the same, that still
doesn't make subtraction commutative over the set of all real numbers.
Further examples: In this more formal sense,
it is correct to say that matrix
multiplication is not commutative
for square matrices. Even though AB = BA for some square matrices
A and B,
commutativity
does
not
hold
for all square matrices. It is also correct to say composition is
not commutative for functions,
even though one-to-one functions commute with their inverses.
See also
Associative
Worked Example
Problem: Determine whether multiplication is commutative by testing whether 7 × 4 = 4 × 7.
Step 1: Write the commutative property for multiplication using the given values.
a×b=b×a⟹7×4=?4×7 Step 2: Compute the left side.
7×4=28 Step 3: Compute the right side.
4×7=28 Step 4: Compare the two results. Since both sides equal 28, the equation holds. This is consistent with the fact that multiplication is commutative over all real numbers — swapping the order never changes the product.
28=28✓ Answer: Yes, 7 × 4 = 4 × 7 = 28. Multiplication is commutative.
Another Example
This example shows how to DISPROVE commutativity. Unlike the first example (which confirmed a commutative operation), here a single counterexample is enough to show that subtraction fails the commutative property.
Problem: Show that subtraction is NOT commutative by finding values of a and b where a − b ≠ b − a.
Step 1: Choose specific values. Let a = 10 and b = 3.
a=10,b=3 Step 3: Compute b − a.
3−10=−7 Step 4: Compare the results. Since 7 ≠ −7, the order matters. We only need one counterexample to prove an operation is not commutative.
Answer: Subtraction is not commutative because 10 − 3 ≠ 3 − 10.
Frequently Asked Questions
What is the difference between commutative and associative properties?
The commutative property says you can swap the order of two inputs: a + b = b + a. The associative property says you can regroup three or more inputs without changing the result: (a + b) + c = a + (b + c). Addition is both commutative and associative. Subtraction is neither commutative nor associative.
Is division commutative?
No, division is not commutative. For example, 12 ÷ 4 = 3 but 4 ÷ 12 = 1/3, and 3 ≠ 1/3. Because changing the order changes the result, division fails the commutative property.
Why does commutativity matter if the answer sometimes stays the same when you swap order?
An operation is commutative only if swapping works for ALL values, not just some. For instance, 5 − 5 = 5 − 5 regardless of order, but 5 − 3 ≠ 3 − 5. One counterexample is enough to prove an operation is not commutative over a given set.
Commutative Property vs. Associative Property
| Commutative Property | Associative Property |
|---|
| Definition | Order of two operands can be swapped: a ∗ b = b ∗ a | Grouping of three operands can be changed: (a ∗ b) ∗ c = a ∗ (b ∗ c) |
| What changes | The order of the inputs | The placement of parentheses |
| Addition | Commutative: 3 + 5 = 5 + 3 | Associative: (3 + 5) + 2 = 3 + (5 + 2) |
| Subtraction | NOT commutative: 3 − 5 ≠ 5 − 3 | NOT associative: (10 − 5) − 2 ≠ 10 − (5 − 2) |
| Number of operands involved | Two: a and b | Three: a, b, and c |
Why It Matters
The commutative property appears throughout algebra whenever you rearrange terms in an expression — for example, rewriting 3x + 7 as 7 + 3x relies on commutativity of addition. In higher math, many structures are not commutative: matrix multiplication and function composition both depend on order, which is why AB ≠ BA in general. Recognizing which operations are commutative helps you know when you can safely reorder terms and when doing so would produce errors.
Common Mistakes
Mistake: Thinking subtraction is commutative because a − a = a − a works.
Correction: Commutativity must hold for ALL pairs of values, not just when a = b. A single counterexample like 8 − 3 ≠ 3 − 8 disproves it.
Mistake: Confusing the commutative property with the associative property.
Correction: Commutative deals with swapping order (a + b = b + a). Associative deals with regrouping parentheses ((a + b) + c = a + (b + c)). They are independent properties — an operation can be one without the other.
