Absolute value

In mathematics, the absolute value function of a real or complex number assigned to their distance to zero. This so-called absolute value, absolute value or simply amount is always a non-negative real number. The value of a number is usually with, more rarely, referred to. The square of the absolute value function is also called the modulus squared.

• 4.1 zero
• 4.2 relative to the sign function
• 4.3 continuity, differentiability and integrability
• 4.4 Archimedean amount

• 5.1 standard
• 5.2 Pseudo amount
• 5.3 magnitude function for body 5.3.1 definition
• 5.3.2 Amount and Characteristics
• 5.3.3 completion
• 5.3.4 Equivalence of amounts
• 5.3.5 The amount functions of rational numbers

Definition

Real value function

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

For a real number applies:

Complex modulus function

For a complex number with real numbers and we define

The complex conjugate of designated. Is real ( ie, that is ), so this definition is in

Above, which is consistent with the definition of the amount of a real number.

Demonstrates to the complex numbers as points of the Gaussian plane, so this definition according to the Pythagorean theorem also corresponds to the distance of the point from the number belonging to the so-called zero point.

Examples

Equation with absolute value: We are looking for all numbers that satisfy the equation.

It is calculated as follows:

So the equation has exactly two solutions, namely 2 and -8.

Amount of standard and metric amount

The value function satisfies the three standard axioms definiteness, absolute homogeneity and subadditivity, making it a standard, called sum norm on the vector space of real or complex numbers. The definiteness follows from the fact that the only zero of the square root function is the zero point, which

Applies. The homogeneity follows from complex

And from the triangle inequality

With the two desired properties result in each case by dragging the (positive) root on both sides. This has been used, that the conjugate of the sum or the product of two complex numbers is the sum and the product of each conjugate numbers. Furthermore, it was used that the two-time conjugation again yields the original number and that the magnitude of a complex number is always at least as large as their real part. In the real case, the three standard properties follow analogously by omitting the conjugation.

The amount norm is induced by the standard scalar product of two real or complex numbers and. In turn, the amount standard itself induces a metric ( distance function ), the amount of metric

By the amount of such difference is taken as the distance between the numbers.

Analytical properties

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

Zero

The only zero of the absolute value function is 0, ie if and only if the following holds. So this is just a slightly different terminology of the previously mentioned definiteness.

Relation to the sign function

For all holds, where the sign function called. Since the complex magnitude function comprising real, the statement also applies to the real absolute value function.

Continuity, differentiability and integrability

The real and complex absolute value function is continuous on its entire domain of definition. From the subadditivity of the function or amount of the ( inverted ) triangle inequality it follows that the two sum ​​functions even are Lipschitz continuous with Lipschitz constant:

The real value function at the location and thus can not be differentiated in their domain not differentiable function. However, it is almost everywhere differentiable, which also follows from the theorem of Rademacher. For the derivation of the real absolute value function is the sign function. As a continuous function, the real absolute value function over limited intervals can be integrated; an antiderivative is.

The complex absolute value function is complex differentiable nowhere, because the Cauchy -Riemann equations are not satisfied.

Archimedean amount

Both amount functions, the real and the complex will be called Archimedean, because there is an integer. But it also follows that for all integers also.

Generalizations

Standard

The absolute value function on the real or complex numbers can be generalized by the properties of definiteness, absolute homogeneity and subadditivity to arbitrary vector spaces. Such a function is referred to as standard. But it is not uniquely determined.

Pseudo amount

Value function for body

Definition

Generalizing speaks of an amount if a function of a integral domain into the real numbers satisfies the following conditions:

The sequel to the quotient field of is unique because of the multiplicativity.

If for all integers, then it is called the amount nichtarchimedisch.

The amount for all ( and is nichtarchimedisch ) is called trivial.

For non- Archimedean amounts shall

It makes the sum at a ultrametric. Conversely, any ultrametric amount is nichtarchimedisch.

Amount and characteristics

• Integral domains with an Archimedean amount have the characteristic 0
• Integral domains with a characteristic different from 0 ( have prime characteristic and ) only accept nichtarchimedische amounts.
• Finite integral domains ( are finite fields of prime characteristic and ) accept only the trivial amount.
• The field of rational numbers as prime field of characteristic 0 and its finite extensions assume both Archimedean as nichtarchimedische amounts.

Completion

The body can be for any amount function, more precisely, for any amount of the induced metric function, complete. The completion is often referred to.

The Archimedean completions are and. Nichtarchimedische are prime numbers.

When trivial amount produced is nothing new.

Equivalence of amounts

Are and amounts of a body, then the following three assertions are equivalent:

The amount functions of rational numbers

By the theorem of Ostrowski represent the amounts mentioned in this article, the non- Archimedean an Archimedean ( and Euclidean ) and the infinite number of prime numbers to be assigned, all classes of sums of rational numbers.

For these values ​​of the approximation theorem applies.