The absolute value of a number $a$ is the distance between $a$ and $0$ on the number line. It is denoted as $|a|$ and it is always a non-negative value. The absolute value is defined for any real number. The absolute value of a negative number is its opposite value, while the absolute value of a positive number is equal to itself.

Absolute Value Properties

There are several properties and identities that are useful when simplifying expressions or solving equations dealing with absolute values. For any two real numbers $a$ and $b,$ the following relationships and identities hold true.

Property	Algebraic Representation
Non-negativity	$|a|\ge 0$
Symmetry	$|\text{-}a|=|a|$
Idempotence	$||a||=|a|$
Positive-definiteness	$|a|=0\iff a=0$
Identity of Indiscernibles	$|a-b|=0\iff a=b$
Multiplicativity	$|ab|=|a|\cdot |b|$
Preservation of Division	$\left|\frac{a}{b}\right|=\frac{|a|}{|b|}\text{ if }b\ne 0$
Subadditivity	$|a+b|\le |a|+|b|$
Triangle Inequality	$|a-b|\le |a-c|+|c-b|$