Mathematics » Properties of Real Numbers » Properties of Identity, Inverses, and Zero

# Key Concepts

## Key Concepts

• Identity Properties
• Identity Property of Addition:For any real number a: $$a+0=a\phantom{\rule{1.5em}{0ex}}0+a=a\phantom{\rule{1.5em}{0ex}}$$ 0 is the additive identity
• Identity Property of Multiplication:For any real number a: $$a\cdot 1=a\phantom{\rule{1.5em}{0ex}}1\cdot a=a\phantom{\rule{1.5em}{0ex}}$$ 1 is the multiplicative identity
• Inverse Properties
• Inverse Property of Addition: For any real number a: $$a+\left(-a\right)=0\phantom{\rule{1.5em}{0ex}}-a$$ is the additive inverse of a
• Inverse Property of Multiplication: For any real number a: $$\left(a\ne 0\right)\phantom{\rule{1.5em}{0ex}}a\cdot \frac{1}{a}=1\phantom{\rule{1.5em}{0ex}}\frac{1}{a}$$ is the multiplicative inverse of a
• Properties of Zero
• Multiplication by Zero: For any real number a,
$$\begin{array}{} a⋅0=0 & 0⋅a=0 & \text{The product of any number and 0 is 0.}\end{array}$$
• Division of Zero: For any real number a,
$$\begin{array}{} \frac{0}{a}=0 & 0+a=0 & \text{Zero divided by any real number, except itself, is zero.}\end{array}$$
• Division by Zero: For any real number a, $$\frac{0}{a}$$ is undefined and $$a÷0$$ is undefined. Division by zero is undefined.

## Glossary

The additive identity is 0. When zero is added to any number, it does not change the value.

The opposite of a number is its additive inverse. The additive inverse of a is $$-a$$.

### Multiplicative Identity

The multiplicative identity is 1. When one multiplies any number, it does not change the value.

### Multiplicative Inverse

The reciprocal of a number is its multiplicative inverse. The multiplicative inverse of a is $$\frac{1}{a}$$.