Properties of Algebraic Operations

The properties of algebraic operations explain when we can rearrange, group or expand expressions without changing their value. They are the reasons a line of algebra follows from the one before it.

Here we work with real numbers. The central properties are commutativity, associativity, distributivity, identities and inverses. These properties justify different changes to an expression.

On this page (6 sections)

Changing order or grouping

Commutativity lets us change the order of addition or multiplication: $a+b=b+a$ and $ab=ba$ . For example, $3x+5x$ can be rearranged as $5x+3x$ without changing its value.

Associativity lets us change grouping: $(a+b)+c=a+(b+c)$ and $(ab)c=a(bc)$ . The order stays the same; the parentheses move. Thus $(2\times5)\times7=2\times(5\times7)=70$ .

Commutative property of multiplication

The arrangements differ; the product remains six.

Diagram explained

The first array has $2$ rows and $3$ columns. The second has $3$ rows and $2$ columns. Each contains $6$ squares. This illustrates commutativity of multiplication, not associativity.

Subtraction and division are neither commutative nor associative. For example, $8-3=5$ but $3-8=-5$ . Also, $(12\div3)\div2=2$ , whereas $12\div(3\div2)=8$ . Do not apply an addition or multiplication property to a different operation.

Distributing multiplication

The distributive property connects multiplication and addition:

a(b+c)=ab+ac

Every term inside the bracket receives the factor. For example, $3(x+4)=3x+12$ , not $3x+4$ . With subtraction, the same reasoning gives $3(x-4)=3x-12$ .

Distributive property shown by two areas

For positive $x$ , the areas add to $3x+12$ .

Diagram explained

The whole rectangle has height $3$ and width $x+4$ . Splitting the width produces areas $3x$ and $12$ . Thus $3(x+4)=3x+12$ . The algebraic identity also holds when $x$ is not a positive length.

Reading distribution backwards gives factorisation: $6x+9=3(2x+3)$ . It also explains combining like terms, since $2x+5x=(2+5)x=7x$ .

These relationships are standard properties of the real numbers. A useful check is to ask which term received each multiplier, especially when that multiplier is negative.

Leaving a value unchanged, or undoing it

An identity element leaves the value unchanged. For addition it is zero: $a+0=a$ . For multiplication it is one: $a\cdot1=a$ .

An inverse combines with a value to give the relevant identity element. The additive inverse of $a$ is $-a$ , because $a+(-a)=0$ . For $a\ne0$ , the multiplicative inverse is $\frac1a$ , because $a\cdot\frac1a=1$ .

Zero has no multiplicative inverse. Also, multiplying by zero does not leave a value unchanged: $a\cdot0=0$ . These small distinctions explain why adding zero, multiplying by one and multiplying by zero have different effects.

Preserving equality

A related rule lets us perform the same addition or subtraction on both sides of an equation. Multiplying or dividing both sides by the same non-zero number preserves an equivalent equation too.

For $2x+3=11$ , subtracting $3$ gives $2x=8$ ; dividing by $2$ gives $x=4$ . Check in the original: $2(4)+3=11$ .

The real numbers are closed under addition, subtraction and multiplication: combining real inputs with those operations gives a real result. Division also gives a real result when the divisor is non-zero. The number system matters; integers, for example, are not closed under division because $1\div2=\frac12$ is not an integer.

Use the property that matches the change you want to make. To explore larger rewrites built from these rules, continue to Algebraic Identities.

Practice questions

Which property changes $4+9$ to $9+4$ ?
Expand $-2(x-3)$ .
What is the multiplicative inverse of $5$ ?

Check your answers

The commutative property of addition. It changes the order.
$-2x+6$ . The factor $-2$ multiplies both $x$ and $-3$ .
$\frac15$ , because $5\cdot\frac15=1$ .

Sources and further reading

OpenStax, Elementary Algebra 2e, §1.9: Properties of Real Numbers. Reference for commutative, associative, distributive, identity and inverse properties. The examples and diagrams in this article were developed for CTS.