Properties of absolute value. Triangle inequality

Absolute value has several key properties:

1. Non-negativity: $|a| \geq 0$
2. Identity: $|a| = 0$ if and only if $a = 0$
3. Multiplicativity: $|ab| = |a| \cdot |b|$
4. Symmetry: $|-a| = |a|$
5. Triangle inequality: $|a + b| \leq |a| + |b|$

The triangle inequality is especially important in algebra and geometry. It says that the absolute value of a sum is at most the sum of the absolute values.

Important properties

• Absolute value is always non-negative.

• Absolute value of a product equals the product of absolute values.

• Absolute value of a sum is at most the sum of absolute values (triangle inequality).

• Absolute value is unchanged by negation.