Number Systems

Naturals and Integers

Natural Numbers

We define the set of natural numbers by the following Peano axioms with successor function .

• is a natural number.
• For every natural number , is a natural number. That is, the natural numbers are closed under .
• For all natural numbers and , if , then . That is, is injective.
• There is no natural number whose successor is .
• If is a set such that , and for every , , then contains every natural number. That is the successor function generates all the natural numbers different from .

Addition on Naturals

Addition is a function , defined recursively by

Multiplication on Naturals

Multiplication is a function . Given addition, it is defined recursively as:

Proposition

is the multiplicative identity.

Total Order on Naturals

The standard total order relation on natural numbers can be defined as follows,

Integers

We define the integers as a field of equivalence classes:where is an equivalence relation defined on as We write as the corresponding equivalence class. And define

Rationals

Rational Numbers

We define the rational numbers as a field of equivalence classes:where is an equivalence relation defined on as We write as the corresponding equivalence class. And thus define

Reals

-Cauchy Equivalence

For two sequences and in , we say that and are -Cauchy equivalent if

Proposition

-Cauchy equivalence is an equivalence relation.

Proof Clearly it is reflexive as . It is symmetric sinceSuppose and . For all , exists such thatLet . Suppose , we have:Hence , proved transitivity.

Real Numbers

The real numbers is defined as a field of equivalence classes of -Cauchy sequences. We write as the corresponding equivalence class. And thus define:

Proposition

Every Cauchy sequence of real numbers converges to a real number.

Interval

A set is an (closed) interval if , and imply .

Archimedean Property

Archimedean Property

In any ordered field , define such that respect and order in . We say that it has Archimedean property if one of the following equivalent properties hold:

• is not bounded above by any element of .
• For all with , there is such that .
• For all with , then there is such that .

We shall check they are equivalent.

Proof Suppose is not bounded above in . Let with . This implies that is also positive. Since is not bounded above, there must exist an integer such that . As , we can multiply both sides by to get . Then, since is a positive integer, exists and is positive, so we can multiply by to obtain . Suppose that for any with , there exists an such that . Now, consider any where .

• If , we can choose . Then , so the property holds.
• If , then the element is also positive. By our assumption (2), there must be an integer such that . Since both and are positive, we can multiply the inequality by to get .

Suppose that for all with , there is an such that . To show that is not bounded above, we must show that for any arbitrary element , there is a natural number such that . Let and . Since , our assumption (iii) guarantees the existence of a such that . Since was an arbitrary element of , no upper bound for can exist in .

Proposition

The Field of Rational Numbers is Archimedean.

Proof By property (1), we must show that is unbounded in . For all , with , pick a representative with in . By Archimedean property of , there exists such that and . Thus Therefore , that is . Hence has the Archimedean property.

Proposition

Archimedean property holds for .

Proof Suppose is the canonical map such that maps to the limit value of in if it is convergent. Clearly is surjective. By Archimedean property of , is not bounded by , thus not bounded by . Therefore Archimedean property holds for as well by (2).

Remark

The Archimedean property is often taken as an axiom for . For any with , the value is a positive real number. A basic property of the real number system is that for any real number, there is a greater integer. Thus, there exists an integer .

Complex Numbers

Although is good enough, it is not quite sufficient, for example, it is not algebraically closed. So complex numbers come into play. There are mainly two ways to construct complex numbers. The first one is to define them as pairs of real numbers (See here), and the second one is to define them as equivalence classes of polynomials (See here). The first one is more intuitive (geometric), while the second one is more algebraic.

Following convention, we take the first as our definition:

The field of complex numbers, denoted by , is the usual Euclidean space endowed with the additional operation of multiplication of vectors defined as follows: And we write for , for . Therefore denotes .

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