Properties of Real Numbers

The four properties

Commutative property: you can change the order. $a + b = b + a$ and $ab = ba$. It works for addition and multiplication — not for subtraction or division.

Associative property: you can change the grouping. $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$. The numbers stay in the same order; only the parentheses move.

Identity property: adding $0$ or multiplying by $1$ changes nothing. $a + 0 = a$ and $a \cdot 1 = a$.

Inverse property: every number has a partner that brings it back to the identity. $a + (-a) = 0$, and for any nonzero $a$, $a \cdot \dfrac{1}{a} = 1$.

Commutative vs. associative: what moved?

These two get confused more than any other pair, and one question settles it: did the numbers change order, or did the parentheses change position?

In $4 \cdot 9 = 9 \cdot 4$ the factors swapped places — commutative. In $(2 + 5) + 8 = 2 + (5 + 8)$ the numbers sit in the same order and only the grouping moved — associative. If both the order and the grouping changed, the equation is using both properties at once.

Using the properties for mental math

Take $4 \cdot 13 \cdot 25$. Multiplied left to right it is clumsy: $52 \cdot 25$. But the commutative property lets you reorder to $4 \cdot 25 \cdot 13$, and the associative property lets you group $(4 \cdot 25) \cdot 13 = 100 \cdot 13 = 1300$. Hunting for pairs that make $10$, $100$, or $0$ is the whole trick.

Worked examples

Try one yourself

Common questions