Absolute value (algebra)

In algebra, the absolute value of a real number x, denoted |x|, is its distance from zero on the number line. Consequently, the absolute value is always non-negative. It can be defined formally as follows:

| x | = x, if x ≥ 0 | x | = -x, if x < 0

This means if x is a positive number or zero, the absolute value of x is simply x. However, if x is a negative number, the absolute value of x is the negation of x, making it positive.

The absolute value can also be thought of as the magnitude of a real number, disregarding its sign.

The absolute value function satisfies several important properties:

• Non-negativity: |x| ≥ 0 for all real numbers x.
• Definiteness: |x| = 0 if and only if x = 0.
• Multiplicativity: |xy| = |x| |y| for all real numbers x and y.
• Triangle Inequality: |x + y| ≤ |x| + |y| for all real numbers x and y.

The concept of absolute value is fundamental in various areas of mathematics, including analysis, geometry, and number theory. It is used to define distance, norms, and metrics in different mathematical spaces. It also plays a critical role in solving equations and inequalities involving real numbers.