Lesson 5: 1.5 Properties of Real Numbers

New Concept

This lesson covers algebra's foundational rules. You'll use the Commutative, Associative, and Distributive properties to strategically rearrange and simplify expressions, making complex equations easier to solve.

What’s next

Now, let's put these rules into practice. You'll master simplifying expressions through a series of interactive examples, practice cards, and challenge problems.

Property

of Addition: If $a$ and $b$ are real numbers, then $a + b = b + a$.
of Multiplication: If $a$ and $b$ are real numbers, then $a \cdot b = b \cdot a$.
When adding or multiplying, changing the order gives the same result. Subtraction and division are not commutative.

Examples

• To simplify $25x + 8y + 15x + 3y$, use the Commutative Property to group like terms: $25x + 15x + 8y + 3y$, which simplifies to $40x + 11y$.
• To simplify $\frac{4}{11} \cdot \frac{9}{10} \cdot \frac{11}{4}$, reorder the factors to $\frac{4}{11} \cdot \frac{11}{4} \cdot \frac{9}{10}$. This becomes $1 \cdot \frac{9}{10} = \frac{9}{10}$.
• Adding $12 + 7$ gives $19$, and adding $7 + 12$ also gives $19$. Similarly, multiplying $6 \cdot 5$ gives $30$, and $5 \cdot 6$ also gives $30$.

Explanation

Think 'commute,' like traveling. The order you add or multiply numbers doesn't change the final destination (the answer). This rule only works for addition and multiplication, not for subtraction or division, so be careful with those operations!

Property

of Addition: If $a$, $b$, and $c$ are real numbers, then $(a + b) + c = a + (b + c)$.
of Multiplication: If $a$, $b$, and $c$ are real numbers, then $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
When adding or multiplying, changing the grouping gives the same result. Subtraction and division are not associative.

Examples

• To simplify $(\frac{7}{15} + \frac{2}{9}) + \frac{7}{9}$, change the grouping to $\frac{7}{15} + (\frac{2}{9} + \frac{7}{9})$. This simplifies to $\frac{7}{15} + 1 = \frac{22}{15}$.
• To simplify $(15 \cdot \frac{1}{4}) \cdot 4$, regroup the factors to $15 \cdot (\frac{1}{4} \cdot 4)$. This becomes $15 \cdot 1 = 15$.
• Calculating $(4 + 6) + 3$ gives $10 + 3 = 13$. Regrouping as $4 + (6 + 3)$ gives $4 + 9 = 13$. The result is the same.

Explanation

Think 'associate,' like changing the group of friends you talk to first. The final result of the operation is the same. It's all about regrouping with parentheses, but this property only applies to addition and multiplication.

Property

Additive Inverse: For any real number $a$, there is an opposite, $-a$, such that their sum is zero.

Multiplicative Inverse: For any non-zero real number $a$, there is a reciprocal, $\frac{1}{a}$, such that their product is one.

These properties are essential for solving equations and simplifying expressions.

Examples

• To simplify $52y + (-15z) + (-52y)$, reorder to group the opposites: $52y + (-52y) + (-15z)$. This becomes $0 + (-15z) = -15z$.
• The multiplicative inverse of $8$ is $\frac{1}{8}$ because $8 \cdot \frac{1}{8} = 1$.
• The additive inverse of $-15$ is $15$ because $-15 + 15 = 0$.

Explanation

Inverses are like 'undo' buttons in math. The additive inverse (the opposite) brings you back to $0$. The multiplicative inverse (the reciprocal) brings you back to $1$. Use them to cancel terms and simplify your work.

Property

Multiplication by Zero: The product of any real number and $0$ is $0$.

Division involving Zero: Division by $0$ is undefined. For any real number $a$, $\frac{a}{0}$ is undefined. Zero divided by any non-zero real number is $0$. For any real number $a \neq 0$, $\frac{0}{a} = 0$.

Examples

• The expression $\frac{0}{x+10}$, where $x \neq -10$, simplifies to $0$ because zero is divided by a non-zero number.
• The expression $\frac{15-2y}{0}$ is undefined because division by zero is not allowed.
• The product $25 \cdot (9 - 9)$ simplifies to $25 \cdot 0$, which equals $0$.

Explanation

Zero has unique rules. Multiplying any number by zero always gives zero. But you can never divide by zero—it's an impossible operation in math. If zero is on top of a fraction (and the bottom is not zero), the answer is just $0$.

Property

If $a$, $b$, and $c$ are real numbers, then:

This property allows you to multiply a sum by multiplying each addend separately and then adding the products. You 'distribute' the factor outside the parentheses to each term inside.

Examples

• To simplify $6(x + 7)$, distribute the $6$ to both $x$ and $7$: $6 \cdot x + 6 \cdot 7$, which equals $6x + 42$.
• To simplify $-5(2y - 3)$, distribute the $-5$: $(-5) \cdot 2y - (-5) \cdot 3$, which simplifies to $-10y - (-15)$ or $-10y + 15$.
• To simplify $5(x - 4) - (x + 1)$, distribute first to get $5x - 20 - x - 1$. Then combine like terms to get $4x - 21$.

Explanation

The distributive property lets you 'share' the number outside the parentheses with every term inside. This is a key tool for removing parentheses, which allows you to combine like terms and simplify expressions. Always be careful with negative signs!