Lesson 1.9: Properties of Real Numbers

New Concept

Discover the foundational rules of algebra: the properties of real numbers. You'll learn how the commutative, associative, and distributive properties allow you to strategically reorder, regroup, and simplify expressions, making complex calculations much more manageable.

What’s next

Next, you’ll dive into interactive examples and practice cards that break down each property. Master simplifying expressions from the commutative to the distributive property.

Property

of Addition: If $a, b$ are real numbers, then $a+b=b+a$
of Multiplication: If $a, b$ 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

• Adding $11+5$ gives $16$, and $5+11$ also gives $16$. So, $11+5=5+11$.
• Multiplying $4 \cdot 9$ gives $36$, and $9 \cdot 4$ also gives $36$. So, $4 \cdot 9=9 \cdot 4$.
• However, for subtraction, $12-5=7$ while $5-12=-7$. The results are not the same, so subtraction is not commutative.

Explanation

Think 'commute' like moving around! The Commutative Property means you can swap the order of numbers when you add or multiply, and the answer stays the same. It's why $2+5$ is the same as $5+2$.

Property

of Addition: If $a, b, c$ are real numbers, then $(a+b)+c=a+(b+c)$
of Multiplication: If $a, b, c$ 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. This property does not apply to subtraction or division.

Examples

• For addition, $(10+5)+3 = 15+3 = 18$ is the same as $10+(5+3) = 10+8 = 18$.
• For multiplication, $(4 \cdot 2) \cdot 6 = 8 \cdot 6 = 48$ is the same as $4 \cdot (2 \cdot 6) = 4 \cdot 12 = 48$.
• To simplify $(\frac{2}{7} + \frac{3}{10}) + \frac{7}{10}$, it's easier to regroup: $\frac{2}{7} + (\frac{3}{10} + \frac{7}{10}) = \frac{2}{7} + 1 = 1\frac{2}{7}$.

Explanation

Think of 'associating' with friends. You can change which numbers you group together first in a long addition or multiplication problem, and the final result won't change. This trick can make calculations much easier!

Property

Identity Property of addition: For any real number $a$: $a+0=a \quad 0+a=a$. 0 is the additive identity.
Identity Property of multiplication: For any real number $a$: $a \cdot 1=a \quad 1 \cdot a=a$. 1 is the multiplicative identity.
Inverse Property of addition: For any real number $a$, $a+(-a)=0$. $-a$ is the additive inverse of $a$.
Inverse Property of multiplication: For any real number $a, a \neq 0$, $a \cdot \frac{1}{a}=1$. $\frac{1}{a}$ is the multiplicative inverse of $a$.

Examples

• The additive inverse of $21$ is $-21$ because $21 + (-21) = 0$, the additive identity.
• The multiplicative inverse of $5$ is $\frac{1}{5}$ because $5 \cdot \frac{1}{5} = 1$, the multiplicative identity.
• For the number $-\frac{2}{7}$, its additive inverse is $\frac{2}{7}$ and its multiplicative inverse is $-\frac{7}{2}$.

Explanation

The identity properties are about staying the same: add 0 or multiply by 1, and the number's identity doesn't change. Inverse properties are about getting back to the identity: add a number's opposite or multiply by its reciprocal.

Property

Multiplication by Zero: For any real number $a$, $a \cdot 0 = 0$ and $0 \cdot a = 0$. The product of any real number and 0 is 0.
Division of Zero: For any real number $a$, except $0$, $\frac{0}{a}=0$ and $0 \div a = 0$.
Division by Zero: For any real number $a$, $\frac{a}{0}$ and $a \div 0$ are undefined.

Examples

• Any number multiplied by zero equals zero: $150 \cdot 0 = 0$.
• Zero divided by any non-zero number is zero: $\frac{0}{9} = 0$.
• Division by zero is not possible: the expression $\frac{13}{0}$ is undefined.

Explanation

Zero has special powers! Anything multiplied by 0 becomes 0. If you have zero items to share, everyone gets zero. But you can never, ever divide by zero—it's undefined, like a math rule that cannot be broken.

Property

If $a, b, c$ are real numbers, then
$a(b+c) = ab+ac$
$(b+c)a = ba+ca$
$a(b-c) = ab-ac$
$(b-c)a = ba-ca$

Examples

• To simplify $7(x+2)$, distribute the $7$: $7 \cdot x + 7 \cdot 2 = 7x + 14$.
• To simplify $-5(y-4)$, distribute the $-5$: $(-5) \cdot y - (-5) \cdot 4 = -5y + 20$.
• To simplify $9 - 3(x+1)$, first distribute the $-3$: $9 - 3x - 3$. Then combine like terms to get $6 - 3x$.

Explanation

The distributive property lets you 'distribute' or 'pass out' the number outside the parentheses to every term inside. It's the key to removing parentheses and simplifying expressions in algebra. Think of it as sharing the multiplication.