Math Core

You already know that $3 + 5$ and $5 + 3$ give the same answer. Facts like this are called properties. They're the rules that let you rearrange and simplify expressions with confidence, and every step you take in algebra is justified by one of them.

Commutative properties: order

Commute means to travel back and forth. The commutative properties say you can swap the order of two numbers when adding or multiplying.

a + b = b + a \qquad\qquad a \cdot b = b \cdot a

For example, $7 + (-2) = (-2) + 7$ and $4 \cdot x = x \cdot 4$ . That's why we can write $x \cdot 4$ as $4x$ .

Associative properties: grouping

The associative properties say you can change the grouping (where the parentheses go) when adding or multiplying three numbers. The order stays the same; only the partners change.

(a + b) + c = a + (b + c) \qquad\qquad (a \cdot b) \cdot c = a \cdot (b \cdot c)

For example, $(6 + 9) + 1 = 6 + (9 + 1)$ , and $3(5x) = (3 \cdot 5)x = 15x$ .

Tip

To tell them apart, look at the order of the numbers. Order changed means commutative. Same order, parentheses moved means associative.

Identity properties

An identity is a number that leaves others unchanged.

Additive identity: $a + 0 = a$ . Adding $0$ changes nothing.
Multiplicative identity: $a \cdot 1 = a$ . Multiplying by $1$ changes nothing.

Inverse properties

An inverse undoes a number, bringing you back to the identity.

Additive inverse: $a + (-a) = 0$ . The additive inverse of a number is its opposite. The opposite of $8$ is $-8$ , and the opposite of $-\tfrac{2}{3}$ is $\tfrac{2}{3}$ .
Multiplicative inverse: $a \cdot \dfrac{1}{a} = 1$ for $a \ne 0$ . The multiplicative inverse is the reciprocal. The reciprocal of $5$ is $\tfrac{1}{5}$ , and the reciprocal of $\tfrac{3}{4}$ is $\tfrac{4}{3}$ .

Zero has no reciprocal, because no number times $0$ equals $1$ .

Zero property of multiplication

Any number times zero is zero: $a \cdot 0 = 0$ .

Properties of real numbers

For all real numbers $a$ , $b$ and $c$ :

property	addition	multiplication
commutative	$a + b = b + a$	$ab = ba$
associative	$(a + b) + c = a + (b + c)$	$(ab)c = a(bc)$
identity	$a + 0 = a$	$a \cdot 1 = a$
inverse	$a + (-a) = 0$	$a \cdot \dfrac{1}{a} = 1 \ (a \ne 0)$

Also: the zero property, $a \cdot 0 = 0$ , and the distributive property, $a(b + c) = ab + ac$ , which gets its own lesson next.

Common mistake

Subtraction and division are not commutative or associative. $9 - 4 = 5$ but $4 - 9 = -5$ . And $(12 \div 6) \div 2 = 1$ while $12 \div (6 \div 2) = 4$ . If you want to rearrange a subtraction, first rewrite it as adding the opposite: $9 - 4 = 9 + (-4) = (-4) + 9$ .

Worked example: Naming the property

Name the property shown.

$(-3) + 3 = 0$
$m \cdot (n \cdot 2) = (m \cdot n) \cdot 2$
$y + 5 = 5 + y$
$1 \cdot \dfrac{7}{9} = \dfrac{7}{9}$

Solutions.

A number plus its opposite is $0$ : additive inverse.
Same order, new grouping: associative property of multiplication.
Order swapped: commutative property of addition.
Multiplying by $1$ changes nothing: multiplicative identity.

Using properties for mental math

The real payoff is choosing a convenient order and grouping.

Worked example: Making friendly numbers

Compute $28 + 49 + 72$ and $4 \cdot 17 \cdot 25$ in your head.

For the sum, swap the last two numbers (commutative) and group $28$ with $72$ (associative):

28 + 49 + 72 = (28 + 72) + 49 = 100 + 49 = 149.

For the product, rearrange so $4$ and $25$ are partners:

4 \cdot 17 \cdot 25 = (4 \cdot 25) \cdot 17 = 100 \cdot 17 = 1700.

Worked example: Justifying each step

Simplify $6 \cdot (x \cdot 5)$ and give a reason for each step.

\begin{aligned} 6 \cdot (x \cdot 5) &= 6 \cdot (5 \cdot x) && \text{commutative property of multiplication} \\ &= (6 \cdot 5) \cdot x && \text{associative property of multiplication} \\ &= 30x && \text{multiply} \end{aligned}

Practice

Practice 1

Which property is shown by $8 \cdot w = w \cdot 8$ ?

Practice 2

Which property is shown by $(a + 4) + 9 = a + (4 + 9)$ ?

Practice 3

What is the additive inverse of $-\dfrac{11}{2}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What is the multiplicative inverse (reciprocal) of $-\dfrac{8}{3}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Which equation is false for some real numbers?

Practice 6

Use the properties to compute $36 + 147 + 53$ in your head.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Use the properties to compute $25 \cdot 13 \cdot 8$ in your head.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

In the steps below, which property justifies Step 2?

\begin{aligned} (7 + x) + (-7) &= (x + 7) + (-7) && \text{Step 1} \\ &= x + (7 + (-7)) && \text{Step 2} \\ &= x + 0 && \text{Step 3} \\ &= x && \text{Step 4} \end{aligned}