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 [latex]a[/latex]: [latex]a+\left(-a\right)=0-a[/latex] is the additive inverse of [latex]a[/latex]
  • Inverse Property of Multiplication: For any real number [latex]a[/latex]: [latex]\left(a\ne 0\right)a\cdot {\Large\frac{1}{a}}=1[/latex].   The multiplicative inverse of [latex]a[/latex] is [latex]{\Large\frac{1}{a}}[/latex].
• Properties of Zero
  • Multiplication by Zero: For any real number [latex]a[/latex],
    [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 [latex]a[/latex],
    [latex]\begin{array}{ccccccc}\hfill {\Large\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 [latex]a[/latex], [latex]{\Large\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 [latex]a[/latex] is [latex]{\Large\frac{1}{a}}[/latex] .