The Commutative Property

Why addition and multiplication can be reordered while subtraction and division cannot, and how reordering speeds up mental arithmetic.

--

Some operations do not care about order, and some care a great deal. The commutative property tells you which is which — and that distinction is what makes it worth studying, even though the idea itself sounds obvious at first.

Concept What the property says

Swapping the order of the terms does not change the result.

For addition:

a + b = b + a

.

For multiplication:

a \times b = b \times a

.

These two operations are commutative. As you will see, not every operation is.

Concept Multiplication and addition

Three rows of five and five rows of three are the same fifteen dots, just viewed differently. Addition behaves the same way:

7 + 2 = 9

and

2 + 7 = 9

.

Concept Where it fails

This is the part worth remembering. Subtraction and division are not commutative.

8 - 3 = 5

, but

3 - 8 = -5

.

10 \div 2 = 5

, but

2 \div 10 = 0.2

.

The answers differ, so the order genuinely matters. Knowing that only addition and multiplication may be reordered is exactly what stops you from rearranging an expression that must not be touched.

Example Reordering to multiply easily

Work out $2 \times 17 \times 5$ mentally.

Multiplying 2 by 17 first is awkward.

Reorder to bring 2 and 5 together:

2 \times 5 \times 17

.

2 \times 5 = 10

, and

10 \times 17 = 170

.

⟹ 170

Nothing about the expression changed — only the order in which the work was done. That freedom is what the property grants.

Example Reordering to add easily

Work out $8 + 37 + 2$ .

Pair the numbers that make a round total:

8 + 2 = 10

.

Then

10 + 37 = 47

.

⟹ 47

Looking for friendly pairs before you start is a habit that makes mental arithmetic noticeably faster.

Example Testing an operation

Is subtraction commutative? Test it with a single pair of numbers.

8 - 3 = 5

.

3 - 8 = -5

.

The two results disagree.

⟹ subtraction is not commutative

One counterexample is enough to settle it. A property must hold in every case, so a single failure disproves it.

Note Mistakes to avoid

Reordering a subtraction:

8 - 3

and

3 - 8

are different.

Reordering a division:

10 \div 2

and

2 \div 10

are different.

Assuming every operation must be commutative because addition is.

Confusing this with the distributive property, which concerns brackets.

Missing the chance to pair friendly numbers before calculating.

Summary

The commutative property says order does not affect the result.
It holds for addition, a + b = b + a, and multiplication, a × b = b × a.
It fails for subtraction and division, where order changes the answer.
A single counterexample is enough to show an operation is not commutative.
Reordering lets you pair friendly numbers and calculate faster.

231. Arithmetic Sequences and Series:Next

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