Summary: Properties of Identity, Inverses, and Zero

• Multiplying and Dividing Involving Zero

Key Concepts

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

Glossary

Additive Identity

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

Additive Inverse

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 $\frac{1}{a}$ .

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