Real number

In Mathematical, the real numbers (Unicode characters: ℝ) can very informellement be conceived like all the Nombre S associated with physical Length S or sizes. These are the numbers, that they are positive, negative or null, having a finished or infinite decimal representation. In other words, in fact the rational (can be written in the form of fraction) supplemented by the numbers of which the decimal representation is infinite not periodical, the such square Racine of 2 and π. The latter are called irrational numbers. Among the real numbers one also distinguishes the algebraic numbers and the transcendent numbers.

The term of real number appears for the first time at Georg Cantor in 1883 in its publications on the bases of the set theory . It is a rétronyme, given in answer to discovered imaginary numbers. The real numbers are in the center of the mathematical discipline of the real Analyze, to which they owe a great part of their history.

The original notation of the whole of the real numbers is $\ textbf {R}$ . However, the fatty letters being difficult to write on a table or a sheet, the notation $\ mathbb {R}$ was essential.

In the everyday life

The real numbers can represent any physical measurement such as: the price of a product, duration between two events, the altitude (positive or negative) of a geographical site, mass of an atom or the distance from most remote from the galaxies. Part of the real numbers is used the every day, for example in economy, data processing, mathematics, physics or engineering.

Most of the time, only certain subsets of realities are used:

positive whole ,
relative whole ,
the rational numbers, exprimables in the form of Fraction S with Numerator S and Denominator S entireties,
the decimal numbers, which are realities that one can write exactly in Base 10;
the algebraic numbers, which include/understand in particular all the numbers that one can write by using the four elementary operations and the root S.
the calculable numbers, which include/understand the near total of the numbers used in science and engineering (in particular E and π).

Although all these subsets of realities are of infinite cardinal, they all are Dénombrable S and thus represent only one negligible part of the whole of realities. They have each clean property. Two are particularly studied: the rational numbers and the algebraic numbers; one calls “irrational” realities which are not rational and “transcendent” those which are not algebraic.

In science

The Physique uses the real numbers as unit of measurement for two essential reasons:

the results of a calculation of physics frequently use numbers which are not rational, without the physicists not taking into account the nature of these values in their reasoning.
science uses concepts like instantaneous speed or acceleration. These concepts result from mathematical theories for which the whole of realities is a theoretical need. Moreover, these concepts have strong and essential properties if the whole of measurements is the space of the real numbers.

On the other hand, the physicist cannot carry out measurements of infinite precision. The digital representation of the result of a calculation can be approximate as precisely as it wishes it by a decimal number. In the actual position of physics, it is even theoretically impossible to carry out measurements of infinite precision. This is why, as well for experimental needs as theoretical, if the physicist calculates the measures to $\ R$ , it expresses the numerical results in the form of decimal numbers.

Thus physicist uses properties of real numbers which makes it possible to give a direction to measurements which it carries out and offers of the powerful theorems to show its theories. For the numerical values, it is satisfied with the decimal numbers. When it measures the distance which a solid traverses on a complete circle, it uses the value π without putting question about its existence, but an often small number of decimals is enough for him for calculations.

Lastly, although the real numbers can represent any physical sizes, and although this space has often more measurements than it is not possible to use some, the real numbers are not adapted to work on very many physical problems. “Supersets” built around realities were created to be able to handle certain physical spaces. For example:

space $\ mathbb {R} ^n$ , to model spaces, for example of dimension 2, 3 (or more);
the whole of the Complex numbers of which the structure has properties stronger than that of the whole of the real numbers.

Technological considerations

The real numbers can be represented in the form of a decimal Développement infinite. In theory, any size can thus be represented kind. In practice, these numbers with infinite decimal development are not adapted to calculations and on Ordinateur S. the economists are not representable and the engineers use them in a round form, while truncating or by rounding the infinite decimal development. Typically the commercial make a round-off with two digits after the comma.

The Data processing specialist S, although having of the standard of data such as the Floating decimal point ( float or double in Pseudo code English) and the Fixed point use also only approximations adapted to data-processing calculations. To represent exactly certain realities on a computer, it would be necessary to have of a infinite memory or a Processeur dedicated to the calculations symbolic systems.

First remarks on the concept of “infinite decimal development”

Any real number can be represented in the form of “number to decimal Développement infinite”. This definition can seem simpler than others usually used by the mathematicians. However, it quickly seems little adapted and implies definitions and demonstrations much more complex. Indeed the real numbers are interesting for the structure and the properties of the unit which they form: addition, multiplication, relation of order, and the properties which bind these concepts. These properties are badly reflected by the definition “infinite decimal development” and of the theoretical problems appear:

Certains numbers has two representations.

For example, the x=0,9999 number… (the 9 continue ad infinitum), the equation 10x = 9+x checks. The y=1,000000 number… (the 0 continue ad infinitum) is also solution. However the existence and the unicity of solution to this equation are two essential properties for a univocal definition of realities. To rectify this situation, it becomes necessary to identify the decimal representations which are solutions of the same equation: the definition becomes more complex.

Utiliser a decimal development makes play a part particular to the bases 10.

This difficulty is not insurmountable. It is solved by the use of an unspecified base: one speaks then about developments in base p. It is then possible to show that the units built starting from these bases are isomorphous and that the properties of the real numbers are valid in all these bases. However the demonstrations become heavy, and the definition loses its simplicity.

Finally the natural algorithms to carry out a Addition or a Multiplication, find their limit because of double representation of the decimal numbers.

Indeed, “reserves” are calculated line towards the left, and an effective algorithm requires to treat only one finished number of decimals, i.e. to truncate the numbers on which one calculates: it may be thus that while truncating also far one wants, one never has the least exact decimal, for example on calculation 0,33… +0,66… =1 . To overcome this difficulty requires to call upon concepts of Convergence, which naturally bring worms of other modes of definition of realities.

However, once established the structure of the whole of the real numbers, the notation by decimal development allows effective calculations, while keeping in mind that in fact so much the exact decimals of a number count, that the position of the number with respect to other realities.

Historical aspect

Origin of the numbers

Installation of the fractions

Since the Antiquité the representation of a measurable size met a need. The first answer was the construction of the Fraction S (quotient of two positive entireties). This solution, installation very early at the Egyptian Sumériens and the , is finally powerful. It makes it possible to approach an unspecified length with all the desired precision.

Correspondence with lengths

The first formalization built in system which one knows is the fruit of the work of Euclide to the III E Its construction, registered in the Éléments of Euclide , brings two great ideas of a major contribution in the history of mathematics.

mathematics is formalized with axioms , Théorème S and Démonstration S. One can then build a system, with theorems whose demonstrations are based on other theorems. Mathematics is classified of categories, the Géométrie and the Arithmétique is two larger. To speak about construction takes all its direction then.

a bridge is built between the two main categories. This step, making it possible to use results of one of the branches of mathematics to light another branch is more fertile. The Nombre S are then put in correspondence with lengths of segments.

Problems of incomplétude

Irrationality of the square root of 2

The approach of Euclide highlights the first contradiction between the concept of number of the time - the fractions - and the role which is allotted to them, the representation of a measurable size.

a length whose square is equal to 2 exists . A geometrical reasoning, already old at the time of Euclide, watch which it is possible to build a square B of surface double of that of an initial square has that one chooses side equal to 1. If one notes $l$ the length on the side of the square B, which is equal to the length of the diagonal of the square has, the equality $l^2=2$ is then checked.

a length whose square is equal to 2 does not exist in the form of fraction. Some results are already known into arithmetic, for example the Lemme of Euclide. Starting from this lemma one shows that no number can be the square Racine of 2. Here, number means nonnull Fraction positive because no other formalization is yet conceivable.

The Elements of Euclide are based on axiomatic which seems to make it possible to prove at the same time that a proposal is true and false. More than two millenia will be necessary to humanity to solve this apparent contradiction, to explain why the rational ones represent only imperfectly the real line and to find how to represent them well.

It should be noted that three centuries before Euclide, Pythagore probably knew the irrationality of certain roots. On the other hand, the first formalization in true a mathematical Corpus built comes us from Euclide.

Unlimited decimal development nonperiodic

If the fractions make it possible indeed to express any length with the desired precision, it should nevertheless be understood that the operations and particularly division become complex if the Numbering system is not adapted. The problem is described by the article Egyptian Fraction which proposes some concrete examples.

It is necessary to wait the 5th century to see the Indian school discovering the concept of the Zero and developing a positional Numbering system decimal and .

A second problem appears then. All the fractions have a decimal Développement insofar as this development is infinite and periodic, i.e. the continuation of the decimals does not stop but buckles on a finished number of values. The question of knowing which direction to give to an object characterized by a nonperiodic succession of decimals. For example, the number with infinite decimal development which is expressed as

0,1010010001… where the number of 0 between figures 1 grows indefinitely, does it correspond to a length?

Continuations and series

In second half of the 17th century, one witnesses an extraordinary blooming of mathematics in the field of the calculation of the series and the continuations.

Nicolaus Mercator, the Bernoulli, James Gregory, Godfried Leibniz, and others work on series which seem to converge but whose limit is not rational. It is the case for example:

of the series of Mercator: $\ sum_ {k=1} ^ \ infty {(- 1) ^k \ over K} = 1 - \ frac 12+ \ frac 13 - \ frac 14 + \ cdots$ which converges towards $\ ln (2) \,$
of the series of Grégory: $\ sum_ {k=0} ^ \ infty {(- 1) ^k \ over {2k+1}} = 1 - \ frac 13+ \ frac 15 - \ frac 17 + \ cdots$ which converges towards $\ pi/4 \,$

Worse, Liouville in 1844, proves the existence of transcendent numbers i.e. not root of a polynomial with whole coefficients. It is not thus enough to supplement the rational ones by adding the algebraic numbers to it to obtain the whole of all the numbers.

of the series of the type $\ sum_ {k=1} ^ \ infty \ frac {a_k} {10^ {K!}} = \ frac {a_1} {10^ {1}} + \ frac {a_2} {10^ {2}} + \ frac {a_3} {10^ {6}} + \ frac {a_4} {10^ {24}} + \ cdots$ representing the numbers of Liouville, where $(a_n)$ is a succession of entireties ranging between 0 and 9.

Infinitesimal calculus

During the second part of the 17th century, Isaac Newton and Gottfried Wilhelm von Leibniz invent a all new branch of mathematics. It is called now the analyzes, at the time it was known under the name of Infinitesimal calculus. This branch acquires an immense fame almost immediately because it is the base of a very new universal physical theory: the theory of the Newtonian revolved. One of the reasons of this fame is the resolution of an old question, namely if the Ground turns around the Sun or the reverse.

However the infinitesimal calculus cannot be shown rigorously in the whole of the rational numbers. If calculations are right, they are expressed in a language of a great complexity and the evidence proceed more geometrical intuition than of a rigorous clarification within the meaning of our time.

The impossibility of the construction of the analysis in the whole of the fractions lies in the fact that this branch of mathematics is based on the analysis of the infinitely smalls. However, one can compare the rational numbers with an infinity of small grains of sand (of infinitely small size) on the real line leaving infinitely more holes that of matter. The analysis cannot be satisfied with such a support. She asks for support a complete Espace. The word is used here in a double direction, the intuitive direction which means that the infinity of small holes must be stopped and the direction which the mathematicians give today more abstract but rigorously formalized.

This concept is so important that it will become at the dawn of the 20th century a broad branch of mathematics called Topologie.

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