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Absolute Value

Concept

Absolute Value

The absolute value of a number, a,a, expressed as a,|a|, is the non-negative value of a.a. Therefore, the absolute value of a negative number changes its sign whereas the absolute value of a non-negative number is equal to the number itself. -3=33=3 |\text{-} 3|=3 \quad \quad |3|=3 A compact way to write the absolute value of a real number is by using a curly bracket. a={aif a0-aif a<0 |a| = \begin{cases} a & \quad \text{if } a\geq0\\ \text{-} a & \quad \text{if } a<0 \end{cases} The absolute value of a number aa can be thought of as the distance between aa and 00 on the number line. Since the absolute value of -3\text{-}3 and 33 are equal, these numbers are also the same distance from 0.0.


Concept

Properties and Identities of Absolute Value

There are several properties and identities that can be useful when simplifying expressions or solving equations including absolute values. For real values of aa and b,b, these relationships are true.

  • a0|a|\geq0
  • -a=a|\text{-} a|=|a|
  • ab=ab|a\cdot b|=|a|\cdot |b|
  • ab=ab\left|\dfrac{a}{b}\right|=\dfrac{|a|}{|b|}
  • a+ba+b|a+b|\leq |a|+|b|
  • a=a2|a|=\sqrt{a^2}