Amount function

Course of the amount function ${\displaystyle \mathbb {R} }$

In mathematics , the absolute value function assigns a real or complex number its distance from zero. This so-called absolute amount, absolute amount, absolute value or simply amount is always a non-negative real number. The amount of a number is usually referred to as , less often , as. The square of the amount function is also called the amount square . ${\displaystyle x}$ ${\displaystyle |x|}$ ${\displaystyle \operatorname {abs} (x)}$

definition

Real amount function

The absolute amount of a real number is obtained by omitting the sign . On the number line , the amount means the distance between the given number and zero .

For a real number : ${\displaystyle x}$

${\displaystyle |x|:={\begin{cases}\ \;\;\;\ x&\mathrm {\;\;f{\ddot {u}}r\;\;} x\geq 0\\\ -x&\mathrm {\;\;f{\ddot {u}}r\;\;} x<0\end{cases}}}$

Complex amount function

For a complex number with real numbers and one defines ${\displaystyle z=x+\mathrm {i} y}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle |z|={\sqrt {z\cdot {\bar {z}}}}={\sqrt {(x+\mathrm {i} y)\cdot (x-\mathrm {i} y)}}={\sqrt {x^{2}+y^{2}}}}$,

where denotes the complex conjugate of . If real (i.e. , so ) this definition goes into ${\displaystyle {\bar {z}}}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle y=0}$ ${\displaystyle z=x}$

${\displaystyle |x|={\sqrt {x^{2}}}}$

about what matches the definition of the magnitude of a real number . ${\displaystyle x}$

If one visualizes the complex numbers as points of the Gaussian plane of numbers , this definition according to the Pythagorean theorem also corresponds to the distance of the point belonging to the number from the so-called zero point . ${\displaystyle z}$

Examples

The following numerical examples show how the amount function works.

${\displaystyle |7|=7}$

${\displaystyle |{-8}|=-(-8)=8}$

${\displaystyle |3+4\mathrm {i} |={\sqrt {(3+4\mathrm {i} )\cdot (3-4\mathrm {i} )}}={\sqrt {3^{2}-(4\mathrm {i} )^{2}}}={\sqrt {3^{2}+4^{2}}}={\sqrt {25}}=5}$

Absolute equations

For real numbers, it follows or . However , then is . ${\displaystyle |a|=b}$ ${\displaystyle a=b}$ ${\displaystyle a=-b}$ ${\displaystyle b<0}$ ${\displaystyle |a|=-b}$

As an example, we are looking for all numbers that satisfy the equation . ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle |x+3|=5}$

One calculates as follows:

${\displaystyle |x+3|=5}$

${\displaystyle \Leftrightarrow x+3=5{\text{ oder }}x+3=-5}$

${\displaystyle \Leftrightarrow x=5-3{\text{ oder }}x=-5-3}$

${\displaystyle \Leftrightarrow x=2{\text{ oder }}x=-8}$

The equation has exactly two solutions for , namely 2 and −8. ${\displaystyle x}$

Inequalities with absolute amount

The following equivalences can be used for inequalities:

${\displaystyle |a|\leq b\Leftrightarrow -b\leq a\leq b}$

${\displaystyle |a|\geq b\Leftrightarrow a\leq -b{\text{ oder }}b\leq a}$

For example, we are looking for all numbers with the property . ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle |x-3|\leq 9}$

Then one calculates:

${\displaystyle |x-3|\leq 9}$

${\displaystyle \Leftrightarrow -9\leq x-3\leq 9}$

${\displaystyle \Leftrightarrow -9+3\leq x\leq 9+3}$

${\displaystyle \Leftrightarrow -6\leq x\leq 12}$

So the solution is all from the interval . ${\displaystyle x}$ ${\displaystyle [-6,12]}$

In general, for real numbers , and : ${\displaystyle x}$ ${\displaystyle m}$ ${\displaystyle r}$

${\displaystyle |x-m|\leq r\iff x\in [m-r,m+r]}$.

Amount norm and amount metric

The absolute value function fulfills the three norm axioms of definiteness , absolute homogeneity and subadditivity and is therefore a norm , called the absolute value norm, on the vector space of real or complex numbers. The definiteness follows from the fact that the only zero of the root function lies in the zero point, which means

${\displaystyle |z|=0\;\Leftrightarrow \;{\sqrt {z{\bar {z}}}}=0\;\Rightarrow \;z{\bar {z}}=0\;\Leftrightarrow \;z=0}$

applies. The homogeneity follows for complex ones ${\displaystyle w,z}$

${\displaystyle |w\cdot z|^{2}=(w\cdot z){\overline {(w\cdot z)}}=(w\cdot z)({\bar {w}}\cdot {\bar {z}})=(w\cdot {\bar {w}})(z\cdot {\bar {z}})=|w|^{2}\cdot |z|^{2}}$

and the triangle inequality

${\displaystyle {\begin{aligned}|w+z|^{2}&=(w+z){\overline {(w+z)}}=(w+z)({\bar {w}}+{\bar {z}})=w{\bar {w}}+w{\bar {z}}+z{\bar {w}}+z{\bar {z}}=\\&=|w|^{2}+|z|^{2}+w{\bar {z}}+{\overline {w{\bar {z}}}}=|w|^{2}+|z|^{2}+2\operatorname {Re} (w{\bar {z}})\\&\leq |w|^{2}+|z|^{2}+2\,|w{\bar {z}}|=\\&=|w|^{2}+|z|^{2}+2\,|w|\,|z|=(|w|+|z|)^{2},\end{aligned}}}$

The two properties we are looking for result from taking the (positive) root on both sides. It was used here that the conjugate of the sum or the product of two complex numbers is the sum or the product of the respective conjugated numbers. It was also used that the double conjugation results in the initial number and that the absolute value of a complex number is always at least as large as its real part. In the real case, the three standard properties follow analogously by omitting the conjugation.

The amount standard is the standard scalar of two real or complex numbers and induced . The amount norm itself induces a metric (distance function), the amount metric ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle d(x,y):=|x-y|}$,

by taking the amount of their difference as the distance between the numbers.

Analytical properties

In this section properties of the absolute value function are given, which are of particular interest in the mathematical area of analysis .

Zero

The only zero of the two absolute value functions is 0, that is, applies if and only if applies. This is a different terminology from the previously mentioned definiteness. ${\displaystyle |z|=0}$ ${\displaystyle z=0}$

Relationship to the sign function

The following applies to all , where denotes the sign function. Since the real one is only the restriction of the complex amount function to , the identity also applies to the real amount function. The derivative of the restricted amount function is the restricted sign function. ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle z=|z|\cdot \operatorname {sgn}(z)}$ ${\displaystyle \operatorname {sgn} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} \setminus \{0\}}$ ${\displaystyle \mathbb {R} \setminus \{0\}}$

Continuity, differentiability and integrability

The real absolute value function and the complex one are continuous over their entire domain . From the subadditivity of the absolute value function or from the (inverse) triangle inequality it follows that the two absolute value functions are even Lipschitz continuous with Lipschitz constant : ${\displaystyle |\cdot |\colon \mathbb {R} \to \mathbb {R} _{0}^{+}}$ ${\displaystyle |\cdot |\colon \mathbb {C} \to \mathbb {R} _{0}^{+}}$ ${\displaystyle L=1}$

${\displaystyle {\bigl |}|z|-|w|{\bigr |}\leq |z-w|}$.

The real amount function is not differentiable at this point and therefore no differentiable function in its domain of definition . However, it can be differentiated almost everywhere , which also follows from Rademacher's theorem. For the derivative of the real amount function is the sign function . As a continuous function, the real amount function can be integrated over limited intervals ; is an antiderivative . ${\displaystyle 0}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle x\neq 0}$ ${\displaystyle \operatorname {sgn} }$ ${\displaystyle x\mapsto {\tfrac {1}{2}}x^{2}\operatorname {sgn}(x)}$

The complex absolute value function is nowhere complex differentiable because the Cauchy-Riemann differential equations are not fulfilled. ${\displaystyle |\cdot |\colon \mathbb {C} \to \mathbb {R} _{0}^{+}}$

Archimedean amount

Both absolute functions, the real and the complex, are called Archimedean because there is an integer with . But it also follows that for all integers is also . ${\displaystyle n}$ ${\displaystyle |n|>1}$ ${\displaystyle m>1}$ ${\displaystyle |m|>1}$

Generalizations

Amount function for body

definition

In general, one speaks of a magnitude if a function from an integrity domain into the real numbers fulfills the following conditions: ${\displaystyle \varphi }$ ${\displaystyle D}$ ${\displaystyle \mathbb {R} }$

	${\displaystyle \varphi (x)\geq 0}$	(0)	Non-negativity
	${\displaystyle \varphi (x)=0\iff x=0}$	(1)	Definiteness
			(0) and (1) together are called positive definiteness
	${\displaystyle \varphi (x\cdot y)=\varphi (x)\cdot \varphi (y)}$	(2)	Multiplicativity, absolute homogeneity
	${\displaystyle \varphi (x+y)\leq \varphi (x)+\varphi (y)}$	(3)	Subadditivity , triangle inequality

The continuation to the quotient field of is unique because of the multiplicativity. ${\displaystyle K:=\operatorname {Quot} (D)}$ ${\displaystyle D}$

comment

An amount function for a body is an evaluation of that body. ${\displaystyle \varphi }$ ${\displaystyle \varphi }$

If natural for all , then the amount (or valuation) is called non-Archimedean. ${\displaystyle \varphi (n)\leq 1}$ ${\displaystyle n:=\underbrace {1+\dotsb +1} _{n{\text{-mal}}}}$

The amount for all (is non-Archimedean and) is called trivial . ${\displaystyle \varphi (x)=1}$ ${\displaystyle x\neq 0}$

For non-Archimedean amounts (or valuations) applies

${\displaystyle \varphi (x+y)\leq \max(\varphi (x),\varphi (y))}$

(3 ')

the tightened triangle inequality.

It makes the amount an ultrametric . Conversely, any ultrametric amount is non-Archimedean.

Amount and characteristics

• Integrity areas with an Archimedean amount have the characteristic 0.
• Integrity domains with a characteristic different from 0 (have prime number characteristic and) only accept non-Archimedean amounts.
• Finite domains of integrity are finite fields with prime number characteristics and only take on the trivial amount.
• The field of rational numbers as the prime field of characteristic 0 and its finite extensions take both Archimedean and non-Archimedean amounts. ${\displaystyle \mathbb {Q} }$

completion

The body can be for any amount function, or more precisely for the induced by any amount function (or review) metric complete . The completion of is often referred to with . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle {\hat {K}}}$

Archimedean completions of rational numbers are and , non-Archimedean are for prime numbers . ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\hat {\mathbb {Q} }}=\mathbb {R} }$ ${\displaystyle {\widehat {\mathbb {Q} (\mathrm {i} )}}={\hat {\mathbb {Q} }}(\mathrm {i} )=\mathbb {C} }$ ${\displaystyle {\hat {\mathbb {Q} }}=\mathbb {Q} _{p}}$ ${\displaystyle p}$

Nothing new arises with the trivial amount.

Equivalence of amounts

If and are amounts (or ratings) of a body , then the following three statements are equivalent: ${\displaystyle \varphi }$ ${\displaystyle \psi }$ ${\displaystyle K}$

1. Any sequence that is below a null sequence , i.e. H. , is also under a zero sequence - and vice versa. ${\displaystyle \{x_{\nu }\}}$ ${\displaystyle \varphi }$ ${\displaystyle \lim \limits _{\nu \to \infty }\varphi (x_{\nu })=0}$ ${\displaystyle \psi }$
2. From follows . ${\displaystyle \varphi (x)<1}$ ${\displaystyle \psi (x)<1}$
3. ${\displaystyle \psi }$is a power of , i. H. for everyone with a solid . ${\displaystyle \varphi }$ ${\displaystyle \psi (x)=\varphi (x)^{\epsilon }}$ ${\displaystyle x}$ ${\displaystyle \epsilon >0}$

The absolute value functions of the rational numbers

According to Ostrowski's theorem , the amounts mentioned in this article, the one Archimedean (and Euclidean) and the infinite number of non-Archimedean amounts that can be assigned to a prime number, represent all classes of amounts (or evaluations) of the rational numbers . ${\displaystyle \mathbb {Q} }$

The approximation rate applies to these amounts .

standard

→ Main article : Norm (mathematics)

The absolute value function on the real or complex numbers can be generalized to any vector spaces through the properties definiteness, absolute homogeneity and subadditivity. Such a function is called a norm. But it is not clearly defined.

Pseudo amount

→ Main article : Pseudo amount

Web links

Wikibooks: Math for Non-Freaks: Amount  - Study and Teaching Materials

• Eric W. Weisstein : Absolute Value . In: MathWorld (English).

Individual evidence

1. ↑ van der Waerden : Algebra . Part 2. Springer-Verlag, 1967, Evaluated Bodies, p. 203, 212 .