The intuitive idea of ​​number is apparently as old as humanity itself, although it is in principle impossible to trace all the early stages of its development with certainty. Before a person learned to count or invented words for numbers, he undoubtedly possessed a visual, intuitive idea of ​​\u200b\u200bnumber, which allowed him to distinguish between one person and two people, or two and many people. What primitive people at first they knew only "one", "two" and "many", is confirmed by the fact that in some languages, for example, in Greek, there are three grammatical forms: singular, dual number and plural. Later, man learned to distinguish between two and three trees and between three and four people. Counting was originally associated with a very specific set of objects, and the very first names of numbers were adjectives. For example, the word "three" was used only in the combinations "three trees" or "three people"; the idea that these sets have something in common - the concept of trinity - requires high degree abstraction. About the fact that the account arose before the advent This level of abstraction is evidenced by the fact that the words “one” and “first”, as well as “two” and “second”, in many languages ​​have nothing in common with each other, while the words “one” that lie outside the primitive account , “two”, “many”, the words “three” and “third”, “four” and “fourth” clearly indicate the relationship between cardinal and ordinal numbers.

The names of numbers, expressing very abstract ideas, undoubtedly appeared later than the first crude symbols for denoting the number of objects in a certain population. AT ancient times primitive numerical records were made in the form of notches on a stick, knots on a rope laid out in a row of pebbles, and it was understood that there was a one-to-one correspondence between the elements of the set being counted and the symbols of the numerical record. But to read such numerical records, the names of the numbers were not directly used. Now we recognize at a glance sets of two, three, and four elements; sets consisting of five, six or seven elements are somewhat more difficult to recognize at a glance. And beyond this limit, it is almost impossible to establish their number by eye, and analysis is needed either in the form of an account or in a certain structuring of elements. Counting on tags seems to have been the first technique used in similar cases: notches on the tags were located certain groups just as when ballots are counted, they are often grouped in bundles of five or ten. Finger counting was very widespread, and it is quite possible that the names of some numbers originate precisely from this method of counting.

An important feature of the account is the connection of the names of numbers with a certain counting scheme. For example, the word “twenty-three” is not just a term meaning a well-defined (by the number of elements) group of objects; it is a compound term meaning "two times ten and three." Here the role of the number ten as a collective unit or foundation is clearly visible; and indeed, many people count by tens, because, as Aristotle noted, we have ten fingers on our hands and on our feet. For the same reason bases five or twenty were used. At very early stages in the development of human history, the numbers 2, 3 or 4 were taken as the bases of the number system; sometimes bases 12 and 60 were used for some measurements or calculations.

A person began to count long before he learned to write, so no written documents have survived that testify to the words that denoted numbers in ancient times. Nomadic tribes are characterized by oral names of numbers, but as for written ones, the need for them appeared only with the transition to a settled way of life, the formation of agricultural communities. There was also a need for a system for recording numbers, and it was then that the foundation was laid for the development of mathematics.

Basic types of numbers

Unlike octaves, sedenions S do not have the property of alternativeness, but retain the property of power associativity.

To represent a positive integer x in computer memory, it is converted to the binary number system. The resulting number in binary number system x 2 is a machine notation of the corresponding decimal number x 10. To write negative numbers, the so-called. an additional code of a number, which is obtained by adding one to the inverted representation of the modulus of a given negative number in the binary number system.

Representation of real numbers in computer memory (in computer science the term floating point number is used to denote them) has some limitations associated with the number system used, as well as the limited amount of memory allocated for numbers. Thus, only some of the real numbers can be accurately represented in computer memory without loss. In the most common scheme, a floating-point number is written as a block of bits, some of which are the mantissa of the number, some are the degree, and one bit is allocated to represent the sign of the number (if necessary, the sign bit may be absent).

Number is an abstraction used for quantitative characteristics objects. Arising back in primitive society from the needs of the account, the concept of number changed and enriched and turned into the most important mathematical concept. By written characters(symbols) numbers are used to write numbers.

Basic types of numbers

Received with a natural account; natural numbers is denoted by . That. (sometimes zero is also included in the set of natural numbers, that is). Natural numbers are closed under addition and multiplication (but not subtraction or division). Natural numbers are commutative and associative under addition and multiplication, and multiplication of natural numbers is distributive under addition.

Whole numbers, obtained by the union of natural numbers with the set of negative numbers and zero, are denoted by . Integers are closed under addition, subtraction, and multiplication (but not division).

Rational numbers are numbers represented as m/n (n≠0), where m is an integer and n is a natural number. For rational numbers, all four "classical" arithmetic operations are defined: addition, subtraction, multiplication, and division (except division by zero). The sign is used to denote rational numbers.

Real (real) numbers represent an extension of the set of rational numbers, closed under some (important for mathematical analysis) operations passage to the limit. The set of real numbers is denoted by . It can be viewed as a completion of the field of rational numbers with the help of a norm, which is the usual absolute value. In addition to rational numbers, it includes a set of irrational numbers that cannot be represented as a ratio of integers. In addition to the division into rational and irrational, they are also divided into algebraic and transcendental. Moreover, every transcendental number is irrational, every rational number is algebraic.

Complex numbers, which are an extension of the set of real numbers. They can be written in the form z = x + iy, where i- so-called. imaginary unit for which i 2 = − 1. Complex numbers are used in problem solving quantum mechanics, hydrodynamics, elasticity theory, etc.

For the listed sets of numbers, the following expression is true:

Natural numbers that have only themselves and one as factors. Row prime numbers has the form: Any natural number N can be represented as a product of powers of prime numbers: 121968=2^4*3^2*5^0*7^1*11^2. This property is widely used in practical cryptography.

Numbers - types, concepts and operations, natural and other types of numbers.

Number - fundamental concept mathematics, which serves to determine the quantitative characteristics, numbering, comparison of objects and their parts. Various arithmetic operations are applicable to numbers: addition, subtraction, multiplication, division, exponentiation and others.

The numbers involved in the operation are called operands. Depending on the action performed, they receive different names. AT general case The operation scheme can be represented as follows:<операнд1> <знак операции> <операнд2> = <результат>.

In the division operation, the first operand is called the dividend (this is the name of the number that is being divided). The second (by which it is divided) is a divisor, and the result is a quotient (it shows how many times the divisible is greater than the divisor).

Types of numbers

The division operation may involve various numbers. The result of the division can be an integer or a fraction. In mathematics there are the following types numbers:

• Natural numbers are used in counting. Among them, a subset of prime numbers stands out, having only two divisors: one and itself. All others, except 1, are called composite and have more than two divisors (examples of prime numbers: 2, 5, 7, 11, 13, 17, 19, etc.);
• Integers - a set consisting of their negative, positive numbers and zero. When dividing one integer by another, the quotient can be integer or real (fractional). Among them, a subset of perfect numbers can be distinguished - equal to the sum all its divisors (including 1) except itself. The ancient Greeks knew only four perfect numbers. The sequence of perfect numbers: 6, 28, 496, 8128, 33550336… Until now, not a single odd perfect number is known;
• Rational - representable as a fraction a / b, where a is the numerator and b is the denominator (the quotient of such numbers is usually not calculated);
• Real (real) - containing an integer and a fractional part. The set includes rational and ir rational numbers(represented as a non-periodic infinite decimal fraction). The quotient of such numbers, as a rule, is a real value.

There are several features associated with the implementation arithmetic operation- divisions. Understanding them is important to get the right result:

• You cannot divide by zero (in mathematics, this operation does not make sense);
• Integer division is an operation that calculates only whole part(the fractional is discarded);
• The calculation of the remainder of an integer division allows you to get as a result the integer remaining after the operation is completed (for example, when dividing 17 by 2, the integer part is 8, the remainder is 1).

In order to better understand the sacred nature of the number, it is useful to break away for a moment from a purely esoteric approach and see how it combines with ideas. conventional science about the form of numbers. encyclopedic Dictionary writes the following about the number: "Number, one of the basic concepts of mathematics; originated in ancient times and gradually expanded and generalized. In connection with the account individual items the concept of positive integer (natural) numbers arose, and then the idea of ​​the infinity of the natural series of numbers: 1, 2, 3, 4 ... The tasks of measuring lengths, areas, as well as highlighting the shares of named quantities led to the concept of a rational (fractional) number . The concept of a negative number arose among the Indians in the VI-XI centuries. Need in exact expression relations of quantities (for example, the ratio of the diagonal of a square to its side) led to the introduction of irrational numbers, which are expressed through rational numbers only approximately; rational and irrational numbers constitute the set of real numbers. The theory of real numbers received its final development only in the second half of the 19th century in connection with the needs of mathematical analysis. In connection with the solution of square and cubic equations introduced in the 16th century complex numbers". Mathematics divides numbers into several groups or varieties, each of which can be considered from an ordinary, or maybe from a metaphysical point of view.

The relationship of numbers

Real numbers, which are the union of the set of rational and the set of irrational numbers. Any real number can, in principle, be depicted on a coordinate line in such a way that every real number and every point on this line correspond to each other. A real number can be any positive or negative number, or zero. From a metaphysical point of view this group numbers corresponds to the material material plane of being and is a sign of quantity. With the help of real numbers, measurements of all physical quantities are expressed. Numbers are rational, which can be represented as an infinite decimal fraction. They are of the form m/n, where m and n are integers and u is not equal to 0. Each infinite decimal is a rational number. The sum, difference, product, and quotient of rational numbers are also considered rational. Rational numbers include whole numbers, fractional numbers, positive numbers, negative numbers, and even zero. From a metaphysical point of view, rational numbers refer to those quantities that can be measured with certainty and precision.

Types of numbers

Irrational numbers refer to the group of real numbers that can be expressed in the form of an infinite decimal non-periodic fraction. They cannot be expressed exactly as m/n, where m and n are integers. Examples of such irrational numbers are square root of 2; 0.1010010001; lg2; cos20±; .... From a metaphysical point of view, irrational numbers belong to the field of those elusive phenomena subtle world which cannot be measured with absolute accuracy. Valid View numbers are considered a kind of complex numbers, which include numbers of the form x + iy, where x and y are real numbers, and i is the so-called imaginary unit (a number whose square is -1); x is called the real part and y is called the imaginary part of the complex numbers. Complex numbers that are not real (for<>0), are sometimes called imaginary numbers, for x=0 complex numbers are called purely imaginary. In other words, imaginary numbers are those complex numbers whose real part is equal to zero and which are denoted by z=bi. From a metaphysical point of view, complex numbers are such quantities that carry a sacred plan. Numbers are also subdivided into positive ones, which include real numbers greater than zero and negative numbers, opposite to positive ones, is less than zero. From a metaphysical point of view, everything positive numbers refer to physical world, and negative ones - to the subtle plane of being, that is, to the astral-mental area.

However, above it was only about the external, devoid of sacredness, purely quantitative nature of number. However, there is also a purely internal sacred aspect of the number, unknown to modern mathematics and predetermining the nature of the manifestation of numbers. X speaks well of this.

"Numbers in symbolism are not just an expression of quantity, but ideas - forces, each with its own special character. Numbers in modern understanding are only outer shell. All numbers are derived from one (which is equivalent to the mystical, unrevealed and dimensionless dot). Further, the number that has arisen from unity is immersed deeper and deeper into matter, into increasingly complex processes, into the "world". The first ten digits in the Greek system (or twelve in Eastern tradition) are related to the spirit: they are, in essence, archetypes and symbols. The rest is the product of a combination of these basic numbers. The ancient Greeks were very interested in the symbolism of numbers. For example, Pythagoras noted that "everything is arranged according to numbers." Plato considered number as the essence of harmony, and harmony as the basis of the cosmos and man, arguing that the rhythms of harmony are "of the same kind as the periodic oscillations of our soul." The philosophy of numbers was further developed by the Jews, the Gnostics, and the Kabbalists, including also the alchemists. The same basic universal concepts are found in Eastern thinking- for example, in Lao Tzu: "One gives birth to two, two gives birth to three, and one comes from the three" - a new unity or new order- like four. Modern symbolic logic and group theory return to the idea quantitative measurement as the basis of quality. Pire believed that the laws of nature and the human spirit are based on general principles and can be located along the same lines".

Types of numbers. Real numbers are also subdivided into algebraic and non-algebraic numbers. An algebraic number is one that satisfies algebraic equation with integer coefficients. These numbers include numbers: the root of 2; root of Z; Non-algebraic or transcendental numbers are numbers that do not satisfy any algebraic equation with integer coefficients. Transcendental numbers belong to the group of irrational numbers, although not always irrational numbers are transcendental. A number a^b is considered transcendental if the numbers a and b are algebraic numbers, but at the same time<>0; a<>1 and in - irrational number. The transcendental numbers are the sines of many rational quantities, as well as decimal logarithms integers not represented by one followed by zeros. Most famous examples The transcendental numbers are s (whose approximate value is 2.718281) and PI (whose approximate value is 3.1415296...)

P. D. Uspensky divides mathematics as a science of numbers into two types:

a) mathematics of finite and constants, which is an artificial discipline created to solve specific tasks on conditional data;

b) mathematics of infinite and variables, which is a more accurate knowledge of the real world. Examples of mathematics of the second type that violate the artificial axioms of mathematics of the first type are the so-called "transfinite numbers" lying beyond infinity.