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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. |a| = a &if a≥ 0 - a &if a < 0
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| ≥ 0 |
| Symmetry | | - a| = |a| |
| Idempotence | ||a|| = |a| |
| Positive-definiteness | |a| = 0 ⇔ a = 0 |
| Identity of Indiscernibles | |a − b| = 0 ⇔ a = b |
| Multiplicativity | |ab| = |a| * |b| |
| Preservation of Division | |a/b|=|a|/|b| if b ≠ 0 |
| Subadditivity | |a + b| ≤ |a| + |b| |
| Triangle Inequality | |a − b| ≤ |a − c| + |c − b| |