Summary: Properties of Identity, Inverses, and Zero

• Multiplying and Dividing Involving Zero

Key Concepts

• Identity Properties
  • Identity Property of Addition: For any real number [latex]a[/latex]: [latex]a+0=a(0)+a=a[/latex] [latex]0[/latex] is the additive identity
  • Identity Property of Multiplication: For any real number [latex]a[/latex]:
    • [latex]a\cdot 1=a[/latex]
    • [latex]1\cdot a=a[/latex]
    • [latex]1[/latex] is the multiplicative identity
• Inverse Properties
  • Inverse Property of Addition: For any real number a: [latex]a+\left(-a\right)=0-a[/latex] is the additive inverse of a
  • Inverse Property of Multiplication: For any real number a: [latex]\left(a\ne 0\right)a\cdot \frac{1}{a}=1\frac{1}{a}[/latex] is the multiplicative inverse of a
• Properties of Zero
  • Multiplication by Zero: For any real number a, [latex]\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}[/latex]
  • Division of Zero: For any real number a, [latex]\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}[/latex]
  • Division by Zero: For any real number a, [latex]\frac{0}{a}[/latex] is undefined and [latex]a\div 0[/latex] is undefined. Division by zero is undefined.

Glossary

Additive Identity

The additive identity is [latex]0[/latex]. 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 [latex]-a[/latex] .

Multiplicative Identity

The multiplicative identity is [latex]1[/latex]. 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 [latex]\frac{1}{a}[/latex] .

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