Payment range extension

In mathematics , a number (s) range expansion means the construction of a new set of numbers from a given set of numbers, mostly to generalize certain algebraic operations , but also, as in the case of real numbers, to generalize topological operations. Typically, number range expansions are not taught completely because they are neither particularly interesting nor particularly difficult, but require a lot of repetition and detailed work.

overview

The usual order of the extension of the number range is that the natural numbers are extended to the whole numbers , the whole numbers to the rational numbers , the rational numbers to the real numbers and the real numbers to the complex numbers , see for example ( Lit .: Landau, 1948). But other procedures would also be possible, for example, instead of the whole numbers, one could first construct the positive rational numbers and the positive real numbers and only then introduce negative numbers. In addition, there are other number range extensions such as the quaternions , the hyper-real numbers and the surreal numbers .

Procedure for payment range extensions

Definition of the new number range

The first step in expanding a number range is to construct a new set from the existing number set. Mostly they are ordered pairs , so the whole numbers are defined as pairs of natural numbers, the rational numbers as pairs of whole numbers and the complex numbers as pairs of real numbers. An exception are the real numbers, which are usually defined as Cauchy sequences of rational numbers or as Dedekind cuts . In a second step, an equivalence relation is introduced on this new set and the new numbers are each defined as an equivalence class . The selection of the equivalence relation depends essentially on the operation that is to be expanded, so two pairs are defined in the construction of the integers and as equivalent if they represent the same difference : ${\ displaystyle (m_ {1}, n_ {1})}$ ${\ displaystyle (m_ {2}, n_ {2})}$

{\ displaystyle (m_ {1}, n_ {1}) \ sim (m_ {2}, n_ {2}): \ iff m_ {1} + n_ {2} = m_ {2} + n_ {1}}

,

in the construction of the rational numbers, two pairs and are defined as equivalent if they represent the same quotient : ${\ displaystyle (p_ {1}, q_ {1})}$ ${\ displaystyle (p_ {2}, q_ {2})}$

{\ displaystyle (p_ {1}, q_ {1}) \ sim (p_ {2}, q_ {2}): \ iff p_ {1} q_ {2} = p_ {2} q_ {1}}

,

and in the construction of the complex numbers, two pairs and are defined as equivalent if they coincide component-wise ${\ displaystyle (u_ {1}, v_ {1})}$ ${\ displaystyle (u_ {2}, v_ {2})}$

{\ displaystyle (u_ {1}, v_ {1}) \ sim (u_ {2}, v_ {2}): \ iff u_ {1} = u_ {2} {\ mbox {and}} v_ {1} = v_ {2}}

.

In the construction of real numbers, two Cauchy sequences and are defined as equivalent if their difference is a zero sequence: ${\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}}}$ ${\ displaystyle (y_ {n}) _ {n \ in \ mathbb {N}}}$

{\ displaystyle (x_ {n}) _ {n \ in \ mathbb {N}} \ sim (y_ {n}) _ {n \ in \ mathbb {N}}: \ iff \ lim _ {n \ to \ infty} | x_ {n} -y_ {n} | = 0}

.

After defining the respective relation, it must still be shown that this relation is actually an equivalence relation, that it is reflexive, symmetrical and transitive.

Definition of the operations in the new number range

The next step in a number range extension is to transfer the algebraic operations defined on the initial set to the new number set. The operation is initially defined for individual representatives of the equivalence class; the result is then also the corresponding equivalence class. For example, the addition of whole numbers is called

{\ displaystyle (m_ {1}, n_ {1}) + (m_ {2}, n_ {2}): = (m_ {1} + m_ {2}, n_ {1} + n_ {2})}

and the addition of rational numbers as

{\ displaystyle (p_ {1}, q_ {1}) + (p_ {2}, q_ {2}): = (p_ {1} q_ {2} + p_ {2} q_ {1}, q_ {1 } q_ {2})}

Are defined.

In detail, this means that the result of adding the equivalence class represented by plus the equivalence class represented by is the equivalence class represented by , that is ${\ displaystyle (m_ {1}, n_ {1})}$ ${\ displaystyle (m_ {2}, n_ {2})}$ ${\ displaystyle (m_ {1}, n_ {1}) + (m_ {2}, n_ {2})}$

{\ displaystyle [(m_ {1}, n_ {1})] _ {\ sim} + [(m_ {2}, n_ {2})] _ {\ sim}: = [(m_ {1} + m_ {2}, n_ {1} + n_ {2})] _ {\ sim}}

where the square brackets denote the equivalence classes.

For this definition to actually make sense, it must be shown that the operations defined in this way are independent of the respective representative of the equivalence class, that is, for example

from and follows that .

{\ displaystyle (m_ {1}, n_ {1}) \ sim (j_ {1}, k_ {1})}

{\ displaystyle (m_ {2}, n_ {2}) \ sim (j_ {2}, k_ {2})}

{\ displaystyle (m_ {1}, n_ {1}) + (m_ {2}, n_ {2}) \ sim (j_ {1}, k_ {1}) + (j_ {2}, k_ {2} )}

Then the respective calculation laws of the respective mathematical structure such as B. the associative law and the commutative law for the newly defined operations are shown. In a further step it can now be shown that the new number range has properties that were missing in the old one. For example, in contrast to natural numbers, whole numbers form an additive group ; in particular, every whole number has an inverse element with regard to addition, which can be defined as follows:

{\ displaystyle - [(m_ {1}, n_ {1})] _ {\ sim}: = [(n_ {1}, m_ {1})] _ {\ sim}}

.

When expanding the number range to the real numbers, it can be shown, for example, that in contrast to the rational numbers, every Cauchy sequence is convergent and that every limited set has an infimum and a supremum .

Embedding the old in the new number range

The last step now consists in showing that the old number range is isomorphic to a subset of the new number range. For this purpose, an injective function is defined from the old to the new number range, for example when the natural numbers are embedded in the integers, the natural number is assigned the equivalence class of the pair . Now we have to show that this function is actually an isomorphism, that is, for example ${\ displaystyle f}$ ${\ displaystyle n}$ ${\ displaystyle (n, 0)}$

{\ displaystyle f (m + n) = f (m) + f (n)}

applies, so

{\ displaystyle (m + n, 0) \ sim (m, 0) + (n, 0)}

.

It should be noted that the old number range is not simply a subset of its extension, but is only isomorphic to a subset of the extension. For example, strictly speaking, the natural numbers are not a subset of the whole numbers, but are only isomorphic to a subset of the whole numbers. In most cases, however, this distinction does not play a role, so that statements of the kind that one set of numbers is a subset of another set of numbers are permissible simplifications.

Generalizations

The basic procedure for the expansion of the number range is also found in more general cases, so the expansion of the whole numbers to the rational numbers is a construction of a quotient field ; the extension of the rational numbers to the real numbers corresponds to the completion of a metric or more generally a uniform space .

literature

Edmund Landau : Fundamentals of Analysis Chelsea Publ. New York 1948