Properties of Real Numbers

Why Do These Matter?

Every time you simplify an expression or solve an equation, you’re using certain rules.

These rules aren’t arbitrary. They’re the properties that make algebra work.

Closure

When you add or multiply two real numbers, you always get a real number.

You can’t “escape” the real numbers through addition or multiplication. The result is always real.

Commutative Property

Order doesn’t matter.

$a + b = b + a$ $a \times b = b \times a$

Commute means to travel. The numbers can swap places.

Associative Property

Grouping doesn’t matter.

$(a + b) + c = a + (b + c)$ $(a \times b) \times c = a \times (b \times c)$

Associate means to group. You can regroup however you like.

Identity Property

There’s a “do nothing” number for each operation.

• Additive identity: $a + 0 = a$
• Multiplicative identity: $a \times 1 = a$

Adding zero or multiplying by one leaves the number unchanged.

Inverse Property

Every number has an “undo” number.

• Additive inverse: $a + (-a) = 0$
• Multiplicative inverse: $a \times \frac{1}{a} = 1$ (where $a \neq 0$)

The inverse brings you back to the identity.

Distributive Property

Multiplication distributes over addition.

$a \times (b + c) = (a \times b) + (a \times c)$

This is the bridge between multiplication and addition. It’s why you can expand brackets.

Using These Properties

Why can you rewrite $2x + 3x$ as $5x$?

Distributive property: $(2 + 3) \times x = 5x$

Why can you rearrange $3 + x + 7$ to $x + 10$?

Commutative: $3 + x + 7 = x + 3 + 7$ Associative: $x + (3 + 7) = x + 10$

Every algebraic manipulation uses these six properties.