Properties of Real Numbers

By the end of this lesson, you will be able to name the five properties that justify algebraic moves, use the distributive property to expand and to factor, and say which property makes each step of a simplification legal.

The Rules That Say Which Moves Are Legal

When you rearrange terms, regroup factors, or multiply out a bracket, you are not doing whatever looks convenient. Each of those moves is allowed by a property: a statement about numbers that is true for every number there is.

Think of them as the grammar of algebra. Grammar is what makes a sentence legal rather than just familiar, and these five properties are what make an algebraic step legal rather than just plausible.

$$8 - 3 \text{ can be rewritten, } 3 - 8 \text{ cannot be swapped in}$$

Knowing which is which is the difference between simplifying an expression and quietly changing it into a different one.

Order and Grouping

Let $a$, $b$ and $c$ stand for any real numbers.

Commutative: order does not matter

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

$$7 + 3 = 3 + 7 = 10$$

$$4 \times 5 = 5 \times 4 = 20$$

Only for addition and multiplication. Subtraction and division depend on the order:

$$8 - 3 = 5 \qquad 3 - 8 = -5$$

Associative: grouping does not matter

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

$$(2 + 3) + 4 = 2 + (3 + 4) = 9$$

This is why a long addition can be done in whatever order is easiest, and why you never have to write brackets in $2 + 3 + 4$.

Again, subtraction is not included:

$$(8 - 3) - 2 = 3 \qquad 8 - (3 - 2) = 7$$

The two are different moves. Commutative changes the order of things. Associative changes the grouping of things. Mixing the names up is the most common slip in this topic.

Distributive: the one you will use most

$$a(b + c) = ab + ac \qquad a(b - c) = ab - ac$$

The factor outside multiplies every term inside, not just the first one.

$$3(x + 5) = 3x + 15$$

$$-2(4y - 3) = -8y + 6$$

It is the only property here that connects multiplication to addition, which is why nearly every simplification uses it somewhere.

It runs both ways. Left to right it expands. Right to left it factors.

$$5(2x - 3) = 10x - 15 \qquad \text{expanding}$$

$$10x - 15 = 5(2x - 3) \qquad \text{factoring}$$

When a negative is out front, every sign inside flips:

By the end of this lesson, you will be able to name the five properties that justify algebraic moves, use the distributive property to expand and to factor, and say which property makes each step of a simplification legal.

The Rules That Say Which Moves Are Legal

Think of them as the grammar of algebra. Grammar is what makes a sentence legal rather than just familiar, and these five properties are what make an algebraic step legal rather than just plausible.

$$8 - 3 \text{ can be rewritten, } 3 - 8 \text{ cannot be swapped in}$$

Knowing which is which is the difference between simplifying an expression and quietly changing it into a different one.

Order and Grouping

Let $a$, $b$ and $c$ stand for any real numbers.

Commutative: order does not matter

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

$$7 + 3 = 3 + 7 = 10$$

$$4 \times 5 = 5 \times 4 = 20$$

Only for addition and multiplication. Subtraction and division depend on the order:

$$8 - 3 = 5 \qquad 3 - 8 = -5$$

Associative: grouping does not matter

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

$$(2 + 3) + 4 = 2 + (3 + 4) = 9$$

This is why a long addition can be done in whatever order is easiest, and why you never have to write brackets in $2 + 3 + 4$.

Again, subtraction is not included:

$$(8 - 3) - 2 = 3 \qquad 8 - (3 - 2) = 7$$

The two are different moves. Commutative changes the order of things. Associative changes the grouping of things. Mixing the names up is the most common slip in this topic.

Distributive: the one you will use most

$$a(b + c) = ab + ac \qquad a(b - c) = ab - ac$$

The factor outside multiplies every term inside, not just the first one.

$$3(x + 5) = 3x + 15$$

$$-2(4y - 3) = -8y + 6$$

It is the only property here that connects multiplication to addition, which is why nearly every simplification uses it somewhere.

It runs both ways. Left to right it expands. Right to left it factors.

$$5(2x - 3) = 10x - 15 \qquad \text{expanding}$$

$$10x - 15 = 5(2x - 3) \qquad \text{factoring}$$

When a negative is out front, every sign inside flips:

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	$a + (-a) = 0$	$a \cdot \dfrac{1}{a} = 1$, $a \neq 0$
Distributive	$a(b + c) = ab + ac$	the same statement

The Rules That Say Which Moves Are Legal

Order and Grouping

Commutative: order does not matter

Associative: grouping does not matter

Distributive: the one you will use most

Properties of Real Numbers

The Rules That Say Which Moves Are Legal

Order and Grouping

Commutative: order does not matter

Associative: grouping does not matter

Distributive: the one you will use most

Identity and Inverse

Identity: the numbers that change nothing

Inverse: the numbers that undo

Example 1: Multiplying Out a Bracket

Worked Example 1: A Negative Outside

Example 2: A Bracket in the Middle of an Expression

Worked Example 2: The Same Shape, Bigger Numbers

Example 3: Running It Backwards to Factor

Worked Example 3: Factor Out the Common Number

Worked Example 4: Two Brackets, One Expression

Example 4: Naming the Move at Every Step

Common Mistakes

Where This Goes Next

Check Your Understanding

Summary