Commutativity Formula

The Formula

a + b = b + a, \quad a \times b = b \times a

When to use: 3 + 5 = 5 + 3 and 3 \times 5 = 5 \times 3. Swapping the order doesn't change the answer.

Quick Example

Addition: 7 + 9 = 9 + 7 = 16 Multiplication: 4 \times 6 = 6 \times 4 = 24

Notation

Commutative law: the order of operands around + or \times may be swapped

What This Formula Means

A property where swapping the order of two operands does not change the result: a \ \star\ b = b\ \star\ a.

3 + 5 = 5 + 3 and 3 \times 5 = 5 \times 3. Swapping the order doesn't change the answer.

Formal View

\forall a, b \in \mathbb{R}: a + b = b + a \text{ and } a \cdot b = b \cdot a

Worked Examples

Example 1

easy
Show that \(6 + 9 = 9 + 6\). Does the order of adding matter?

Solution

  1. 1
    Calculate \(6 + 9\): count on from 9 to 15. \(6 + 9 = 15\).
  2. 2
    Calculate \(9 + 6\): count on from 9 to 15. \(9 + 6 = 15\).
  3. 3
    Both equal 15, so \(6 + 9 = 9 + 6\).
  4. 4
    The order does NOT matter for addition.

Answer

Both equal 15; order does not matter
The commutative property of addition states \(a + b = b + a\). You get the same sum no matter which number you start with.

Example 2

medium
Verify that \(4 \times 7 = 7 \times 4\) using a rectangular array. Explain what changes and what stays the same.

Common Mistakes

  • Assuming subtraction is commutative: writing 3 - 7 = 7 - 3
  • Assuming division is commutative: writing 2 \div 6 = 6 \div 2
  • Thinking commutativity means you can rearrange terms in any expression, ignoring that it only applies to a single operation

Why This Formula Matters

Allows flexibility in calculation order, which simplifies mental math and algebraic manipulation. Knowing which operations are commutative (addition, multiplication) and which are not (subtraction, division) prevents errors in computation.

Frequently Asked Questions

What is the Commutativity formula?

A property where swapping the order of two operands does not change the result: a \ \star\ b = b\ \star\ a.

How do you use the Commutativity formula?

3 + 5 = 5 + 3 and 3 \times 5 = 5 \times 3. Swapping the order doesn't change the answer.

What do the symbols mean in the Commutativity formula?

Commutative law: the order of operands around + or \times may be swapped

Why is the Commutativity formula important in Math?

Allows flexibility in calculation order, which simplifies mental math and algebraic manipulation. Knowing which operations are commutative (addition, multiplication) and which are not (subtraction, division) prevents errors in computation.

What do students get wrong about Commutativity?

Subtraction and division are NOT commutative: 5 - 3 \neq 3 - 5.

What should I learn before the Commutativity formula?

Before studying the Commutativity formula, you should understand: addition, multiplication.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Commutative, Associative, and Distributive Properties →
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