Lesson 4: Properties of Identity, Inverses, and Zero

New Concept

Learn foundational algebra rules! Discover how identity numbers ($0$ and $1$), inverses (opposites and reciprocals), and the properties of zero are powerful tools for simplifying expressions and solving equations.

What’s next

Next, you'll see these properties in action through worked examples and apply them yourself in a series of interactive practice cards and challenge problems.

Property

The identity property of addition: for any real number $a$,

$0$ is called the additive identity.

The identity property of multiplication: for any real number $a$,

$1$ is called the multiplicative identity.

Examples

• The expression $42 + 0$ simplifies to $42$ by the identity property of addition.
• Using the identity property of multiplication, the expression $1 \cdot (-5y)$ simplifies to $-5y$.
• When simplifying $0 + (x+y)$, the identity property of addition shows the result is just $x+y$.

Property

Inverse Property of Addition for any real number $a$,

$-a$ is the additive inverse of $a$.

Inverse Property of Multiplication for any real number $a \neq 0$,

$\frac{1}{a}$ is the multiplicative inverse of $a$.

Examples

• The additive inverse of $25$ is $-25$, because their sum is $25 + (-25) = 0$.
• The multiplicative inverse of $-\frac{3}{7}$ is $-\frac{7}{3}$, because their product is $(-\frac{3}{7}) \cdot (-\frac{7}{3}) = 1$.
• To find the multiplicative inverse of $0.5$, we first write it as a fraction, $\frac{1}{2}$. The inverse is its reciprocal, $2$.

Property

Multiplication by Zero: For any real number $a$,

The product of any number and 0 is 0.

Examples

• The simplification of $-15 \cdot 0$ is $0$, as the product of any number and zero is zero.
• In the expression $0 \cdot (3x - 2)$, the result is $0$ because the entire quantity is multiplied by zero.
• No matter the value of $y$, the expression $y \cdot 0$ will always equal $0$.

Explanation

Think of it this way: if you have 8 groups of 0 items, you have a total of 0 items. Any quantity multiplied by zero results in zero, because you are taking none of that quantity.

Property

Division of Zero: For any real number $a$, $a \neq 0$

Zero divided by any real number, except itself, is zero.

Division by Zero: For any real number $a$,

Division by zero is undefined.

Examples

• The expression $\frac{0}{15}$ simplifies to $0$, as zero divided by any non-zero number is zero.
• The expression $\frac{-8}{0}$ is undefined, because division by zero is not a defined operation in mathematics.
• For any non-zero value of $k$, the expression $0 \div k$ will always equal $0$.