Extended real number line

As an extended real numbers is called in mathematics a lot of that out of the field of real numbers by adding new symbols for infinite elements (also: points at infinity ) is created. We distinguish precisely between the affine extended real numbers, in which there are two signed points at infinity, and the projective extended real numbers with only an unsigned improper point. Without the addition of affine or projective, the term is extended real numbers in the literature usually used synonymously with affine extended real numbers, in this article, but that it is used as a generic term for both extensions.

For example, make the affine extended real numbers it possible to see the infinite elements than the limit determined by divergent sequences and thus to deal with such consequences analogous to kovergenten consequences. The definition of extensions is first accordingly topologically motivated. The arithmetic of real numbers can be contrast to the extended real numbers not fully continue.

• 4.1 calculation rules of continuity 4.1.1 Basic arithmetic
• 4.1.2 powers
• 4.1.3 function values

Definition

The real numbers form a locally compact space with its usual topology. By appropriately adding improper points from this result is a compact space.

• In the affine extension is added to two elements and as signed infinities. With any two non- elements of the real numbers are initially referred.
• In the case of the projective extension considering the Einpunktkompaktifizierung with a single designated by the symbol improper point.

Topology

Every open set in is also in or open. In addition, a neighborhood basis for the points at infinity is given.

Affine case

Intended for each

An open neighborhood of and

An open neighborhood of his. This is, for example, by given sequence to a convergent sequence against: almost all followers are included for each, namely those with.

The picture that by

Is given, is a homeomorphism. Topologically is thus completely equivalent to a closed interval.

The affine extended real numbers form a strictly totally ordered set by the order of the real numbers by, for all, and continues. The usual notations for open and semi - open and closed intervals are therefore also useful if and / or is. The topology of is also specified by that order order topology.

The homeomorphism with shows that is metrizable. However, the standard metric can not continue on to a metric on: This would require to be open, so one included; but from this would

Follow.

Projective case

For every positive real number, the complement of an open neighborhood of to be. More generally follows as for the Einpunktkompaktifizierung common that for each compact subset of the complement is an open neighborhood of. This example also by given sequence to a convergent sequence against: almost all sequence elements in the complement of are included for each, ie it applies. Generally in amount from each convergent in against determined according to divergent real sequence.

With this topology, is homeomorphic to the circle

A homeomorphism is for example given by

Just as the Circle Line can therefore in total order with the topology of sound manner. Usually one leaves it at that, with finite numbers is incomparable.

As in the case, the affine projective metrisable extension, but not by continuing the real standard metric.

One can imagine by sticking the points and also emerged as from.

In addition, the real projective line, this corresponds also motivates the name.

Common Features

Both the affine and projective extension form a compact space in which the real numbers are a dense subset. It follows that any sequence of numbers then a convergent subsequence ( and it was against an improper point) contains. Only be determined or magnitude determined divergent consequences in the real affine or projective extension to convergent sequences. A sequence such as by given is divergent in the extended real numbers. In the Stone - Čech compactification of the real numbers, however, converge all bounded sequences.

Simplified spellings

The introduction of the ( affine ) extended real numbers, it first allows to treat the spellings and analog finite, without introducing this separately as its own notation, or speech. Even without this merely symbolic notations as determined for divergent sequences fit seamlessly into the case of convergent sequences.

Arithmetic

This raises the question of how the mathematical basic arithmetic operations are to be adapted to the new infinite places. For the purposes of this Permanenzprinzips to old calculation rules continue to exist, but throughout this is not feasible, since the extended real numbers can not form a complete ordered field - such would have to be re- isomorphic ( and homeomorphic ). The operations thus remain undefined least for some arguments.

For or you want a turn in underlying value for the expressions for as many

Define such that the usual arithmetic laws (particularly associative law and commutative law of addition and multiplication and the distributive property ) remain valid for this extension. More is a sensible requirement: If two expressions in a finite number of variables in the finite always agree, where both sides (including all the used sub-expressions ) are defined, and it is not trivial reasons always a page undefined, so is this equality of the two expressions also apply in the extension, so if even infinite values ​​for the variables are permitted and all subexpressions are defined. Such an equation is for example. In this applies to the finite, that is, once and are defined (the product here is always defined). If it is defined for the case in the extension must either apply or be the product of undefined.

In addition, interested in the basic arithmetic operations nor the power calculation, that is, you want to assign to the expression of as many as possible a value such that the power laws, to always be true if all occurring sub-expressions are defined.

Calculation rules of continuity

The ( algebraically formulated ) conditions mentioned are definitely satisfied if the operations are continued steadily. However, there are, for example, not a continuous image, or that corresponds to the addition. Therefore, the continuous extension principle is only partially possible. "?" Through the widest possible continuous extension, the following calculation rules, which is noted for in this way can not be defined, the value expressions result:

Basic arithmetic

Potencies

In the following, the continuous extension of the exponentiation is indicated only in the affine case. It should be noted that is already defined in the finite only ( real) if ( and arbitrary) or and or and.

The value of negative and finite non- whole remains undefined, since these sites do not include the completion of the domain of the finite power function. The steady sequels with no positive base is to be noted that these bodies, although at the end of the domain include, but are not interior points of the financial statements are. There is, therefore, entirely located outside of the domain of consequences that converge to these points.

Function values

Some standard functions can be steadily continue indefinitely to pictures, such as

• And (but in is not defined ).

In measure theory, a function with a non-empty set is called numerically. Numeric functions can occur as a supremum or infimum of a sequence of real functions.

Undefined expressions

Using the method of steady continuation can be divided into the basic arithmetic expressions

Or in for

Do not specify a value. In principle, it is conceivable that an appropriate - to find Taxation - necessary discontinuous. This is for the expressions referred to but not possible without violating the permanence, ie without contradiction to the usual calculation rules. This shows in detail the following list:

• : Because for all the follows through the permanence that should apply if the expression is defined. However, this leads to a contradiction.
• In: Similarly, there.
• : Because for all to apply. On the other hand, applies where the left side is defined. According to this yields the contradiction
• : Paths and results
• : Here follows.
• In: It follows from that is to apply, therefore. Because follows.

Assign a value to the listed expressions, so it is not possible to " reasonable" manner. Apart from having to be so designated in the undefined expressions as indeterminate expressions.

Notwithstanding the above list is in some areas of mathematics, including measure theory, commonly agreed, as many statements are to be taken more concise in this way. In that case, make sure that the inverse is never used of infinite, or it is to refrain from fixing it. Otherwise, the exceptions to the ordinary rules of calculation would have (namely, that does not always apply ) are regularly considered by case distinctions, and this made the advantage of the shortcut for it.

Algebraic continuation of the exponentiation

Unlike the four basic arithmetic operations, it is also independent of continuity considerations possible, consistent ( but intermittently )

To define. That, at least no other value for these terms can be defined, arises directly from the permanence, as is true in the finite. These determinations are consistent in the sense that the power laws, and apply whenever all sub-expressions are defined.

In the context of limit tests, however, the expressions, and even counted among the vague terms, as in the context of continuity is crucial.

Solving the equations

When solving equations caution when working with infinities, as additional solutions may exist. Particularly evident this is in the equation, which always has exactly one solution for finite. In contrast, none at all and infinitely many. Some other examples that result from the above calculation rules, the following table shows:

During forming of equations can not generally to the reduction property of addition ( follows ) may be used, but only under the condition that is finite. The reduction property of multiplication ( follows ), which also applies in the finite only on the condition is also valid for infinite. The last equation from the table above, can not be equivalently rewritten as, for these, in contrast to the former no infinite solution (the right side is not defined for ).

Complex Numbers

If one starts instead of the real of the complex numbers, we consider mainly the homeomorphic to a sphere Einpunktkompaktifizierung ( Riemann sphere ). The calculation rules for the basic arithmetic agree here essentially the same as those for the Einpunktkompaktifizierung of. There are also alternative approaches, in which compacted to a closed disk or projective plane.