Rules that always work
Arithmetic follows a handful of properties. They’re the reason mental maths shortcuts work, and they become the rules of algebra.
The properties
| Property | Addition | Multiplication |
|---|---|---|
| Commutative property (order) | a + b = b + a | a × b = b × a |
| Associative property (grouping) | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity property | a + 0 = a | a × 1 = a |
| Inverse | a + (−a) = 0 | a × (1/a) = 1 (a ≠ 0) |
| Zero property | — | a × 0 = 0 |
And one that links the two operations:
Distributive property: a × (b + c) = a × b + a × c
Examples with numbers
| Property | Example |
|---|---|
| Commutative | 7 + 5 = 5 + 7 = 12; 4 × 9 = 9 × 4 = 36 |
| Associative | (2 + 8) + 6 = 2 + (8 + 6) = 16 |
| Identity | 13 + 0 = 13; 13 × 1 = 13 |
| Inverse | 6 + (−6) = 0; 6 × 1/6 = 1 |
| Zero | 1,234 × 0 = 0 |
| Distributive | 4 × (10 + 3) = 40 + 12 = 52 |
Subtraction and division don’t play along
Subtraction and division are not commutative or associative:
- 10 − 4 = 6, but 4 − 10 = −6
- (12 ÷ 6) ÷ 2 = 1, but 12 ÷ (6 ÷ 2) = 4
That’s why the order of operations matters.
Using properties for mental maths
| Problem | Property | Quick way |
|---|---|---|
| 25 + 47 + 75 | Commutative + associative | (25 + 75) + 47 = 147 |
| 4 × 17 × 25 | Commutative + associative | (4 × 25) × 17 = 1,700 |
| 8 × 46 | Distributive | 8 × 40 + 8 × 6 = 320 + 48 = 368 |
| 9 × 99 | Distributive | 9 × (100 − 1) = 900 − 9 = 891 |
Properties in algebra
The same rules let you rearrange expressions:
- 3x + 5 + 2x = 3x + 2x + 5 = 5x + 5 (commutative, then combining)
- 2(x + 7) = 2x + 14 (distributive)
Worked examples
Example 1: Which property: 6 × 8 = 8 × 6?
Commutative property of multiplication.
Example 2: Use the distributive property to find 7 × 23.
7 × 20 + 7 × 3 = 140 + 21 = 161.
Example 3: Which property: (3 × 5) × 2 = 3 × (5 × 2)?
Associative property of multiplication.
Common mistakes
- Applying commutative or associative rules to subtraction or division.
- Distributing to only one term: 3(x + 4) ≠ 3x + 4.
- Mixing up commutative (order) and associative (grouping).
Try it yourself
Name the property.
- 9 + 0 = 9
- 5(2 + 6) = 10 + 30
- (7 + 3) + 1 = 7 + (3 + 1)
- Use properties to calculate 50 × 37 × 2 mentally.
- Is 20 ÷ 5 = 5 ÷ 20 true?
Answers: 1) identity 2) distributive 3) associative 4) 3,700 5) No; division isn’t commutative.
Quick recap
- Commutative = order; associative = grouping; both work for + and ×.
- Identity: + 0 and × 1; zero property: × 0 = 0.
- Distributive: a(b + c) = ab + ac.