absolute value — math

The absolute value of a real number $x$ , denoted $|x|$ , is its distance from zero. It is the non-negative value of $x$ without regard to its sign. Namely, $|x| = x$ if $x$ is a positive number, and $|x| = -x$ if $x$ is negative (in which case negating $x$ makes $-x$ positive), and $|0|=0$ . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.

Definition
Absolute value of a real number The absolute value of a real number n, $\|n\|$ , is defined as $\|n\| = \begin{cases} n, & \mbox{if } n \ge 0 \\ -n, & \mbox{if } n < 0. \end{cases}$

Definition
Absolute value of a complex number The absolute value of a complex number $a+bi$ is defined as $\|a + bi\| = \sqrt{a^2 + b^2},$ equal to the distance between the number in the complex number plane and the origin.

The definition may also be extended to vectors, where absolute value is often times called "magnitude".

Definition
Magnitude of a vector The magnitude of a vector $a\hat{i} + b\hat{j} + c\hat{k} + \cdots$ is defined as $\|a\hat{i} + b\hat{j} + c\hat{k} + \cdots\| = \sqrt{a^2 + b^2 + c^2 + \cdots}$ , is equal to the length or magnitude of the vector in question. Notice that magnitude is sometimes denoted with a double stroke absolute value $\|a\hat{i} + b\hat{j} + c\hat{k} + \cdots\| = \|\|a\hat{i} + b\hat{j} + c\hat{k} + \cdots\|\|$

The absolute value signs are also used in set theory to indicate a cardinality, or set size. The operation measures the magnitude of a set, or in other words, the operation counts the number of elements a set contains.

Definition
Cardinality of a set The cardinality of a set $A = \{0,1,2,3,a,b,c\}$ containing seven elements is defined as $\|A\| = 7$ , the number of elements in the set.