Natural valid. Number concept. Types of numbers. Fractions and Decimals

Real number concept: real number- (real number), any non-negative or negative number, or zero. With the help of real numbers, the measurements of each physical quantity are expressed.

Real, or real number arose from the need to measure the geometrical and physical quantities of the world. In addition, to perform operations of root extraction, calculating the logarithm, solving algebraic equations, etc.

Natural numbers were formed with the development of counting, and rational numbers with the need to control parts of a whole, then real numbers (real) are used to measure continuous quantities. Thus, the expansion of the stock of numbers that are considered has led to a set of real numbers, which, in addition to rational numbers, consists of other elements, called irrational numbers.

Lots of real numbers(denoted by R) are the sets of rational and irrational numbers put together.

Real numbers are divided byrational and irrational.

The set of real numbers denotes and is often called material or number line... Real numbers are made up of simple objects: whole and rational numbers.

A number that can be written as a ratio, wherem is an integer, and n- natural number, isrational number.

Any rational number can be easily represented as a finite fraction or an infinite periodic decimal fraction.

Example,

Infinite decimal, it is a decimal fraction with an infinite number of digits after the decimal point.

The numbers that cannot be represented are irrational numbers.

Example:

Any irrational number can easily be represented as an infinite non-periodic decimal fraction.

Example,

Rational and irrational numbers create set of real numbers. All real numbers correspond to one point of the coordinate line, which is called number line.

For numeric sets, the following notation is used:

N- a set of natural numbers;
Z- a set of integers;
Q- a set of rational numbers;
R- a set of real numbers.

The theory of infinite decimal fractions.

A real number is defined as infinite decimal, i.e .:

± a 0, a 1 a 2 ... a n ...

where ± is one of the symbols + or -, the sign of a number,

a 0 - positive integer,

a 1, a 2, ... a n, ... is a sequence of decimal places, i.e. elements of the numerical set {0,1,…9}.

An infinite decimal fraction can be explained as a number that is located on the number line between rational points such as:

± a 0, a 1 a 2 ... a n and ± (a 0, a 1 a 2… a n +10 −n) for all n = 0,1,2, ...

Comparison of real numbers as infinite decimal fractions occurs bit by bit. for instance, suppose given 2 positive numbers:

α = + a 0, a 1 a 2 ... a n ...

β = + b 0, b 1 b 2… b n…

If a 0 0, then α<β ; if a 0> b 0 then α>β ... When a 0 = b 0 we pass to the comparison of the next category. Etc. When α≠β , then after a finite number of steps, the first digit will be encountered n such that a n ≠ b n... If a n n, then α<β ; if a n> b n then α>β .

But at the same time it is boring to pay attention to the fact that the number a 0, a 1 a 2… a n (9) = a 0, a 1 a 2… a n +10 −n. Therefore, if the record of one of the compared numbers, starting from a certain place, is a periodic decimal fraction, which has 9 in the period, then it must be replaced with an equivalent record, with zero in the period.

Arithmetic operations with infinite decimal fractions are a continuous continuation of the corresponding operations with rational numbers. for instance, the sum of real numbers α and β is a real number α+β that satisfies the following conditions:

∀ a ′, a ′ ′, b ′, b ′ ′∈ Q (a ′⩽ α ⩽ a ′ ′)∧ (b ′⩽ β ⩽ b ′ ′)⇒ (a + b⩽ α + β ⩽ a ′ ′ + b ′ ′)

The operation of multiplying infinite decimal fractions is defined similarly.

The numbers in the notation of multidigit numbers are divided from right to left into groups of three numbers each. These groups are called classes... In each class, numbers from right to left represent the units, tens and hundreds of that class:

The first class on the right is called class of units, second - thousand, third - million, fourth - billion, fifth - trillion, the sixth - quadrillion, seventh - quintillion, eighth - sextillion.

For the convenience of reading a multi-digit number, a small space is left between the classes. For example, to read the number 148951784296, select the classes in it:

and read the number of units of each class from left to right:

148 billion 951 million 784 thousand 296.

When reading a class of ones, the word of ones is usually not added at the end.

Each digit in the notation of a multi-digit number occupies a certain place - position. The place (position) in the record of the number on which the digit stands is called discharge.

The digits are counted from right to left. That is, the first digit on the right in the number is called the first digit, the second digit on the right - the second digit, etc. For example, in the first class of the number 148 951 784 296, digit 6 is the first digit, 9 is the second digit, 2 - digit of the third category:

Units, tens, hundreds, thousands, etc. are also called otherwise bit units:
units are called units of the 1st category (or simple units)
tens are called units of the 2nd category
hundreds are called units of the 3rd category, etc.

All units except simple ones are called constituent units... So, ten, hundred, thousand, etc. are composite units. Every 10 units of any rank is one unit of the next (higher) rank. For example, a hundred contains 10 tens, and a dozen contains 10 simple ones.

Any composite unit in comparison with another unit smaller than it is called unit of the highest category, and in comparison with a unit larger than it, it is called unit of the lowest grade... For example, one hundred is the highest-ranked unit relative to ten and the lowest-ranked one relative to a thousand.

To find out how many all units of any category are in a number, you need to discard all the numbers that mean the units of the lowest digits and read the number expressed by the remaining digits.

For example, you need to know how many hundreds are contained in the number 6284, i.e. how many hundreds are contained in thousands and in hundreds of a given number together.

In the number 6284 in the third place in the class of units is the number 2, which means there are two simple hundreds in the number. The next digit to the left is 6, which means thousands. Since each thousand contains 10 hundred, then 6 thousand contain 60. In total, this number contains 62 hundred.

The digit 0 in any digit means that there are no ones in this digit. For example, the digit 0 in the tens place means the absence of tens, in the hundreds place - the absence of hundreds, etc. In the place where 0 stands, nothing is said when reading the number:

172 526 - one hundred seventy two thousand five hundred twenty six.
102,026 - one hundred two thousand twenty six.

Integers

The numbers used in counting are called natural numbers. For example, $ 1,2,3, etc. Natural numbers form the set of natural numbers, which denote $ N $. This notation comes from the Latin word naturalis- natural.

Opposite numbers

Definition 1

If two numbers differ only in signs, they are called in mathematics opposite numbers.

For example, the numbers $ 5 $ and $ -5 $ are opposite numbers, since differ only in signs.

Remark 1

For any number, there is an opposite number, and moreover, only one.

Remark 2

The number zero is the opposite of itself.

Whole numbers

Definition 2

Whole numbers are natural, opposite numbers and zero.

The set of integers includes many naturals and their opposite.

Denote integers $ Z. $

Fractional numbers

Numbers like $ \ frac (m) (n) $ are called fractions or fractional numbers. Fractional numbers can also be written in decimal notation, i.e. as decimal fractions.

For example: $ \ \ frac (3) (5) $, $ 0.08 $, etc.

Just like integers, fractional numbers can be either positive or negative.

Rational numbers

Definition 3

Rational numbers is called a set of numbers containing a set of integers and fractional numbers.

Any rational number, both integer and fractional, can be represented as a fraction $ \ frac (a) (b) $, where $ a $ is an integer and $ b $ is a natural number.

Thus, the same rational number can be written in different ways.

For instance,

Hence it is clear that any rational number can be represented in the form of a finite decimal fraction or an infinite decimal periodic fraction.

The set of rational numbers is denoted by $ Q $.

As a result of performing any arithmetic operation on rational numbers, the resulting answer will be a rational number. This is easy to prove, due to the fact that when adding, subtracting, multiplying and dividing ordinary fractions, you get an ordinary fraction

Irrational numbers

In the course of studying a course in mathematics, you often have to deal with numbers that are not rational.

For example, to make sure that there is a set of non-rational numbers, solve the equation $ x ^ 2 = 6 $. The roots of this equation will be the numbers $ \ surd 6 $ and - $ \ surd 6 $. These numbers will not be rational.

Also, when finding the diagonal of a square with a side of $ 3 $, we apply the Pythagorean theorem and obtain that the diagonal will be equal to $ \ surd 18 $. This number is also not rational.

Such numbers are called irrational.

So, an irrational number is called an infinite decimal non-periodic fraction.

One of the most common irrational numbers is the number $ \ pi $

When performing arithmetic operations with irrational numbers, the result can be both rational and irrational.

Let us prove this by the example of finding the product of irrational numbers. Let's find:

$ \ \ sqrt (6) \ cdot \ sqrt (6) $

$ \ \ sqrt (2) \ cdot \ sqrt (3) $

Decision

$ \ \ sqrt (6) \ cdot \ sqrt (6) = 6 $

$ \ sqrt (2) \ cdot \ sqrt (3) = \ sqrt (6) $

This example shows that the result can be both a rational and an irrational number.

If in arithmetic operations if rational and irrational numbers are involved simultaneously, then the result will be an irrational number (except, of course, multiplication by $ 0 $).

Real numbers

A set of real numbers is a set containing a set of rational and irrational numbers.

The set of real numbers $ R $ is denoted. The set of real numbers can be symbolically denoted $ (-?; +?). $

We said earlier that an irrational number is called an infinite decimal non-periodic fraction, and any rational number can be represented as a finite decimal fraction or an infinite decimal periodic fraction, therefore any finite and infinite decimal fraction will be a real number.

When performing algebraic actions, the following rules will be fulfilled

when multiplying and dividing positive numbers, the resulting number will be positive
when multiplying and dividing negative numbers, the resulting number will be positive
when multiplying and dividing negative and positive numbers, the resulting number will be negative

Also real numbers can be compared with each other.

From a huge variety of all kinds sets of particular interest are the so-called number sets, that is, sets whose elements are numbers. It is clear that for comfortable work with them you need to be able to write them down. We will begin this article with the notation and principles of writing numerical sets. And then we will consider how the numerical sets are depicted on the coordinate line.

Page navigation.

Number set notation

Let's start with the accepted notation. As is known, to denote sets, we use capital letters Latin alphabet. Numerical sets, as a special case of sets, are also denoted. For example, we can talk about numerical sets A, H, W, etc. Of particular importance are the sets of natural, whole, rational, real, complex numbers, etc., for them their own designations were adopted:

N is the set of all natural numbers;
Z is a set of integers;
Q is the set of rational numbers;
J - the set of irrational numbers;
R is a set of real numbers;
C is a set of complex numbers.

Hence, it is clear that you should not denote a set consisting, for example, of two numbers 5 and −7 as Q, this designation will be misleading, since the letter Q usually denotes the set of all rational numbers. It is better to use some other “neutral” letter, for example, A, to denote the indicated numerical set.

Since we are talking about designations, here we also recall the designation of an empty set, that is, a set that does not contain elements. It is denoted by the sign ∅.

Let us also recall the designation of belonging and non-belonging of an element to a set. To do this, use the signs ∈ - belongs and ∉ - does not belong. For example, the notation 5∈N means that the number 5 belongs to the set of natural numbers, and 5.7∉Z - the decimal fraction 5.7 does not belong to the set of integers.

And let us also recall the notation adopted for the inclusion of one set into another. It is clear that all elements of the set N are included in the set Z, thus, the numerical set N is included in Z, this is denoted as N⊂Z. You can also use the notation Z⊃N, which means that the set of all integers Z includes the set N. Relationships not included and not included are denoted by ⊄ and respectively. Also, signs of loose inclusion of the form ⊆ and ⊇ are used, meaning, respectively, included or coincides and includes or coincides.

We've talked about the designations, let's move on to the description of numerical sets. In this case, we will touch upon only the main cases that are most often used in practice.

Let's start with number sets containing finite and a small amount of elements. Numerical sets consisting of a finite number of elements can be conveniently described by listing all their elements. All number elements are written separated by commas and enclosed in, which is consistent with the general set description rules... For example, a set of three numbers 0, −0.25, and 4/7 can be described as (0, −0.25, 4/7).

Sometimes, when the number of elements of a numerical set is large enough, but the elements obey a certain pattern, ellipsis are used to describe. For example, the set of all odd numbers from 3 to 99 inclusive can be written as (3, 5, 7, ..., 99).

So we smoothly approached the description of numerical sets, the number of elements of which is infinite. Sometimes they can be described using all the same ellipsis. For example, let us describe the set of all natural numbers: N = (1, 2. 3,…).

They also use the description of numerical sets by indicating the properties of its elements. In this case, the notation (x | properties) is used. For example, the notation (n | 8 n + 3, n∈N) specifies the set of natural numbers that, when divided by 8, give the remainder 3. The same set can be described as (11,19, 27, ...).

In particular cases, numerical sets with an infinite number of elements represent the known sets N, Z, R, etc. or numeric gaps. Basically, numerical sets are represented as an association their constituent separate numerical intervals and numerical sets with a finite number of elements (which we talked about just above).

Let's show an example. Let the number set be the numbers −10, −9, −8.56, 0, all the numbers of the segment [−5, −1,3] and the numbers of the open number ray (7, + ∞). By virtue of the definition of the union of sets, the indicated numerical set can be written as {−10, −9, −8,56}∪[−5, −1,3]∪{0}∪(7, +∞) ... Such a notation actually means a set containing all elements of the sets (−10, −9, −8.56, 0), [−5, −1.3] and (7, + ∞).

Similarly, by combining various numerical ranges and sets of separate numbers, any numerical set (consisting of real numbers) can be described. Here it becomes clear why such types of number intervals as interval, half-interval, segment, open number ray and number ray were introduced: all of them, together with the designation of sets of individual numbers, allow you to describe any number sets through their union.

Please note that when you write a number set, its constituent numbers and number intervals are sorted in ascending order. This is not a necessary, but desirable condition, since an ordered number set is easier to represent and represent on a coordinate line. Also note that such records do not use numeric gaps with common elements, since such records can be replaced by joining numeric gaps without common elements. For example, the union of numerical sets with common elements [−10, 0] and (−5, 3) is a half-interval [−10, 3). The same applies to the union of numeric intervals with the same boundary numbers, for example, the union (3, 5] ∪ (5, 7] is a set (3, 7], we will dwell on this separately when we learn to find the intersection and union of numeric sets.

Representation of number sets on a coordinate line

In practice, it is convenient to use geometric images of numerical sets - their images on. For example, for solving inequalities, in which it is necessary to take into account the ODV, it is necessary to represent the numerical sets in order to find their intersection and / or union. So it will be useful to understand well all the nuances of the image of number sets on the coordinate line.

It is known that there is a one-to-one correspondence between the points of the coordinate line and real numbers, which means that the coordinate line itself is a geometric model of the set of all real numbers R. Thus, in order to depict the set of all real numbers, it is necessary to draw a coordinate line with shading along its entire length:

And often they do not even indicate the origin and the unit segment:

Now let's talk about the image of numerical sets, which are a certain finite number of separate numbers. For example, let's draw a number set (−2, −0.5, 1.2). Geometrically, this set, consisting of three numbers −2, −0.5 and 1.2, will be three points of the coordinate line with the corresponding coordinates:

Note that it is usually not necessary to make the drawing exactly for practice purposes. Often, a schematic drawing is sufficient, which implies the optional keeping of scale, while it is only important to maintain the relative position of points relative to each other: any point with a smaller coordinate must be to the left of a point with a larger coordinate. The previous drawing will schematically look like this:

Separately, from all kinds of numerical sets, numerical intervals (intervals, half-intervals, rays, etc.) are distinguished, which represent their geometric images, we figured it out in detail in the section. We will not repeat ourselves here.

And it remains only to dwell on the image of numerical sets, which are the union of several numerical intervals and sets consisting of separate numbers. There is nothing tricky here: according to the meaning of the union in these cases, all the constituents of the set of a given numerical set must be depicted on the coordinate line. As an example, we will show the image of a number set (−∞, −15)∪{−10}∪[−3,1)∪ (log 2 5, 5) ∪ (17, + ∞):

And let's dwell on quite common cases when the depicted numerical set represents the entire set of real numbers, with the exception of one or several points. Such sets are often given by conditions like x ≠ 5 or x ≠ −1, x ≠ 2, x ≠ 3.7, etc. In these cases, geometrically, they represent the entire coordinate line, with the exception of the corresponding points. In other words, these points must be “gouged out” from the coordinate line. They are depicted as circles with an empty center. For clarity, we will depict a numerical set corresponding to the conditions (this set is essentially there):

Summarize. Ideally, the information of the previous paragraphs should form the same view of the recording and image of numerical sets, as well as a view of individual numerical intervals: the record of a numerical set should immediately give its image on the coordinate line, and from the image on the coordinate line we should be ready to easily describe the corresponding number set through the union of individual intervals and sets consisting of separate numbers.

Bibliography.

Algebra: study. for 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M.: Education, 2008 .-- 271 p. : ill. - ISBN 978-5-09-019243-9.
A. G. Mordkovich Algebra. Grade 9. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., Erased. - M .: Mnemozina, 2011 .-- 222 p.: Ill. ISBN 978-5-346-01752-3.

What is a number? NUMBER - one of the basic concepts of mathematics, originated in ancient times and gradually expanded and generalized. In connection with the counting of individual objects, the concept of positive integer (natural) numbers arose, and then the idea of the infinity of the natural series of numbers: 1, 2, 3, Natural numbers are numbers used when counting objects. one

Story. On the excavations of the camp of ancient people, they found a wolf bone, on which 30 thousand years ago, some ancient hunter made fifty-five notches. It can be seen that, making these notches, he counted on his fingers. The bone pattern consisted of eleven groups of five notches each. At the same time, he separated the first five groups from the rest by a long line. Also in Siberia and in other places, stone tools and decorations made in the same distant era were found, on which there were also lines and dots, grouped by 3, 5 or 7. Celts are an ancient people who lived in Europe 2,500 years ago. back, being the ancestors of the French and the British, were considered twenty (two hands and two legs gave twenty fingers). Traces of this have survived in the French language, where the word "eighty" sounds like "four times twenty." Other peoples were also considered twenty - the ancestors of the Danes and Dutch, Ossetians and Georgians. 2

Even and odd numbers. An even number is an integer that is divisible by 2 without a remainder: ..., 2, 4, 6, 8, ... An odd number is an integer that is not evenly divisible by 2: ..., 1, 3, 5, 7, 9, ... Pythagoras defining number as energy and believed that through the science of numbers the secret of the Universe is revealed, for number contains the secret of things. Pythagoras considered even numbers to be female, and odd numbers as male: 2 + 3 = 5 5 is a symbol of family, marriage. Even and odd numbers = feminine and masculine numbers. 4

Simple and Composite. A prime number is a natural number that has exactly two different natural divisors: one and itself. A sequence of primes starts like this: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, ... Composite numbers are numbers with 3 or more divisors. Number theory deals with the study of the properties of prime numbers. Thus, all natural numbers greater than one are divided into prime and composite. 5

Perfect and imperfect numbers. Perfect numbers, positive integers equal to the sum of all their correct (i.e. less than this number) divisors. For example, the numbers 6 = and 28 = are perfect. Until now (1976) not a single odd Owl is known. and the question of their existence remains open. Research on Sov. hours were started by the Pythagoreans, who ascribed a special mystical meaning to numbers and their combinations. Imperfect Pythagoras called numbers, the sum of regular divisors, which are less than himself. 6

Magic numbers. The secrets of numbers attract people, make them delve into, understand, compare their conclusions with the real ratio of affairs. To the numbers in ancient world treated very kindly. People who knew them were considered great, they were equated with deities. The simplest example is the absence in many countries of aircraft with tail number 13, floors and rooms in hotels with the number "13". eight
Magic row 2 - the number of balance and contrast, and maintaining stability, mixing positive and negative qualities. 6 - Symbol of reliability. It is a perfect number that is divisible by both an even number (2) and an odd number (3), thus combining the elements of each. 8 - The number of material success. It signifies reliability perfected as it is represented by a double square. Divided in half, it has equal parts (4 and 4). If it is further divided, then the parts will also be equal (2, 2, 2, 2), showing a fourfold balance. 9 - The number of universal success, the largest of all numbers. As three times the number 3, the nine turns imbalance into striving. 10