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.

\ \;\,\ \ x &\mathrm{if}\ x \ge 0\\
\ \;\, - x &\mathrm{otherwise}

Properties[change | change source]

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]

|x| = \begin{cases} x, & \mbox{if }  x \ge 0  \\ -x,  & \mbox{if } x < 0. \end{cases}

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

|a| = \sqrt{a^2} (1)

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

The absolute value has the following four main properties:

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

Other important properties of the absolute value include:

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

Two other useful properties concerning inequalities are:

|a| \le b \iff -b \le a \le b
|a| \ge b \iff a \le -b \mbox{ or } b \le a

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

|x-3| \le 9 \iff -9 \le x-3 \le 9
\iff -6 \le x \le 12

References[change | change source]

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