Absolute value
Absolute value refers to the non-negative value of a real number, regardless of its sign. It is denoted by two vertical lines surrounding the number or expression. For example, the absolute value of -3 and 3 are both 3, represented as |-3| = 3 and |3| = 3.
Definition[edit | edit source]
The absolute value of a real number a is defined as:
- If a is greater than or equal to zero, then the absolute value of a is a.
- If a is less than zero, then the absolute value of a is -a.
In mathematical notation, this is expressed as |a| = a if a ≥ 0, and |a| = -a if a < 0.
Properties[edit | edit source]
The absolute value function has several key properties, including:
- Non-negativity: For any real number a, |a| ≥ 0.
- Identity of Positives: For any positive real number a, |a| = a.
- Symmetry: For any real number a, |-a| = |a|.
- Triangle Inequality: For any real numbers a and b, |a + b| ≤ |a| + |b|.
- Multiplicative: For any real numbers a and b, |a × b| = |a| × |b|.
Applications[edit | edit source]
The concept of absolute value is used in a wide range of mathematical and scientific fields, including algebra, calculus, complex analysis, geometry, and physics. It is particularly useful in situations where only the magnitude of a quantity matters, and not its direction.
See also[edit | edit source]
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