Arithmetic

History

The first arithmetic traces of operations are found in Africa, in: 18000 front J. - C. on the Os of Ishango. This bone would have been notched in order to carry out additions and subtractions.

In the school pythagorician (Pythagore of Samos), with second half of the 6th century before J. - C., the arithmetic one was, with the Géométrie, the Astronomie and the Musique, one of four quantitative or mathematical sciences ( Mathemata ). Those were gathered by the Romains under the name of Quadrivium which was considered, with the rather logical Trivium (Grammaire, Rhétorique, Dialectique), like the liberal septem artes (seven Liberal arts).

Arithmetic elementary

Units used into arithmetic

The totality of the Nombre S were gathered in the most known Ensemble S. are:

$\ mathbb {NR}$ : the whole of the natural whole ( $0; \, 1; \, 2; \, 3; \, 4;$ etc)
$\ mathbb {Z}$ : the whole of the relative whole ( $-12; \, -2; \, 0; \, 5; \, 6;$ etc)
$\ mathbb {D}$ : the whole of the decimal numbers, i.e. which are written with a finished number of decimals $\ left (- \ frac {1} {2}; \, 6,36; \, 0; \, 25; \ mbox {etc} \ right)$ .
$\ mathbb {Q}$ : the whole of the rational numbers, i.e. numbers being able to be written like the fraction of two decimal; the number of decimals can be infinite but must be periodic. $\ left ({1 \ over3}; \, - {5 \ over13}; {22 \ over7} \ mbox {etc} \ right)$ .
$\ mathbb {R}$ : the whole of the real numbers, i.e. those whose imaginary Partie is null ( $\ pi$ , the golden section, $\ sqrt 2$ )
$\ Im$ : the whole of the pure imaginary numbers, i.e. those whose real Partie is null (I for example)
$\ mathbb {C}$ : complex numbers which gather all the numbers, that they are real, imaginary, or a combinaision of both.

Some of these units are Sous-ensemble S of the others; All the elements of $\ mathbb {NR}$ also belong to $\ mathbb {Q}$ , for example. But contrary, an element of $\ mathbb {Q}$ is not inevitably element of $\ mathbb {NR}$ . One can represent this whole by concentric circles: smallest is $\ mathbb {NR}$ , then come $\ mathbb {Z}$ , $\ mathbb {D}$ , $\ mathbb {Q}$ , $\ mathbb {R}$ and $\ mathbb {C}$ .

It is possible to consider only part of a unit. Thus, one will note $\ mathbb {R^+}$ the whole of the positive numbers of $\ mathbb {R}$ . In the same way one will note $\ mathbb {R^*}$ the unit $\ mathbb {R}$ private of 0. It is noticed amongst other things that $\ mathbb {Z^+} \, = \, \ mathbb {NR}$ and that $\ mathbb {Z} \ backslash \ mathbb {NR} \, = \, \ mathbb {Z^ {- *}}$ (it is about $\ mathbb {Z}$ “private of” $\ mathbb {NR}$ .)

Properties

The numbers and their combinations have many remarkable properties.

It is the case of the numbers known as first. They are elements of $\ mathbb {NR}$ having only two distinct positive dividers, namely 1 and them-even. The first prime numbers are 2; 3; 5; 7; 11; 13; 17 etc 1 is not first because it does not have 2 distinct dividers, but only one. There exists an infinity of prime numbers. By supplementing a grid of size 10 $\ times$ 10 with the first 100 nonnull entireties, and by striping those which are not first, one obtains the prime numbers pertaining to $\ {1, \ ldots 100 \}$ by a process called a screen of Eratosthène, name of the Greek scientist who invented it.

The natural entireties are divided into two well-known categories of the players of caster: pars and impairs.
An even entirety $n$ is a multiple of 2 and can be noted $n = 2 \, k$ , with $k \ in \ mathbb {NR}$ An odd $n$ number is not multiple of 2 and notes $n = 2 \, K + 1$ , with $k \ in \ mathbb {NR}$ .

It is shown that entire is either even or odd, and at least one of both, and this for single a $k$ : one notes $\ forall N \ in \ mathbb {NR}, \, \ exists! K \ in \ mathbb {NR}, \, \ left (n=2 \, K \ lor n=2 \, k+1 \ right)$
The first even entireties are 0,2,4,6,8,10… The first odd entireties are 1,3,5,7,9,11…

See too

Addition of the natural entireties
Associativeness
Commutation
Distributivité
Arithmetic Transitivity
Order of the operations
saturated
Prime number

Be-X-old: Арытмэтыка Simple: Arithmetic

Random links: