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

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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