Numeric system

In arithmetic, algebra, and mathematical analysis, a number system is a set provided with two operations that satisfy certain conditions related to the commutative, associative, and distributive properties. The set of integers, rational numbers, or real numbers are examples of number systems, although mathematicians have created many other, more abstract number systems for various purposes. It should also be taken into account that given a number system there are various ways to represent it, for example, in integers we can use decimal, binary, hexadecimal representation, etc. In the rationals we can choose not to express them in a decimal way or as a fraction of integers, etc.

Number systems are characterized by having an algebraic structure (monoid, ring, field, algebra over a field), satisfying order properties (total order, good order) and additional topological and analytical (density, metrizability, completeness) properties.

Introduction

Conventionally various sets endowed with "addition" and "multiplication" They are called number systems. Among these sets are the natural numbers, the integers, the rational, the real and the complex numbers, although there are others that generalize some of the above. Although there is no formal definition of a number system, all sets endowed with binary operations that are conventionally counted among the number systems have common properties.

In all conventional numerical systems there are defined two associative binary operations called addition and multiplication, and it is also fulfilled that multiplication is distributive with respect to the addition. The addition is always commutative, although in some numerical systems multiplication is not always commutative): For a, b and c any elements S{displaystyle mathbb {S} }:

• Switching property of the addition: a+b=b+a{displaystyle a+b=b+a}
• Associative property of the addition: (a+b)+c=a+(b+c){displaystyle (a+b)+c=a+(b+c)}
• Associative property of multiplication: (a⋅ ⋅ b)⋅ ⋅ c=a⋅ ⋅ (b⋅ ⋅ c){displaystyle (acdot b)cdot c=acdot (bcdot c)}
• Distribution property of multiplication on addition: a⋅ ⋅ (b+c)=a⋅ ⋅ b+a⋅ ⋅ c{displaystyle acdot (b+c)=acdot b+acdot c}

Addition and multiplication do not necessarily have to be those of elementary arithmetic.

More formally a numerical system is characterized by a sextupla (S,+,⋅ ⋅ ,A,O,T){displaystyle scriptstyle (mathbb {S}+,cdot{mathcal {A}},{mathcal {O}},{mathcal {T}}}}where:

A{displaystyle scriptstyle {mathcal {A}}} is a set of axioms that define the algebraic properties of operations and conjecture the possible existence of certain types of elements (opposed, inverse, etc.)

O{displaystyle scriptstyle {mathcal {O}}}} is a set of axioms referring to the theory of order, which give account of certain properties associated with the existing relations between the elements.

T{displaystyle scriptstyle {mathcal {T}}} is a set of topological axioms, which possibly include the definition of certain functions (distance) and properties (completeness, density, etc.)

Examples according to algebraic structure

Number systems with ring structure

• The integers Z{displaystyle mathbb {Z} } are one of the simplest examples of rings.
• Integers module n (where) nI was. I was. pq{displaystyle nneq p^{q}, with p an integer prime).
• Gaussian integers Z+iZ{displaystyle mathbb {Z} +imathbb {Z} }

• The p-addic integers Zp{displaystyle mathbb {Z} _{p}} which are much more numerous than ordinary integers.

Number systems with field structure

• The rational numbers (Q{displaystyle mathbb {Q} }), minimum body containing the ring (Z{displaystyle mathbb {Z} }).
• Algebraic numbers (A{displaystyle mathbb {A} }) are an algebraic extension of the rational number (Q{displaystyle mathbb {Q} }).
• The real numbers (R{displaystyle mathbb {R} }), minimum full body containing a Q{displaystyle mathbb {Q} }
• Complex numbers (C{displaystyle mathbb {C} }), minimum algebraicly closed body containing a R{displaystyle mathbb {R} }
• Integers module p (with p cousin, Zp{displaystyle mathbb {Z} _{p}}) or modular arithmetic module p.
• Hyperreal numbers (↓ ↓ R{displaystyle} {^{mathbb {R} }) are an extension of the real numbers (R{displaystyle mathbb {R} }).
• Superreal numbers are a generalization of hyperreal numbers.
• Surreal numbers are the largest possible body that contains the real and remain an orderly body.
• The p-addic numbers (Qp{displaystyle mathbb {Q} _{p}}) are a non-archimedean body that are a metric completion of the rationals, different from the completion of the actual numbers, since they are based on an ultrametric.
• The P-Adic algebraic (Q! ! p{displaystyle {bar {mathbb {Q}}_{p}}}) form the algebraic closing of the p-addic numbers, although the metric completion is lost.
• The p-addic complexes (Cp{displaystyle mathbb {C} _{p}}) form the metric completion of Q! ! p{displaystyle {bar {mathbb {Q}}_{p}}}.

Number systems with algebra structure

• Quaternion numbers
• Octonionic numbers
• The headquarters numbers

All of these sets are examples of hypercomplex numbers.

Discussion of the examples

Intuitive examples

Most examples of simple number systems are related to extensions of the natural numbers:

• The whole numbers, (Z,+,⋅ ⋅ ){displaystyle scriptstyle (mathbb {Z}+,cdot)} generalize the idea of counting and permit formalizing the concept of debt or defect of something, that is, in them can formalize operations like "4 - 7", etc. This numerical system has a unitary connective ring structure, a trivial discrete topology. The order properties are relatively simple as any set is finite has a minimum element and a maximum element belonging to that set.
• The rational numbers, (Q,+,⋅ ⋅ ){displaystyle scriptstyle (mathbb {Q}+,cdot)} In addition to the idea of debt or defect of something, the notion of portion of something, that in turn implies more complicated topological properties, such as the one between two rational numbers always exists at least another rational number. That makes it topologically complicated to the rationals since a coupled group does not have to have a maximum or a minimum (although it is a top and a bottom). Algebraically the rationals have body structure. The main difference with reals is that rationals are not a topologically complete set.

The remains of module 2

The remains of module 2, with the operations of sum and multiplication of remains, form a numerical system. Gauss' congruence is an equivalence relationship. The quotient of the whole Z{displaystyle mathbb {Z} } by a ratio of equivalence divides it into disjoint classes. In the case of module 2 congruences what is done is to divide the integers into pairs and odd numbers. Defined sum and multiplication operations can respond to what parity is the result of a sum or multiplication of pairs or odd numbers, in any combination that is used. The "0" and "1" symbols represent the possible remains of the entire division by 2: 0 for the pair numbers and 1 for the odds. The expression 1 + 1 = 0 is equivalent to: odd + odd = pair.

With the tables it is easy to verify that the operations are commutative, associative and that the product is distributive with respect to the sum. We have, then, a two-symbol number system. For a deeper understanding, see modular arithmetic.

Examples based on order properties

Totally ordered number systems

• Natural N{displaystyle mathbb {N} }, the integers Z{displaystyle mathbb {Z} }the rational Q{displaystyle mathbb {Q} } and the real R{displaystyle mathbb {R} } are examples of totally sorted sets.
• Gaussian integers or complexes are not a fully ordained set, as a total order compatible with arithmetic operations cannot be defined. That fact follows the hypothesis that i  0 as i ≤ 0 lead to a contradiction, if it is admitted that the proposed order is non-trivial and compatible with multiplication.
• Neither integers module n do not admit any total order compatible with the sum since being cyclical groups regarding the sum. Since a  0 should involve two things that their opposite additives - Yeah. 0 plus add a finite number of times a It implies n·a  0, but since (n-1)·a =a, it comes to a contradiction, being the first postivo member and the second negative.

Well-ordered number systems

• Natural numbers N{displaystyle mathbb {N} } are an example of numerical system which is also a well-ordained set.
• The integers are not a well-ordained set, although any integer subset is finite and therefore is also a well-ordained set.
• Rational and real numbers are not a well-ordained set. Not even those of any subset collected from rational or real numbers is a well-ordained set. For example, the open interval (0.1) is a subset arranged in both rational and real but does not have a minimum element belonging to the set, since 0 is not an element of that subset.

Number systems with dense order

• Neither natural numbers, nor integers have a dense order, since two consecutive numbers such that among them there is no other element. For example, there is no other integer between 2 and 3.
• Instead the rational and the real have a dense order, given two different numbers r₁ and r₂ There is always some other number among them for example (r₁+r₂)/2 or (2r₁+r₂)/3