Math Core

Why is $4 \cdot 37 \cdot 25$ easy to do in your head? Because you're allowed to multiply in any order, so you can do $4 \cdot 25 = 100$ first. The properties of operations are the rules that say which rearrangements are allowed. They're also the reasons behind every step you'll take when you simplify expressions and solve equations.

Order and grouping

The commutative properties say you can swap the order of two numbers when you add or multiply:

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

The associative properties say you can change the grouping (where the parentheses go) when you add or multiply three numbers:

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

A quick way to tell them apart: commutative is about commuting, moving around, so the order changes. Associative is about who you associate with, so the parentheses move but the order stays the same.

Worked example: Mental math with the properties

Find $38 + 45 + 62$ and $5 \cdot (-17) \cdot 2$ without a calculator.

Swap and regroup so the friendly pairs go first: $38 + 62 + 45 = 100 + 45 = 145$ .
Multiply $5 \cdot 2 = 10$ first: $10 \cdot (-17) = -170$ .

Identities and inverses

Identity and inverse properties

property	addition	multiplication
identity	$a + 0 = a$	$a \cdot 1 = a$
inverse	$a + (-a) = 0$	$a \cdot \dfrac{1}{a} = 1$ (for $a \ne 0$ )

The additive inverse of a number is its opposite: the additive inverse of $-9$ is $9$ . The multiplicative inverse, or reciprocal, is what you multiply by to get $1$ : the reciprocal of $\dfrac{3}{5}$ is $\dfrac{5}{3}$ . Zero has no reciprocal, because nothing times $0$ equals $1$ .

One more fact is worth naming. The multiplication property of zero says $a \cdot 0 = 0$ for every number $a$ .

Common mistake

Subtraction and division are not commutative or associative. For example, $10 - 4 = 6$ but $4 - 10 = -6$ . And $(12 \div 6) \div 2 = 1$ , but $12 \div (6 \div 2) = 4$ .

If you want to reorder a subtraction, rewrite it as adding the opposite first: $10 - 4 = 10 + (-4) = -4 + 10$ . Now the $-4$ carries its sign with it.

Using properties with variables

These properties work for variables too, because variables stand for numbers. That lets you rewrite expressions without changing their value.

Worked example: Simplifying a product

Simplify $3 \cdot (8x)$ .

\begin{aligned} 3 \cdot (8x) &= 3 \cdot (8 \cdot x) && \\ &= (3 \cdot 8) \cdot x && \text{associative property of multiplication} \\ &= 24x \end{aligned}

Worked example: Justifying each step

Simplify $(n + 7) + (-7)$ and name the property used in each step.

\begin{aligned} (n + 7) + (-7) &= n + (7 + (-7)) && \text{associative property of addition} \\ &= n + 0 && \text{additive inverse} \\ &= n && \text{additive identity} \end{aligned}

This is exactly the reasoning behind "subtract $7$ from both sides" when you solve an equation like $n + 7 = 20$ .

Worked example: Rearranging a sum with variables

Simplify $6 + 5y + 9$ .

Use the commutative property to swap $5y$ and $9$ : $6 + 9 + 5y$ . Then add: $15 + 5y$ . It's standard to write the variable term first, so the answer is $5y + 15$ (another use of the commutative property).

Tip

The distributive property, $a(b + c) = ab + ac$ , is the one property that connects multiplication and addition. It gets its own lesson later in this unit.

Practice

Practice 1

Which property does $9 + (-4) = -4 + 9$ show?

Practice 2

Which property does $(2 \cdot 5) \cdot x = 2 \cdot (5 \cdot x)$ show?

Practice 3

What is the additive inverse of $-13$ ?

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

Practice 4

What is the multiplicative inverse of $\dfrac{2}{7}$ ?

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

Practice 5

Use the properties to find $4 \cdot 17 \cdot 25$ in your head.

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

Practice 6

Find $36 + 19 + 64 + 81$ using the properties of addition.

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

Practice 7

Which statement is not true for all numbers $a$ and $b$ ?

Practice 8

Simplify $\dfrac{1}{4} \cdot (4m)$ and choose the list of properties used, in order.

\frac{1}{4} \cdot (4m) = \left(\frac{1}{4} \cdot 4\right) m = 1 \cdot m = m