1.9 Properties of Real Numbers

Real numbers follow a set of properties that govern how addition, multiplication, and other operations behave. These properties let you rearrange terms, simplify expressions, and solve equations with confidence. They're the foundation for everything else in algebra.

Properties of Real Numbers

Rearranging algebraic terms

The commutative property says you can swap the order of terms when adding or multiplying, and the result stays the same.

• Addition: $a + b = b + a$ (so $3 + 5 = 5 + 3$)
• Multiplication: $a \times b = b \times a$ (so $2 \times 7 = 7 \times 2$)

The associative property says you can regroup terms when adding or multiplying without changing the outcome. The numbers stay in the same order; only the parentheses move.

• Addition: $(a + b) + c = a + (b + c)$ (so $(2 + 3) + 4 = 2 + (3 + 4)$)
• Multiplication: $(a \times b) \times c = a \times (b \times c)$ (so $(2 \times 3) \times 4 = 2 \times (3 \times 4)$)

These two properties work together when you simplify expressions. For example:

$3x + 2y + 5x$

You can rearrange (commutative) and regroup (associative) to combine like terms:

$(3x + 5x) + 2y = 8x + 2y$

Note that neither property applies to subtraction or division. $5 - 3 \neq 3 - 5$, and $12 \div 4 \neq 4 \div 12$.

The closure property means that adding or multiplying any two real numbers always produces another real number. You never "leave" the real numbers through these operations.

Identity and inverse properties

The identity property says there's a special number that leaves any value unchanged under an operation.

• Additive identity: $a + 0 = a$ (so $7 + 0 = 7$)
• Multiplicative identity: $a \times 1 = a$ (so $4 \times 1 = 4$)

The inverse property says every real number has a partner that brings it back to the identity element.

• Additive inverse: $a + (-a) = 0$ (so $5 + (-5) = 0$)
• Multiplicative inverse: $a \times \frac{1}{a} = 1$ for $a \neq 0$ (so $6 \times \frac{1}{6} = 1$)

These properties are exactly how you solve equations. Here's the logic:

1. Solving with the additive inverse: $x + 5 = 10 \rightarrow x + 5 + (-5) = 10 + (-5) \rightarrow x = 5$
2. Solving with the multiplicative inverse: $3x = 12 \rightarrow 3x \times \frac{1}{3} = 12 \times \frac{1}{3} \rightarrow x = 4$

Every time you "subtract from both sides" or "divide both sides," you're using an inverse property.

Zero's unique mathematical properties

Zero behaves differently from every other real number. It's worth knowing each of its special rules:

• Adding zero gives back the original number: $a + 0 = a$ (so $10 + 0 = 10$). This is the identity property of addition.
• Multiplying by zero always gives zero: $a \times 0 = 0$ (so $-4 \times 0 = 0$). No matter how large or small the other number is, the product is 0.
• Dividing by zero is undefined. The expression $\frac{5}{0}$ has no answer because no number multiplied by 0 gives 5. If you see it on a test, the answer is "undefined."
• Zero as an exponent equals one for any non-zero base: $a^0 = 1$ for $a \neq 0$ (so $2^0 = 1$).

Distributive property in algebra

The distributive property connects multiplication and addition. It says that multiplying a factor by a sum is the same as multiplying the factor by each term separately, then adding.

$a(b + c) = ab + ac$

For example: $2(3 + 4) = 2 \times 3 + 2 \times 4 = 6 + 8 = 14$

Expanding expressions means applying the distributive property to remove parentheses:

$2(3x + 4) = 2 \times 3x + 2 \times 4 = 6x + 8$

Simplifying means expanding first, then combining like terms:

$3(2x + 1) + 4(x - 3) = 6x + 3 + 4x - 12 = 10x - 9$

Factoring is the reverse of distributing. You pull out the greatest common factor:

$5x + 10 = 5(x + 2)$

This works because if you distribute the 5 back in, you get $5x + 10$ again.

Additional Properties of Real Numbers

These properties come up less often in elementary algebra, but they round out the picture of how real numbers work:

• The density property states that between any two real numbers, there's always another real number. You can never find two real numbers that are "right next to each other" with nothing in between.
• The trichotomy property states that for any two real numbers $a$ and $b$, exactly one of these is true: $a < b$, $a = b$, or $a > b$. There's no fourth option.
• Together, the commutative, associative, distributive, identity, and inverse properties form what mathematicians call the field axioms. These are the core rules that make arithmetic with real numbers consistent and predictable.