Number

Chinese digits. The second millennium BC.

Emergence of numbers

Intuitive idea of number, apparently, is also old, as well as mankind though with reliability it is impossible to trace all early stages of its development in principle. Before the person learned to count or thought up words for designation of numbers, he, undoubtedly, owned the evident, intuitive idea of number allowing it to distinguish one person and two people or two and many people. The fact that primitive people at first knew only "one" "two" and is "a lot of", is confirmed by the fact that in some languages, for example in Greek, there are three grammatical forms: singular, dual number and plural. After people learned to do differences between two and three trees and between three and four people. The account was initially linked with quite specific set of objects, and the very first names of numbers were adjectives. For example, word "three" was used only in combinations "three trees" or "three persons"; idea that these sets have among themselves something the general – a concept of a troichnost – requires high extent of abstraction. That the account arose before emergence of this abstraction layer, the fact that the words "one" and "first", it is equal as "two" and "second" demonstrates, in many languages have among themselves nothing in common while the lying outside the primitive account "one", "two", "many", words "three" and "third", "four" and "fourth" clearly indicate interrelation between cardinal and ordinal numerals.

The names of numbers expressing very abstract ideas appeared, undoubtedly, later, than the first rough characters for designation of number of objects in some set. In an extreme antiquity primitive numerical entries were made in the form of notches on a stick, the nodes on a rope which are laid out in a row stones and was meant that between the recalculated set members and characters of numerical record there is a univocity. But for reading such numerical records of the name of numbers directly were not used. Nowadays we at first sight distinguish sets from two, three and four elements; the sets consisting of five, six or seven elements will slightly more difficultly be recognized by sight. And behind this border it is almost already impossible to determine approximately their number, and the analysis either in the form of the account, or in a certain structuring elements is necessary. The account on labels, apparently, was the first acceptance which was used in similar cases: notches on labels were located with certain groups just as at calculation of ballots they are often grouped packs on five or ten pieces. The account on fingers was very widely widespread, and it is quite possible that names of some numbers originate from this method of calculation.

The important feature of the account consists in communication of names of numbers with a certain scheme of the account. For example, the word "twenty three" is not just the term meaning quite certain (on a number of elements) an object group; it is the term compound, meaning "two times on ten and three". Here clearly number ten role as collective unit or basis is visible; and it is valid, many consider tens, because as still Aristotle noted, at us ten fingers on hands and standing. The bases five or twenty were for the same reason used. At very early stages of development of history of mankind were accepted to base radixes number 2, 3 or 4; sometimes for some measurements or calculations the bases 12 and 60 were used.

The person began to consider long before he learned to write therefore no written instruments confirming those words by which in the ancient time designated numbers remained. Oral names of numbers are characteristic of nomad tribes, as for written, need for them appeared only with transition to a settled way of life, formation of agricultural communities. There was also a need for a recording system of numbers, and the basis for development of mathematics was put then.

Main types of numbers

The natural numbers received at the natural account. The set of natural numbers is designated by N. Thus N= { 1, 2, 3... } (sometimes also carry zero, i.e. N to a set of natural numbers } = { 0, 1, 2, 3.... }. Natural numbers are closed concerning addition and multiplication (but not subtraction or division). Natural numbers are commutative and associative concerning addition and multiplication, and multiplication of natural numbers distributivno concerning addition.

The integer numbers received by consolidation of natural numbers with a set of negative numbers and zero are designated by Z= {...-2,-1, 0, 1, 2... }. Integer numbers are closed concerning addition, subtraction and multiplication (but not divisions).

Rational numbers are numbers, are representable in the form of fraction of m/n (n≠0) where m and n — integer numbers. For rational numbers all four "classical" arithmetic actions are defined: addition, subtraction, multiplication and division (except division into zero). For designation of rational numbers sign Q is used.

The real (material) numbers represent the expansion of a set of rational numbers closed rather some (important for the mathematical analysis) transactions of limit transition. The set of real numbers is designated by R. Except rational numbers, R the set of the irrational numbers not representable in the form of the relation of whole includes. Except division on rational and irrational, real numbers are also subdivided into algebraic and transcendental. At the same time each transcendental number is irrational, each rational number — algebraic.

Complex numbers of C which are expansion of a set of real numbers. They can be written in the form of z = x + by iy where i — so-called imaginary unit for which equality of i^2=-1 is executed. Complex numbers are used at solving of tasks of a quantum mechanics, hydrodynamics, the theory of elasticity and so forth.

Quaternions the representing kind of hypercomplex numbers. The set of quaternions is designated by H. Quaternions in difference from complex numbers are not commutative concerning multiplication.

In turn O octaves which are expansion of quaternions already lose property of associativity.

Unlike octaves, sedenions of S have no property of alternativeness, but save property of degree associativity.

Representation of numbers in memory of the computer

for more details see. The direct code, Branching code (representation of number), Number from a floating comma

For representation of the integer positive number x in memory of the computer, it is transferred to a binary numeral system. The received number in a binary numeral system _h2 represents machine record of the corresponding decimal number _h10. For record of negative numbers the so-called branching code of number which turns out by addition of unit to the inverted representation of the module of this negative number in a binary numeral system is used.

Representation of real numbers in memory of the computer (in ADP equipment for their designation the term number from a floating comma is used) has some restrictions connected with the used numeration system and also, limitation of the amount of memory selected under numbers. So, only some of real numbers can be provided without loss in accuracy to memories of the computer. In the most widespread scheme the number from a floating comma registers in the form of the block of bits a part of which represent a number mantissa, a part — degree, and one bits is selected for submission of the sign of number (in case of need the sign bit can be absent).