Absolute value

Absolute Value

Graph of the absolute value function.

The absolute value of a real number is its numerical value without regard to its sign. In mathematics, the absolute value of a number is denoted by two vertical bars surrounding the number, for example, \(|x|\). The absolute value of a number is always non-negative.

Definition

For a real number \(x\), the absolute value is defined as:

\[ |x| = \begin{cases}

 x, & \text{if } x \geq 0 \
 -x, & \text{if } x < 0 

\end{cases} \]

This definition can be extended to complex numbers. For a complex number \(z = a + bi\), where \(a\) and \(b\) are real numbers, the absolute value is defined as:

\[ |z| = \sqrt{a^2 + b^2} \]

Properties

The absolute value function has several important properties:

• Non-negativity: \(|x| \geq 0\) for all real numbers \(x\).
• Identity of indiscernibles: \(|x| = 0\) if and only if \(x = 0\).
• Multiplicativity: \(|xy| = |x||y|\) for all real numbers \(x\) and \(y\).
• Subadditivity (Triangle Inequality): \(|x + y| \leq |x| + |y|\) for all real numbers \(x\) and \(y\).

Applications

The concept of absolute value is used in various fields of mathematics, including:

• Algebra: Solving equations and inequalities involving absolute values.
• Calculus: Defining the distance between points on the real number line.
• Complex analysis: Measuring the magnitude of complex numbers.
• Vector spaces: Defining norms and distances.

Graphical Representation

Graphical representation of the absolute value function.

The graph of the absolute value function \(y = |x|\) is a V-shaped curve that intersects the origin \((0, 0)\). It is symmetric with respect to the y-axis.

Complex Numbers

Complex plane representation of a complex number and its conjugate.

In the context of complex numbers, the absolute value represents the distance of the complex number from the origin in the complex plane. It is also related to the complex conjugate \(\overline{z}\) of a complex number \(z\), as \(|z|^2 = z\overline{z}\).

Composition

Composition of absolute value functions.

The composition of absolute value functions can be used to model various real-world phenomena, such as oscillations and reflections.

Related Pages

• Complex number
• Real number
• Norm (mathematics)
• Distance

Gallery

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  Graph of the absolute value function.

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  Graphical representation of the absolute value function.

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  Complex plane representation of a complex number and its conjugate.

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  Composition of absolute value functions.

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  Absolute value

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  Diagram of absolute value

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  Complex conjugate picture

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  Absolute value

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  Composition of absolute value