The associative property lets you regroup terms to make mental arithmetic easier, for example 17 + 25 + 75 = 17 + (25 + 75) = 117. It does not hold for subtraction or division – the calculator shows that too.
How to use the Associative Property Calculator
- Operation
- a
- b
- c
Results update as you type, and everything is calculated in your browser.
How it works
Both groupings are evaluated separately so you can see they agree. For comparison, the same grouping test is applied to subtraction or division, where the results usually differ.
Inputs explained
| Field | What to enter |
|---|---|
| Operation | Choose one: Addition, Multiplication |
| a | A number |
| b | A number |
| c | A number |
Example values
How the main result changes when a changes and everything else stays at the example values above:
| a | (a ∘ b) ∘ c |
|---|---|
| 8.5 | 108.5 |
| 12.75 | 112.75 |
| 17 | 117 |
| 21.25 | 121.25 |
| 25.5 | 125.5 |
| 34 | 134 |
Formula
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
Worked example
Inputs — Operation: Addition; a: 17; b: 25; c: 75.
Results — (a ∘ b) ∘ c = 117; a ∘ (b ∘ c) = 117; Non-associative check: (a − b) − c or (a ÷ b) ÷ c = -83; versus a − (b − c) or a ÷ (b ÷ c) = 67.
Key terms
- Associative property
- Regrouping does not change the result.
Frequently asked questions
Which operations are associative?
Addition and multiplication. Subtraction, division and exponentiation are not.
How is it different from the commutative property?
Associativity is about grouping (brackets); commutativity is about order (a + b = b + a).
Why is it useful?
It lets you add or multiply in whatever grouping is easiest, such as pairing numbers that make 10 or 100.
Do computers always obey it?
Not exactly with floating-point numbers – rounding can make (a + b) + c differ slightly from a + (b + c).