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In
mathematics, the '''absolute value''' (or '''modulus''') of a number is that number without a
negative sign. So, for example, 3 is the absolute value of both 3 and −3.
==Definition==
It can be defined as follows: For any
real number ''a'', the '''absolute value''' of ''a'', denoted |''a''|, is equal to ''a'' if ''a'' ≥ 0, and to −''a'', if ''a'' < 0 (see also:
inequality). |a| is never
negative, as absolute values are always either
positive or
zero. Put another way, |''a''| < 0 has no solution for ''a''.
The absolute value can be regarded as the ''distance'' of a number from zero; indeed the notion of
distance in mathematics is a generalisation of the properties of the absolute value. When real numbers are considered as one-dimensional vectors, the absolute value is the
magnitude, and the
p-norm for any p. Up to a constant factor, every norm in '''R'''
1 is equal to the absolute value: ||x||=||1||.|x|
==Properties==
The absolute value has the following properties:
# |''a''| ≥ 0
# |''a''| = 0
iff ''a'' = 0.
# |''ab''| = |''a''||''b''|
# |''a/b''| = |''a''| / |''b''| (if ''b'' ≠ 0)
# |''a''+''b''| ≤ |''a''| + |''b''| (the
triangle inequality)
# |''a''−''b''| ≥
||''a''| − |''b''|
|
# <math>\left| a \right| = \sqrt{a^2}</math>
# |''a''| ≤ ''b'' iff −''b'' ≤ ''a'' ≤ ''b''
# |''a''| ≥ ''b'' iff ''a'' ≤ −''b'' '''or''' ''b'' ≤ ''a''
The last two properties are often used in solving inequalities; for example:
:|''x'' − 3| ≤ 9
:−9 ≤ ''x''−3 ≤ 9
:−6 ≤ ''x'' ≤ 12
For real arguments, the absolute value function ''f''(''x'') = |''x''| is
continuous everywhere and
differentiable everywhere except for ''x'' = 0. For
complex arguments, the function is continuous everywhere but differentiable ''nowhere'' (One way to see this is to show that it does not obey the
Cauchy-Riemann equations).
For a
complex number ''z'' = ''a'' + ''ib'', one defines the absolute value or ''modulus'' to be |''z''| = √(''a''
2 + ''b''
2) = √ (''z'' ''z''
*) (see
square root and
complex conjugate). This notion of absolute value shares the properties 1-6 from above. If one interprets ''z'' as a point in the plane, then |''z''| is the distance of ''z'' to the origin.
It is useful to think of the expression |''x'' − ''y''| as the ''distance'' between the two numbers ''x'' and ''y'' (on the
real number line if ''x'' and ''y'' are real, and in the complex plane if ''x'' and ''y'' are complex). By using this notion of distance, both the set of real numbers and the set of complex numbers become
metric spaces.
The function is not
invertible, because a negative and a positive number have the same absolute value.
== Absolute value and
complex numbers ==
<math>\mbox{if }c = a + bi \mbox{ then }|c| = \sqrt{a^2 + b^2}\,\!</math> (the modulus)
== Algorithm ==
If the absolute value would not be a standard function '''Abs''' in
Pascal it could be easily computed using the following code:
program absolute_value;
var n: integer;
begin
read (n);
if n < 0 then n := -n;
writeln (n)
end.
In the
C programming language, the
abs(),
labs(),
llabs() (in C99),
fabs(),
fabsf(), and
fabsl() functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input:
int abs(int i)
{
if (i < 0)
return -i;
else
return i;
}
The floating-point versions are trickier, as they have to contend with special codes for
infinity and
not-a-numbers.
de:Absoluter Betrag
es:Valor absoluto
fr:Valeur absolue
is:Algildi
ja:絶対値
nl:Absolute waarde
pl:Wartość bezwzględna
sv:Absolutbelopp
zh:绝对值
