Absolute Value

In Mathematical, the absolute value (sometimes called module ) of a Real number is its numerical value without taking account of sound sign. In data-processing Programming, the Identifying used to indicate the absolute value is usually ABS . There exist many generalizations of the absolute value in more abstract spaces (Complex number, vector Space or body), to see for example the article standard. The concept of absolute value is close to that of Distance, Magnitude in many branches of the Physique and Mathématiques.

Absolute value of a real number

First approach

A Real number consists of two parts: a sign about and a absolute value .

+ 7 is consisted of the sign + and the absolute value 7.

- 5 is consisted of the sign - and of the absolute value 5.

The absolute value of (+ 7) is thus 7, the absolute value of (- 5) is thus 5.

As it is frequent to remove the sign when this one is +, one obtains then

• the absolute value of 7 is 7.
• the absolute value of (- 5) is 5, i.e. the opposite of (- 5).

From where the following definition.

Definition

For all Real number $x$, the absolute value of $x$ (noted $|X|$) is defined by:

• $|X|\; =\; X$, if $X\; >\; 0$
• $|X|\; =\; -\; x$, if
• $|X|\; =\; 0$, if $x=0$

We notice that $|X|=\; \backslash \; max\; (-\; X,\; X)$

Properties

The absolute value has the following properties:

• $\backslash \; forall\; has\; \backslash \; in\; \backslash \; R,\; \backslash \; |has|\; \backslash \; geq\; 0$
• $\backslash \; forall\; has\; \backslash \; in\; \backslash \; R,\; \backslash \; |has|=0\; \backslash \; Leftrightarrow\; has\; =\; 0$
• $\backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; R,\; \backslash \; |ab|=|has|\; \backslash \; times\; |B|$
• $\backslash \; forall\; has\; \backslash \; in\; \backslash \; R,\; \backslash \; \backslash \; forall\; B\; \backslash \; in\; \backslash \; R^*,\; \backslash \; \backslash \; left|\backslash \; frac\; \{has\}\; \{B\}\; \backslash \; right|\; =\; \backslash \; frac$
• $\backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; R,\; \backslash \; |a+b|\; \backslash \; Leq\; |has|\; +\; |B|$ (triangular inequality)
• $\backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; R,\; \backslash \; |-\; B\; has|\; \backslash \; geq\; ||has|-|B||$ (second triangular inequality, rises from the first)
• $\backslash \; left|\backslash sum\_\{k=1\}^n\; a\_k\backslash right|\backslash \; Leq\; \backslash \; sum\_\; \{k=1\}\; ^n\; |a\_k|$ (triangular inequality generalized with a finished family $(a\_i)$)
• Is $f:\; I\; \backslash \; subset\; \backslash \; R\; \backslash \; longrightarrow\; \backslash \; R$ continues on $I$, $\backslash \; left|\backslash \; int\_I\; F\; (T)\; \backslash \; mathrm\; \{D\}\; T\; \backslash \; right|\backslash \; Leq\; \backslash \; int\_I|F\; (T)|\backslash \; mathrm\; \{D\}\; t$
• $\backslash \; forall\; has\; \backslash \; in\; \backslash \; R,\; \backslash \; |has|=\; \backslash \; sqrt\; \{a^2\}$
• $\backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; R,\; \backslash \; |has|\; \backslash \; Leq\; B\; \backslash \; Leftrightarrow\; -\; B\; \backslash \; Leq\; has\; \backslash \; Leq\; b$
• $\backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; R,\; \backslash \; |has|\; \backslash \; geq\; B\; \backslash \; Leftrightarrow\; has\; \backslash \; Leq\; -\; b$ or $has\; \backslash \; geq\; b$

These last properties are often used in the resolution of the inequations; for example for X real:

$\backslash begin\{array\}\{lcll\}$

&|x-3|& \ Leq 9 & \ \ \ Longrightarrow &-9 & \ Leq X - 3 & \ Leq 9 \ \ \ Longrightarrow &-6 & \ Leq X & \ Leq 12 \ end {array}

Absolute value and distance

It is useful to interpret the expression $|X\; -\; there|$ like the distance between the two $x$ numbers and $y$sur real line.

By providing the unit with the real numbers of the distance absolute value, it becomes a metric Espace.

The resolution of an inequation such as $|X\; -\; 3|\; \backslash \; Leq\; 9$ is solved then simply using the concept of distance. The solution is the whole of realities whose distance to reality 3 is lower or equal to 9. It is the interval of center 3 and 9. It is the interval $-\; 9;\; 3\; +\; 9\; =;\; 12$.

Extension to the Complex numbers

The same notation gets busy for the module of a complex. This choice is legitimate because the two concepts coincide for the complexes of which the imaginary part is null. Moreover, the module $\backslash \; left|z\_2\; -\; z\_1\; \backslash \; right|$ of the difference of two complex numbers $z\_1\; =\; x\_1\; +\; I\; y\_1$ and $z\_2\; =\; x\_2\; +\; I\; y\_2$ is the Euclidean distance two points $\backslash \; left\; (x\_1,\; y\_1\; \backslash \; right)$ and $\backslash \; left\; (x\_2,\; y\_2\; \backslash \; right)$.

• $|a+ib|\; =\; \backslash \; sqrt\; \{a^2+b^2\}$

• If B is null, module of has = $\backslash \; sqrt\; \{a^2\}$ = absolute value of has

The function absolute value

This function makes correspond to all $x$, $x$ if this one is positive or $-x$ if this one is negative. The function absolute value is with positive values, even.

The function absolute value $f$ defined by $f\; (X)\; =\; |X|$ is continuous on $\backslash \; mathbb\; R$ and derivable on $\backslash \; mathbb\; R^*$ but is not derivable into 0.

If $f$ is a function,

• the function $g$ defined by $g\; (X)\; =\; F\; (|X|)$ is an even function coinciding with $f$ for all $x$ of $D\_f\; \backslash \; course\; \backslash \; mathbb\; \{R\}\; \_+$.
• the function $h$ defined by $h\; (X)\; =\; |F\; (X)|$ is a function coinciding with $f$ for all $x$ such as $f\; (X)\; \backslash \; geq\; 0$ and coinciding with $-f$ for all $x$ such as $f\; (X)\; \backslash \; Leq\; 0$

Absolute value in a body

An absolute value definite on a body $\backslash \; mathbb\; \{K\}$ is an application which with any element $x$ of $\backslash \; mathbb\; \{K\}$ makes correspond a positive real number noted $|X|$ so that   :

* $\backslash \; forall\; X\; \backslash \; in\; \backslash \; mathbb\; \{K\}:\; |X|\; =\; 0\; \backslash \; yew\; X\; =\; 0$ (separation) ;

* $\backslash \; forall\; (X,\; there)\; \backslash \; in\; \backslash \; mathbb\; \{K\}\; ^2:\; |X\; +\; there|\; \backslash \; Leq\; |X|\; +\; |there|$ (triangular inequality) ;

* $\backslash \; forall\; (X,\; there)\; \backslash \; in\; \backslash \; mathbb\; \{K\}\; ^2:\; |X\; \backslash \; times\; there|\; =\; |X|\; |there|$

An absolute value is known as ultrametric if

* $\backslash \; forall\; (X,\; there)\; \backslash \; in\; \backslash \; mathbb\; \{K\}\; ^2:\; |X\; +\; there|\; \backslash \; Leq\; \backslash \; max\; (\; |X|\; ,\; |there|\; )$

One can use absolute values on a ring or a group thanks to the absolute value induced on this group or this body.

Examples

• the definite Module on $\backslash \; mathbb\; \{C\}$ is well an absolute value from where the fact that the same notation is used.

• the absolute value p-adic definite on the body $\backslash \; mathbb\; \{Q\}\; \_p$ (p a Prime number) is a ultrametric absolute value.

Zh-classical: 絕對值

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