Expanding of fields

Numeric fields

Numeric fields - endless, single-component:
  N - natural numbers (i.e. positive integers excluding zero)
  N0 - positive integers including zero
  P - prime numbers
  Z - whole numbers
  Q - rational numbers p/q, pεZ, qεN
  R – real numbers

Basic numeric fields

A set of numbers of certain properties is called a numeric field. Primordial series of natural numbers N = { 1,2,3,4,5 , ....} was first supplemented by a zero to N₀={0,1,2,3,4,5,....} ( zero began as the first to use Chinese, Indians , and Mayans ...). Series of integers Z = ={..,-3,-2,-1,0,1,2,...} arised from N₀ by adding of the negative values. The integers supplemented by mutual ratios (fractions) makes rational numbers Q = p / q where pεZ , qεN. Natural numbers that do not have divisors other than number 1 and themselves are called primes P.

Square root of numbers is generally not possible to express in the form of ratios: there are irrational numbers (eg. √2, √3, π, ...) apart rational numbers. All the complex of (single-component) numbers (rational+irrational) got designation the real numbers (R).

Writing numbers

Basics of modern mathematical symbolism was laid by Arab mathematician al-Chvárizmí and the Italian mathematician Fibonacci. In the decimal system (ie. the system base 10) with an enrollment of 243 means (2,4,3)₁₀ = 2∙10²+4∙10+3. In the system based on 5 the number 243 would mean (2,4,3)₅ = 2∙5²+4∙5+3 = 73, i.e.(7,3)₁₀. Generally, the number (2,4,3) _n can be written as 2n²+4n+3, where n is the base of the system (nεN). Exponents t of the members n^t indicate the orders of the particular symbols. In the number 243 has symbol 2 order 2 (hundreds), symbol 4 order 1 (ten) and symbol 3 order 0 (units).

Numerous mathematicians participated in the next completation, simplification and clarification of the symbolism - Greats (F.Viéte, R.Descartes, G.W.Leibnitz, L.Euler,...) as well as less known scholars (R.Recorde, c. 1510-1558, introduced the sign '=', T.Harriot, 1560-1621, the signs '>' and '<',...).
Field of natural numbers extends to whole numbers (using sign before writing number), rational numbers (by permission of negative orders), and real numbers (allowing an infinite number of negative orders).

E.g. number −243,67 = −(2∙10²+4∙10+3+6∙10^-1+7∙10^-2).

Algebraic structures

Algebraic structures:
   G - structures with one operation (groups, groupoids, ..)
   T - structures with two operations (bodies, circuits ..)

In modern algebra a set of elements with permissible operations are being used instead of fields. Relationships defined on the sets are called a relations. If the relation has assignment function, it is called mapping. Operation is a special type of mapping - it assigns result to specified numbers by certain prescription.

Algebraic structures with one operation are derived from the elementary structure, which is called grupoid [Borůvka]. Each result (v) of the given operation with arbitrary elements of the groupoid must be specified clearly and belong to groupoid (vεG). For structures satisfying other conditions, special (sometimes quirky) names (semigroup, quasigroups, loupe, monoid, ...) are used. Conditions means e.g. having a neutral element, associative operations, operations of reduction and division and so on.

Structures with operations of addition are called additive, with operations of multiplication are called multiplicative.

Basic algebraic structure with two operations is a circuit. Circuit O(+ ∙) is formed by two pervading grupoids (additive and multiplicative).

Circuit which meet other conditions is called body of integrity or solid. Numbers whose product is 0 are called a zero divisors. The existence of divisors of zero excludes that the circuit can be a field of integrity (and thus the solid).
Number fields R, Q and C with the operations of addition and multiplication are solids.

Operations

 Associative operations - operations that does not depend on bracketing of elements,
       i.e. (ab)c=a(bc), resp. a(b(cd)) =a((bc)d) =(ab)(cd) =(a(bc))d =((ab)c)d, and the like.
 
 Commutative operations - operations that does not depend on the order of elements,
       i.e. ab = ba, for every two elements a, b.
 
 Alternative operation - a special type of operation,
       it applies: (aa)b = a(ab), resp. (ba)a = b(aa).

Various operations in different algebraic structures (numeric fields, ...) may behave differently. For example the following properties of operations were named:

Usual operations - addition, subtraction, multiplication, division, expansion, reduction - are in the fields of (Z, R, ..) associative and commutative.
Generally, however, such operations may not have this property, even some operations (shortening, cutting, ..) become irrelevant in certain structures. If operation is not commutative, we distinguish the left operation and right operation.

Functions

Display is a function (f) if there is just one result to the given numbers (the arguments of the function).

  • Zero element 0 - acts as the number 0 in addition
  • Unit element - acts as the number 1 in multiplication. It is denoted usually as 1, e or I, ...
  • Annihilating element has properties similar to 0 in the multiplication 0 (0 * x = 0).
  • Idempotent element with its own product (such as 1 at multiplication) or sum (such as zero in addition).
  • Contrary element (contrasting element) to the element x is such an element x', for that is x + x' = 0
  • Inverse element to element x is such an element x', for that is x * x' = 1 

Special elements

Operation on the algebraic structure need no to be addition nor multiplication. Therefore more general terms are introduced:

Neutral element means a generalization of zero and unitary element.

(Otherwise, often, even though it is sometimes confusing, uses concepts derived from the multiplication operation).

Expansion of fields

Superior fields

When solving algebraic equations a∙x^k+ b∙x^k-1+ c∙x^k-2+ ...+p∙x +q = 0 shows that when the coefficients a, b, c, .. z are from a numeric field T it is not always possible to find x from this field. The equation can be in such cases satisfied only by values, which belongs to another field U, which contains the T (U is a superset of T).

E.g. equation 10x-3 = 0 can not be granted in the field of whole numbers T = Z, but it has a solution in the field of rational numbers Q Z. In doing so, all rational numbers are not necessary for the solution. Is sufficient to supplement to the field Z the number of 0.1 (ie a solution equation 10x-1 = 0) and all its combination (products) with all integers. Such over-field of T=Z represents T(0.1) and we speaks about supplement (adjunctions) of element 0.1 to the field T.

What mathematical field covers all possible solutions?

Complex numbers

Intuitive estimate that the largest over-field must be real numbers fails.

E.g. equation x²+1 = 0 has no solution in the field R and defines a new element x = √(-1) = i. Supplementing it to the field R, we can write all the numbers in the form:

 a₀ + a₁x = a₀ + a₁i 

This creates field C=R(√−1)=R(i).

Numbers of the form a+bi are called complex numbers, numbers of the form a+b√d (where a,b,d εR) quadratic numbers.

Complex numbers were introduced in the theory of cubic equations, only later they began to be used also in quadratic equations.

Generally numbers that which may be derived as roots of algebraic equation, are called algebraic numbers, their opposite are transcendental numbers. Algebraic numbers are subsets of complex numbers.

According to so-called fundamental theorem of algebra (see Chapter X) field of complex numbers cover all existing solutions of algebraic equations.

Numeric fields - infinite, two-component:
  C - complex numbers a+bi, i = √-1
  A - quadratic integers a+b√d

Number a+bi is called whole (racional) complex number, when each of numbers a,b is whole (racional). Similarly in case a,b,d εZ numbers of the form a+b√d are called quadratic whole numbers. In the case, when a,b are of equal parity, quadratic whole numbers can have also form (a+b√d)/2 (a,b,d εZ).

Multicomponent numbers

Complex numbers are - in contrast to all the previously mentioned numbers - bi-component numbers. After adding the element given by x³−2 = 0 to field Q, we can rewrite all the numbers to form a₀ + a₁x + a₂x². 

E.g. x⁴+x³ +x²+x+1 = x³ (x+1)+ x²+x+1=2(x+1)+ x²+x+1= x²+3x+3 = [1,3,3].

Numeric fields - endless, multicomponent:
     Hamilton's quaternions
     K - Kummer's cyclical numbers α+bα+cα2+..

More successful generalization of complex numbers proved to be cyclical Kummer (cyklotomic) numbers f(α) =[a,b,c,..q]_k =a+bα+cα²+..qα^k−1,  where α= .

Values α^j make up points on the unit circle in the plane of complex numbers, divides the circle into k parts. (see Binomial equation).

Kummer was able to continue in Gauss generalization - some basic relations (divisibility, law of reciprocity, ...) which was generalized and transferred from field N to field C by Gauss, Kummer moved further into the field of cyclic numbers K.

Numeric associations

Units

Two units are in the field of real numbers: +1,−1, that divide numbers into two main groups - positive and negative. Positive numbers are considered to be primary, primary shape of the given number is its absolute value. Four units are in the field of complex numbers: +1,−1,+i,−i, i.e. while writing into pairs (a,b): (1,0),(−1,0),(0,1),(0,−1). Both cases can be united with introduction of the so-called norm N of complex numbers:

 N=a²+b² 

Complex number a+bi is unit, when its norm has value 1. The size of the complex number means the square root of its norm. In real numbers (a + bi, where b = 0) corresponds size √a² to absolute value.

For unification with quadratic numbers a + b√d is necessary to modify the calculation of the norm:

 N=a²−b²d 

Here for d=−1, ie. a+b√d = a+bi the previous relation holds. Thus derived units of quadratic numbers look at least strange, eg. for quadratic whole numbers:

Number (9,4) is unit among numbers of the form a+b√5 because N(9,4) = 9²−4²5 = 81−80 = 1.
In the case of whole numbers (a+b√d)/2 is moreover necessary to count with alternatives N(a,b) = +4, and N(a,b) = -4, whence comes the next set of units.

Associated and conjugated numbers

All numbers that arise from the given numbers by multiplication by units are associated with each other. Associate numbers is at most as many as units:

· In the field of real numbers are two units: +1 a −1, each two real numbers +a, −a are associated.

· In the field of complex numbers are four units: +1,−1,+i,−i. Associated are complex numbers a+bi, −a−bi, −b+ai, b−ai.

(number zero is associated with itself, special case is also a=0, b=0 or a≡±b)

Generally, there are eight different numbers with the same norm: a+bi, −a−bi, −b+ai, b−ai, a−bi, −a+bi, −b−ai, b+ai. Numbers a+bi and a−bi are called conjugated.

Bound numbers (numbers with the same norm) are all numbers associated with two conjugated numbers a+bi and a−bi:

  Complex conjugated numbers 
 ─────────────────────────
  a+bi  =>   a−bi  *
 −b+ai  =>  −b−ai  *  Associated 
 −a−bi  =>  −a+bi  *  numbers 
  b−ai  =>   b+ai  *  

Norm of complex numbers is product of conjugated numbers a+bi and a−bi:

 N=a²+b²=(a+bi)(a−bi) 

Similarly norm of kvadratic number

 N=a²−b²d =(a+b√d)(a−b√d) 

Primary numbers

In the field of real numbers is product (and also division) of the two primary (ie positive) numbers always primary (positive) number. A similar attempt - to allocate one group of four associated groups - with complex numbers fails. Product of every two primary numbers should belong to the group of primary numbers, primary numbers must form a subgroup (see chapter X) of all complex numbers. Such a subgroup can be found only for a whole complex numbers and it acquires meaning only for odd numbers (Gauss, II). Gauss presents these two options for choosing of primary numbers   a+bi:

The product of two numbers of the same form (I/ or II/) does not change the form.

Association of cyclic numbers

   2  0  0  2  5
   0  2  2  0  5
  ──────────────
  10  0  0 10 25
   0  0  0  0  0
   0  4 10  4  0
   4 10  4  0  0
   0  0  0  0  0
  ──────────────
  14 14 14 14 25

Cyclical numbers are equal when their coefficients are identical, or differ by a constant.

 [a₀,a₁,a₂,..]=[a₀+c,a₁+c,a₂+c,..] 

All cyclical numbers F=f(α), f(α²),.. f(α^k−1) are associated, there is always k-1 associated cyclic numbers.

Norm of cyclical numbers has been introduced as a product of all the associated numbers:

 N(F)= f(α) f(α²),.. f(α^k−1) 

The value of norm is always an integer.

E.g. number norm F=[2,1,0,0,0]_k is calculated:

N(F)=(α+2)(α²+2)(α³+2)(α⁴+2)=(2α⁴+2α+5)(2α³+2α²+5) N(F)= 14α⁴+ 14α³+ 14α²+ 14α + 25 = 11

For cyclic numbers [a,b,0,0,..]_k it is possible simplify the calculation of the norm:

 N(F)= (a^k+b^k)/(a+b) 

ie. in our example:

   N(F)= (2⁵+1⁵)/(2+1) = 33/3 = 11