Analcime

number the concept of number in linguistics is treated with the article grammatical Nombre. ---- A number is a Concept making it possible to evaluate and compare quantities or ratios of Grandeur S, but also to order elements by a classification. The complexity of this concept is such as certain dictionaries are acknowledged impotent with the définir.
In Mathématiques, this difficulty is partially solved by the proposal: “a number is an element of a whole of numbers”. These Together S or classes of numbers extends the basic concept of number Entier naturalness in various ways to define operations, to solve equation S, to give a direction to certain approximations or to apprehend the Infini.

The number is to be distinguished from its writing, made up of one or more Chiffre S and dependant on the Numbering system employed.

Concepts to be distinguished

; Quantify A Chiffre is a character used with the writing of a number or a number. The most frequent error is to confuse the figure with the number. The figures generally used for the writing of the numbers are: {0, 1,2,3,4,5,6,7,8,9}.

; Number The numbers are used to solve problems utilizing values. But attention, a number is not a quantity. It is a mathematical Objet which answers precise laws.

; Cardinal number A Cardinal number is a type of particular number used for the enumeration of the units. They should not be confused with the numeral adjective cardinals.

; Ordinal number A ordinal Nombre is a type of particular number used to mark the order of the elements of a unit. They should not be confused with the ordinal numeral adjective . The enumeration with the ordinal numbers starts with “0”, while with the ordinal numeral adjectives it starts with first or “1”.

; Number A number is simply a combination of figures which does not respect necessarily an enumeration and generally plays the part of a numerical label

Types of numbers

There exist various types of numbers. The most familiar numbers are the whole natural: 0,1,2,3,… elements of the unit $\backslash \; N$, and used for the Enumeration.

If the negative entireties are included, one obtains the whole of the relative integers $\backslash \; mathbb\; \{Z\}$. There exists also the whole of the decimal numbers noted $\backslash \; mathbb\; \{D\}$. If D belongs to $\backslash \; mathbb\; \{D\}$, then $d\; =\; has\; \backslash \; cdot\; 10^p$ where has belongs to $\backslash \; mathbb\; \{Z\}$ and p belongs to $\backslash \; mathbb\; \{Z\}$.

The division of a relative entirety by a relative entirety not no form a rational number. The whole of all the rational numbers is noted $\backslash \; mathbb\; \{Q\}\; \backslash ,$. It results from the meeting of the whole of the numbers with decimal Développement finished (decimal numbers) and of that of the periodic numbers.

If, as a whole, in addition to the elements of $\backslash \; mathbb\; \{Q\}$, one includes all the decimal developments infinite and not periodicals, one obtains the whole of the real numbers, noted $\backslash \; R$. All the real numbers which are not rational are called irrational numbers. $\backslash \; R$ is the meeting of the whole of the algebraic numbers (roots of polynomials with rational coefficients) and of the unit of the transcendent numbers.

The real numbers can be wide with the complex numbers, whose whole is noted $\backslash \; mathbb\; \{C\}$, which is a Corps algebraically closed in which each Polynôme with complex coefficients can be completely factorized.

We thus have a hierarchy of units:

$\backslash \; NR\; \backslash \; sub\; \backslash \; mathbb\; \{Z\}\; \backslash \; sub\; \backslash \; mathbb\; \{D\}\; \backslash \; sub\; \backslash \; mathbb\; \{Q\}\; \backslash \; sub\; \backslash \; R\; \backslash \; sub\; \backslash \; mathbb\; \{C\}$

The complex numbers can, in their turn, being extended to the Quaternion S, but the multiplication of the quaternions is not any more commutative. The Octonion S, in their turn, extend the quaternions, but this time, the Associativité is lost. The Sédénion S extend in their turn the whole of octonions.

In fact, only the Algèbre S of division associative to dimension finished on $\backslash \; R$ is the real numbers, the complex numbers and the quaternions.

The elements of the Corps of algebraic functions of finished characteristic were often interpreted several manners like a kind of numbers by the theorists of the numbers.

$1+\; \backslash \; cfrac\; \{2\}\; \{$

3+ \ cfrac {4} {

5+ \ cfrac {6} {7+ \ dotsb}}} =

\ frac {1} {\ sqrt E - 1}

History

The numbers appeared in this order:

• natural whole ,
• the rational numbers positive,
• the invention of the zero,
• relative whole ,
• the rational numbers,
• the irrational numbers and the real numbers,
• the complex numbers,
• the numbers hypercomplexes (Quaternion S),
• the p-adic numbers,
• the transcendent real numbers and the algebraic real numbers,
• the transfinite numbers, consisted of the ordinal and cardinal
• the numbers hyperréels,
• the calculable real numbers,
• the surreal numbers and pseudo-realities.

It is not fortuitous: one passes in the way simplest to measure with techniques much more elaborate.

The comprehension of the limits of the rational numbers and the need of the irrational numbers was particularly painful for the pythagorician S; one even says that sealed the end of this School.

The complex numbers were essential initially like a specious but effective argument to solve the polynomial equations (from where the term of “imaginary” to indicate some of them), before finally being recognized like numbers with whole share.

The numbers hypercomplexes were invented by Hamilton (Quaternion S) then by Cayley (Octonion S) and the Sédénion S by the Construction of Cayley-Dickson. With each component of a hypercomplexe number, one can associate a bases with several dimensions (4 for the quaternions, 8 for octonions and 16 for sédénions). There exist also the Biquaternion S.

The appearance of the p-adic numbers is related to the concept of absolute Value, and are very much used in Théorie of the numbers.

The numbers hyperréels were conceived to solve certain problems of the analysis and their creation by Abraham Robinson allowed the development of the Analyze not-standard. The numbers pseudo-realities are very similar to the vaster whole of the hyperréels, but construction is different.

The arithmetic operations on the numbers, such as the Addition, the Subtraction, the Multiplication and the Division are generalized in the branch of the Mathématiques called abstract Algèbre in which one obtains the group S, the rings and the body.

See too

• Numeration;
• Construction of the number in the child
• Numbering system;
• Mathematical;
• Fraction;
• the first ten integers or figures, which are used to form all the numbers in the decimal classification: zero, a, two, three, four, five, six, seven, eight, nine;
• Prime number;
• Gogol ;
• Numbers in French;
• Name of the great numbers;
• Numbers in the world;
• List of the numbers;
• the ordinal numbers and cardinal;
• Table of the dividers
• grammatical Number.

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