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.


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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