# Definition:Absolute Value/Definition 1

## Definition

Let $x \in \R$ be a real number.

The absolute value of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows:

$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

## Also presented as

Because $0 = -0$, the value of $\size x$ at $x = 0$ is often included in one of the other two cases, most commonly:

$\size x = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases}$

but this can be argued as being less symmetrically aesthetic.

## Also known as

The absolute value of $x$ is sometimes called the modulus or magnitude of $x$, but note that modulus has a more specialized definition in the domain of complex numbers, and that magnitude has a more specialized definition in the context of vectors.

Some sources refer to it as the size of $x$.

Some sources call it the numerical value.

## Examples

### Absolute Value of $3$ and $-3$

$\size 3 = 3 = \size {-3}$

### Absolute Value of $-2$

$\size {-2} = 2$

### Absolute Value of $-6$

$\size {-6} = 6 = \size 6$

### Absolute Value of $\dfrac 3 4$

$\size {\dfrac 3 4} = \dfrac 3 4$

### Absolute Value of $3 - 5$

$\size {3 - 5} = \size {5 - 3} = 2$

### Absolute Value of $x - a$

Let $x, a \in \R$.

Then:

$\size {x - a} = \begin {cases} x - a & : x \ge a \\ a - x & : x < a \end {cases}$

### Absolute Value $\size x \le 2$

$\size x \le 2 \iff -2 \le x \le 2$

### Absolute Value of $0$

$\size 0 = 0$

## Also see

• Results about the absolute value function can be found here.