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Absolute value

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The graph of the absolute value function for real numbers.

In mathematics, the absolute value of a real number is the number without the sign. The absolute value of 2 is 2, the absolute value of -2 is also 2. This notation is to express a numbers distance from zero on a number line. The absolute value of 10 would be 10 since the number 10 is 10 numbers away from zero, same follows with negatives.

[math] \begin{cases} \ \;\,\ \ x &\mathrm{if}\ x \ge 0\\ \ \;\, - x &\mathrm{otherwise} \end{cases} [/math]


For any real number x the absolute value or modulus of x is denoted by | x | (a vertical bar on each side of the quantity) and is defined as[1]

[math]|x| = \begin{cases} x, & \mbox{if } x \ge 0 \\ -x, & \mbox{if } x \lt 0. \end{cases} [/math]

The absolute value of x is always either positive or zero, but never negative.

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line. The absolute value of the difference of two real numbers is the distance between them.

The square-root notation without sign represents the positive square root. So, it follows that

[math]|a| = \sqrt{a^2}[/math] (1)

which is sometimes used as a definition of absolute value.[2]

The absolute value has the following four main properties:

[math]|a| \ge 0 [/math] (2) Non-negativity
[math]|a| = 0 \iff a = 0 [/math] (3) Positive-definiteness
[math]|ab| = |a||b|\,[/math] (4) Multiplicativeness
[math]|a+b| \le |a| + |b| [/math] (5) Subadditivity

Other important properties of the absolute value include:

[math]||a|| = |a|\,[/math] (6) Idempotence (the absolute value of the absolute value is the absolute value)
[math]|-a| = |a|\,[/math] (7) Symmetry
[math]|a - b| = 0 \iff a = b [/math] (8) Identity of indiscernibles (equivalent to positive-definiteness)
[math]|a - b| \le |a - c| +|c - b| [/math] (9) Triangle inequality (equivalent to subadditivity)
[math]|a/b| = |a| / |b| \mbox{ (if } b \ne 0) \,[/math] (10) Preservation of division (equivalent to multiplicativeness)
[math]|a-b| \ge ||a| - |b|| [/math] (11) (equivalent to subadditivity)

Two other useful properties concerning inequalities are:

[math]|a| \le b \iff -b \le a \le b [/math]
[math]|a| \ge b \iff a \le -b \mbox{ or } b \le a [/math]

These relations may be used to solve inequalities involving absolute values. For example:

[math]|x-3| \le 9 [/math] [math]\iff -9 \le x-3 \le 9 [/math]
[math]\iff -6 \le x \le 12 [/math]


  1. Mendelson, p. 2.
  2. Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1 ., p. A5