Why is there a set of real ones? Lesson "set of real numbers"

The set of real numbers is the collection of the complement of rational numbers by irrational ones. This set is denoted by the letter R, and it is customary to use the notation (-∞, +∞) or (-∞,∞) as a symbol.

The set of real numbers can be described as follows: it is a set of finite and infinite decimals, finite decimal fractions and infinite decimal periodic fractions are rational numbers, and infinite decimal and non-periodic fractions - irrational numbers.
Any real number can be indicated on a coordinate line. The converse statement is also appropriate: any point on the coordinate line has a real coordinate. On mathematical language it sounds like this: a one-to-one relationship can be established between the set of points on the coordinate line and the set R of real numbers. For the coordinate line itself, the term “number line” is often used, since the coordinate line is a geometric model of the set of real numbers.
It turns out that your acquaintance with the coordinate line was a long time ago, but you will only begin to use it now. Why? You can find the answer in the example from the video tutorial.

It is known that for real numbers a and b the laws of addition and multiplication that are already well known to you are satisfied: the communicative law of addition, the commutative law of multiplication, the associative law of addition, the distributive law of multiplication relative to addition, and others. Let's illustrate some of them:
a + b = b + a;
ab = ba;
a + (b + c) = (a + b) + c;
a(bc) = (ab)c;
(a + b)c = ac + bc
Also performed following rules:
1. As a result of the product (quotient) of two negative numbers, a positive number is obtained.
2. As a result of the product of a (quotient) negative and positive number, a negative number is obtained.
You can compare real numbers with each other based on the definition:
A real number a is greater or less than a real number b, in the case when the difference a - b is a positive or negative number.
It is written like this: a > b, a< b.
This means that a is positive number, and b is negative.
That is, in the case when a > 0 => a is positive;
a< 0 =>a negative;
a > b, then a - b is positive => a - b > 0;
a< b, то a - b отрицательное =>a-b< 0.
In addition to the signs (<; >) strict inequalities, signs of weak inequalities are also used - (≤;≥).
For example, for any number b, the inequality b2 ≥ 0 holds.
You can see examples of comparing numbers and arranging them in ascending order in the video tutorial.
Thanks to geometric model set of real numbers - the number line, the comparison operation looks especially clear.

Main property algebraic fraction

We continue our acquaintance with algebraic fractions. If the previous lesson talked about basic concepts, then in this lesson you will learn about the main property of an algebraic fraction. The definition of the basic property of a fraction is known from the 6th grade mathematics course (reducing fractions). What does it consist of? Often, when solving problems or equations, it becomes necessary to transform one “inconvenient” fraction for calculations into another, “convenient” one. It is to perform such transformations that you need to know its main property and the rules for changing signs, which you will become familiar with by watching the video tutorial.

The value of a common fraction will remain the same when the numerator and denominator are multiplied or divided by the same number (except zero). This is the main property of a fraction.
Let's look at an example:
7/9 = 14/18
We have two fractions that are identically equal to each other. Numerator and denominator in in this case multiplied by 2, but the value of the fraction did not change.
You will learn from the video lesson what happens to a fraction when the numerator and denominator are divided by the same number.
An algebraic fraction is, in principle, the same common fraction, you can perform the same actions on it as on an ordinary one.
An expression in the numerator and an expression in the denominator of a fraction can be multiplied or divided by the same alphanumeric expression (polynomial or monomial), the same number (except zero: if the expression or number in the denominator fractions, multiplied by zero, it will take a zero value; and, as you know, you cannot divide by zero). This transformation of an algebraic fraction is called its reduction. This is the main property of an algebraic fraction. You can learn how it is implemented in practice from the video tutorial.
Converting fractions to fractions with same denominators is called reducing fractions to a common denominator. To perform of this action it is necessary to perform a certain sequence of actions, consisting of the following:

Having factorized all the denominators, we determine the LCM for numerical coefficients.
. We write down the product, taking into account the LCM coefficients and all letter factors. If the multipliers are the same, take the multiplier once. Of all the degrees that identical grounds, take the multiplier with maximum indicator degrees.
. We find the values ​​that are additional factors for the numerator of each fraction.
. For each fraction, we determine a new numerator - as the product of the old numerator and an additional factor.
. We write down fractions with a new numerator that we have determined and a common denominator.

Example 1: Reduce the following fractions a/4b2 b a2/6b3 to a common denominator.
Solution:
To begin with, let's define common denominator. (It is equal to 12b2).
Then, following the algorithm, we determine an additional factor for each of the fractions. (For the first - 3b, for the second - 2).
After performing the multiplication, we get the result.
(a*3b)/(4b2*3b) = 3ab/12b3 and (a2*2)/(6b2*2) = 2a2/12b2.
Example 2: Reduce the fractions c/(c - d) and c/(c + d) to a common denominator.
Solution:
(c+d)(c-d)=c2-d2
c*(c + d)/(c - d)(c + d) = (c2 + cd)/(c2 - d2)
c*(c - d)/(c + d)(c - d) = (c2 - cd)/(c2 - d2)

More detailed solution similar examples you will find in the video tutorial.
The main property of an algebraic fraction has a consequence in the form of a rule for changing signs:
a - b/c - d = b - a/d - c
In this case, the numerator and denominator of the fraction were multiplied by -1. Similar actions can be performed not with the entire fraction, but only with the numerator or only with the denominator. How the result will change if, for example, only the numerator or only the denominator is multiplied by -1, you will find out by watching the video tutorial.
Now, having studied the main property of an algebraic fraction and the rule that follows from it, we are able to solve more complex tasks, namely: subtraction and addition of fractions. But this is the topic of the next lesson.

Historically, natural numbers $N$ were the first to emerge as a result of the recalculation of items. The set of these numbers is infinite and forms the natural series $N=\(1, 2, 3, ..., n, ...\)$. The operations of addition and multiplication are feasible in this set. New numbers were required to perform the subtraction operation, resulting in a set of integers: $Z$. $Z=N_+\cup N_- \cup \(0\)$. Thus, in the set of integers, the operations of addition, multiplication, and subtraction are always performed.

Rational numbers

The need to perform division led to the set of rational numbers $Q$. $Q=\(\frac(m)(n), m\in Z, n\in N\)$.

Definition. Two rational numbers are equal: $\frac(m_1)(n_1)=\frac(m_2)(n_2)$ - if $m_1\cdot n_2=n_1\cdot m_2$. This means that every rational number can be represented in a unique way in the form of an irreducible fraction $\frac(m)(n)$. $GCD(m, n)=1$.

Properties of the set of rational numbers

1. As a result of arithmetic operations on rational numbers (addition, multiplication, subtraction, division, except division by zero), a rational number is obtained.

2. The set of rational numbers is ordered, that is, for any pair of rational numbers $a$ and $b$ or $a b$.

3. The set of rational numbers is dense, that is, for any pair of rational numbers $a$ and $b$ there is a rational number $c$ such that $a

Any positive rational number can always be represented as a decimal fraction: either finite or infinitely periodic. For example: $\frac(3)(5)=0.6$, $\frac(1)(3)=0.333...=0,(3)$.

$\frac(m)(n)=a_0,a_1a_1...a_kb_1b_2b_3...b_nb_1b_2b_3...b_n...$.

$b_1b_2b_3...b_n...$ - called the period of the decimal fraction, where not all $b_i=0$.

Note that a finite fraction can be written as an infinite periodic fraction with zero in the period. $\frac(m)(n)=a_0,a_1a_1...a_k000000...$, $a_k\ne0$.

However, another representation of rational numbers in the form of a decimal fraction is more common: $\frac(m)(n)=a_0,a_1a_1...(a_k-1)999...$.

Negative rational numbers $-\frac(m)(n)$ are written as a decimal expansion rational number of the form $\frac(m)(n)$, taken with the opposite sign.

The number $0$ is represented as $0,000...$.

Thus, any rational number is always representable as an infinite decimal periodic fraction that does not contain $0$ in the period, except for the number $0$ itself. This is the only representation.

Irrational numbers

The set of rational numbers is closed under four arithmetic operations. However, in the set of rational numbers there is not always a solution to the simplest equation of the form $x^2-n=0$. Therefore, there is a need to introduce new numbers.

Let us show that among the rational numbers there is no number whose square is equal to three. We will carry out the proof by contradiction.

Suppose that there is a rational number $\frac(m)(n)$ such that its square is equal to three: $\left(\frac(m)(n)\right)^2=3\;\;\;( 1)$.

$\frac(m^2)(n^2)=3$,

$m^2=3n^2.\;\;\;(2)$

The right side of equality (2) is divisible by 3. This means that $m^2$ is also divisible by 3, therefore $m$ is divisible by 3, which means that $m=3k$. Substitute into equality (2), we get:

$3k^2=n^2.\;\;\;(3)$

The left side of the equality $(3)$ is divisible by $3$, which means the right side is also divisible by $3$. Therefore $n^2$ is divisible by $3$, which means $n$ is divisible by $3$, hence $n=3p$. As a result, we obtain: $\frac(m)(n)=\frac(3k)(3p)$, that is, the fraction $\frac(m)(n)$ turned out to be reducible, which contradicts the assumption. This means that among the rational numbers there is no number whose square is equal to three.

But a number whose square is three exists. It can be represented as an infinite non-periodic fraction. And we got a new kind of numbers. Let's call them irrational.

Definition. An irrational number is any infinite number periodic fraction.

The set of all infinite non-periodic fractions is called the set of irrational numbers and is denoted by $I$.

Real numbers

The union of the set of rational numbers $Q$ and irrational numbers $I$ gives the set of real numbers $R$: $Q\cup I=R$.

Thus, every real number can be represented as an infinite decimal fraction: periodic in the case of a rational number and non-periodic in the case of an irrational number.

Comparison of real numbers

For real numbers $a=a_0,a_1a_2a_3\ldots a_n\ldots$, $b=b_0,b_1b_2b_3\ldots b_n\ldots$ comparison is carried out as follows:

1) Let $a$ and $b$ be both positive: $a>0$, $b>0$, then:

$a=b$, if for any $k$ $a_k=b_k$;

$a>b$ if $\exists s$ $\forall k b_s$.

2) Let $a>0$, $b<0$, или иначе: $b<0

3) Let $a$ and $b$ both be negative: $a<0$, $b<0$, тогда:

$a=b$, if for $-a=-b$;

If the set of rational numbers is supplemented with a set of irrational numbers, then together they form the set of real numbers. The set of real numbers is usually denoted by the letter R; They also use symbolic notation (-oo, +oo) or (-oo, oo).

The set of real numbers can be described as follows: it is a set of finite and infinite decimal fractions; finite decimals and infinite decimal periodic fractions are rational numbers, and infinite decimal non-periodic fractions are irrational numbers.

Every real number can be represented by a point on a coordinate line. The converse is also true: every point on a coordinate line has a real coordinate. Mathematicians usually say this: a one-to-one correspondence has been established between the set R of real numbers and the set of points on the coordinate line. The coordinate line is a geometric model of the set of real numbers; For this reason, the term number line is often used for the coordinate line.

Think about this term: doesn’t it seem unnatural to you? After all, a number is an object of algebra, and a straight line is an object of geometry. Is there a “mixing of genres” here? No, everything is logical, everything is thought out. This term once again emphasizes the unity of various areas of mathematics and makes it possible
identification of the concepts “real number” and “point on the coordinate (numeric) line.”

Please note: you have been using the coordinate line since the 5th grade. But it turns out that there was a completely justified gap in your knowledge: not for any point on the coordinate line you would have been able to find the coordinate - the teacher simply protected you from such trouble.

Let's look at an example. A coordinate line is given, a square is constructed on its unit segment (Fig. 100), the diagonal of the square OB is plotted on the coordinate line from point O to the right, the result is point D. What is the coordinate of point D? It is equal to the length of the diagonal of the square, i.e. This number is like
We now know that it is not a whole or a fraction. This means that neither in the 5th, nor in the 6th, nor in the 7th grade would you be able to find the coordinate of point D.

That’s why we have so far said “coordinate line” and not “number line”.

Note that there was another justifiable gap in your algebra knowledge. When considering expressions with variables, we always meant that variables could take any valid values, but only rational ones, because there were no others. In fact, variables can take
any valid valid values. For example, in the identity
(a + b)(a-b) = a 2 -b 2 any numbers can act as a and b, not necessarily
rational. We already used this at the end of the previous paragraph. We used the same in § 18 - in particular, in examples 6, 7, 8 from this paragraph.

For real numbers a, b, c, the usual laws apply:
a + b = b + a;
ab = ba;

a + (b + c) = (a + b) + c

a(bc) =(ab)c
(a + b) c = ac + bc, etc.
The usual rules also apply: the product (quotient) of two positive numbers is a positive number;
the product (quotient) of two negative numbers is a positive number;
the product (quotient) of a positive and a negative number is a negative number.

Real numbers can be compared to each other using the following definition.

Definition . A real number a is said to be greater (less than) a real number b if their difference a - b is a positive (negative) number. Write a > b (a< b).

From this definition it follows that every positive number a is greater than zero (since the difference a - 0 = a is a positive number), and every negative number b is less than zero (since the difference b - 0 = b is a negative number).

So, a > 0 means that a is a positive number;
A< 0 означает, что а — отрицательное число;
a>b means that a -b is a positive number, i.e. a - b > 0;
a those. a - b< 0.
Along with the signs of strict inequalities (<, >) use signs of weak inequalities:
a 0 means that a is greater than zero or equal to zero, that is, a is a non-negative number (positive or 0), or that a is not less than zero;
and 0 means that a is less than zero or equal to zero, that is, a is a non-positive number (negative or 0), or that a is not greater than zero;
and b means that a is greater than or equal to b, that is, a - b is a non-negative number, or that a is not less than b; a - b 0;
and b means that a is less than or equal to b, that is, a - b is a non-positive number, or that a is not greater than b; a - b 0.
For example, for any number a the inequality a 2 0 is true;
for any numbers a and b the inequality (a - b) 2 0 is true.
However, to compare real numbers, it is not necessary to make up their difference each time and find out whether it is positive or negative. You can draw the appropriate conclusion by comparing numbers in the form of decimal fractions.

The geometric model of the set of real numbers, i.e. the number line, makes the operation of comparing numbers especially clear: of two numbers a, b, the one located on the number line to the right is greater.

Thus, comparing real numbers needs to be approached quite flexibly, which is what we use in the next example.

Example 1. Compare numbers:

Example 2. Arrange in ascending order of numbers

On the third line there are three numbers for each cubic equation, respectively. ordered fours, etc.

That. we obtain a matrix that can be traversed using the Cantor diagonal process. If some of the roots of an algebraic equation are complex, we simply skip them when numbering. That. every algebraic number will receive a corresponding number, and this confirms the fact that the set of algebraic real numbers countably .

Fact efficient enumerability set A directly follows from the given method of numbering elements with natural numbers, since at the same time an effective procedure for numbering sets of rational numbers that uniquely define algebraic equations of the corresponding degree is indicated. It is important that the algebraic equation of the nth degree has an effective solution algorithm, i.e. the procedure is completely effective. So, the set of algebraic real numbers is countable and effectively enumerable, Q.E.D.

Sets made up of all pairs, triplets, etc. of algebraic numbers will also be countable.

2.3.7. Countable number sets: generalization

T.2 Theorem (without proof)

The set of elements that can be represented using a finite number of countable symbols is countable.

In real life, we use various finite sign systems, such as numbers, letters, notes.

Let's consider a system of signs, for example, numbers in any finite number system, say decimal. Having 10 characters at our disposal: 0,1,2,3,4,5,6,7,8,9, we can create two types of sets: fixed length and arbitrary length.

In the first case, we are talking about a purely combinatorial problem, for example, you can create 105 different sequences of five characters. This is a rather large number, but it is a natural number and the cardinality of the considered set of all possible sequences of this kind is expressed by a natural number. In the second case, the set of such sequences will be countably infinite, by analogy with the sets of complexes of natural numbers, and its cardinality is the number aleph-zero.

It can be generalized that the set obtained as a result of applying Theorem 2.3.(7) will be countably infinite if, in the case of a finite system of signs, arbitrarily long complexes of signs are allowed (as long as desired, but still finite!).

Countably infinite are, for example:

· a set of “words” that can be composed using a finite alphabet (“a word” here is a complex of letters, no matter whether they have meaning or not),

· the set of all books written in any or even all languages,

· set of all symphonies, etc.

2.4.1. Uncountability of the set of real numbers (continuum)

We denote the set of real numbers Latin letter R.

T.2 Theorem

The set of real numbers is uncountable.

Proof

Let us assume the opposite, let the set of real numbers be countable. Then any subset of a countable set is also countable. On the set of real numbers, let’s take a subset R1 - the interval (0,1) and remove from this segment numbers that contain zeros or nines in at least one of their digits (examples of such numbers: 0.9, 0.0001, etc.). The set R2, made up of the remaining numbers, is a subset of the set R1. This means that R2 is countable.

From the fact that R2 is countable, it directly follows that some way of enumerating its elements is possible to establish a one-to-one correspondence between the elements of R2 and the elements of the set natural numbers. This follows from the very definition of cardinality of a set, according to which it is assumed that in sets of equal cardinality, each element of one set has a paired element from another set and vice versa. Please note that the fundamental difference between this definition and the definition of effective enumerability is that in this case we are not even talking about the presence of any enumeration algorithm, we simply claim that it is possible to give a list of real numbers from the set R2 and a list of corresponding natural numbers from the set N. In this case, we are not interested in the algorithm for constructing the connection N ↔ R2; it is enough that such a correspondence is possible.

Let's build the following list of numbers from the set R2 and number the numbers in digits:

Now let’s construct the number b=0.b1b2…, and

bi=aii+1, where + denotes the operation of addition, the result of which cannot be the numbers 0 and 9, i.e. if aii=1, then bi=2; if aii=2, then bi=3, ...., if aii=8, then bi=1).

Thus, the constructed number b will differ from each of the numbers in the set R2 in at least one digit, and, therefore, will not be included in the compiled list. However, by its structure, the number b must be contained in the set R2. We get a contradiction, which means the original assumption is incorrect and the set R2 is uncountable.

Since the set R2 is by condition a subset of the set R1, then R1 is uncountable, and since R1 is uncountable, then the set R is uncountable, Q.E.D.

Note: You don’t have to throw away numbers containing 0 and 9. Thus, some numbers will appear in our series twice. This is because finite fractions can be turned into infinite fractions. For example ½=0.5=0.5(0)=0.4(9).

In general, this could be the reason that it was not possible to count the set of real numbers. But the set of numbers that can be represented in two ways (finite fractions) is the set of rational numbers. As was proven earlier, there are a countable number of them. One can even show that this set is effectively enumerable. That. even a double representation of the set of such numbers forms a countable set, therefore the proof is correct even without such a simplification.

A fundamentally new result was obtained - an uncountable set of numbers was found. Its power, according to the proven theorem, is not equal to aleph-zero (À0), which means a new number is needed in the transfinite scale.

Aleph ( À) – the second transfinite number. By definition, this is the power of the continuum (of all real numbers). This is the second highest infinite power. The just proven Theorem 2.4.(1) on the uncountability of the set of real numbers is a convincing proof that the cardinality of this set is greater than aleph-zero (greater than the set of natural numbers). And this is a very important result after a series of proofs of the countability of various sets of numbers.

If we operate with the concept of a cardinal number (power), we obtain that, since each number of the segment (0,1) can be represented by a decimal fraction of the form 0.a1a2a3... at least once and at most twice, then:

À≤10 À0≤ 2À ,

and since 2À=À, we get that 10 À0= À. The same reasoning is valid if we decompose numbers not into decimals, but, for example, into binary fractions, fractions with a base of 3, 15, 10005, or even À0 (if you can imagine that).

That. À =2À0=3À0=…=10À0=…nÀ0=…À0À0

If you think about it, you can discover another not entirely obvious fact from set theory. À2=À À is the power of the set of pairs of real numbers. A pair of real numbers, generally speaking, corresponds to a point on the plane. In turn, À3=À À À is the power of the set of triplets of real numbers, and these are points in space. The reasoning can be continued further up to À0 - a dimensional space or the set of all sequences of real numbers of countable length. That. all finite-dimensional or countably-dimensional spaces have the same cardinality À (here À is the number of points in the space).

For an À0-dimensional real space or the set of all sequences of real numbers of countable length, from the point of view of operations on cardinal numbers, we obtain ÀÀ0=(2À0)À0=2À0∙À0=2À0=À.

At this point it will be interesting to turn to historical events associated with a series of evidence in this area. Mathematicians, although not immediately, eventually came to terms with the fact that there are as many points on an infinite straight line as there are on a segment. But Cantor's next result was even more unexpected. In search of a set that has more elements than a segment on the real axis, he turned his attention to the set of points of a square. Initially, there was no doubt about the result: after all, the entire segment is located on one side of the square, and the set of all segments into which the square can be decomposed itself has the same cardinality as the set of points of the segment. For almost three years (from 1871 to 1874), Cantor sought proof that a one-to-one correspondence between the points of a segment and the points of a square is impossible. And at some point, completely unexpectedly, the exact opposite result turned out: he managed to construct a correspondence that he sincerely considered impossible. Cantor did not believe himself and even wrote to the German mathematician Richard Dedekind: “I see it, but I don’t believe it.” When the shock of this fact passed, it became intuitively clear and soon proven that a cube has the same number of points as a segment. Generally speaking, any geometric figure on a plane (a geometric body in space) containing at least one line has the same number of points as a segment. Such sets were called continuum power sets (from the Latin continuum - continuous). The next step is almost obvious: the dimension of space within certain limits is unimportant. For example, a 2-dimensional plane, 3-dimensional familiar space, 4, 5 and further n-dimensional spaces are of equal power in terms of the number of points contained in the corresponding n-dimensional body. This situation will be observed even in the case of a space with an infinite number of dimensions, it is only important that this number is countable.

At this stage, two types of infinities have been discovered and, accordingly, two transfinite numbers denoting their powers. Sets of the first type have power equivalent to the power of natural numbers (aleph-zero). Sets of the second type have cardinality equivalent to the number of points on the real axis (cardinality of the continuum, aleph). It is shown that sets of the second type have more elements than sets of the first type. Naturally, the question arises: is there an “intermediate” set in nature that would have a cardinality greater than the number of natural numbers, but at the same time less than the set of points on a line? This difficult question is called "continuum problem" . She is also known as "continuum hypothesis" or " Hilbert's first problem". The exact wording is as follows:

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As a result, after much research on the continuum hypothesis, in 1938 the German mathematician Kurt Gödel proved that the existence of intermediate power does not contradict the other axioms of set theory. And later, in Almost simultaneously, but independently of each other, the American mathematician Cohen and the Czech mathematician Vopenka showed that the presence of such an intermediate power cannot be deduced from the other axioms of set theory. By the way, it is interesting to note that this result is very similar to the story with the postulate of parallel lines. As is known, for two thousand years they tried to deduce it from the other axioms of geometry, but only after the work of Lobachevsky, Hilbert and others they managed to obtain the same result: this postulate does not contradict the other axioms, but cannot be deduced from them.

2.4.2. Sets of complex, transcendental and irrational numbers

In addition to the set of real numbers, we present several more uncountable sets.

https://pandia.ru/text/78/390/images/image010_26.gif" width="81" height="76"> T.2.4.(2) Theorem

Many complex numbers uncountable.

Proof

Since the set of real numbers R, uncountable by the previously proven Theorem 2.4.(1), is a subset of the set of complex numbers C, then the set of complex numbers is also uncountable, Q.E.D.

Transcendental number - a real number that is not algebraic.

Many transcendental numbers let us denote it by the Latin letter T. Every transcendental real number is irrational, but the converse is not true. For example, a number is irrational, but not transcendental: it is the root of the equation x 2 − 2=0.

T.2 Theorem

The set of transcendental numbers is uncountable.

Proof

Since real numbers are an uncountable set, and algebraic numbers are countable, and the set A is a subset of R, then R \ A (the set of transcendental numbers) is an uncountable set, Q.E.D.

This simple proof of the existence of transcendental numbers was published by Cantor in 1873 and made a great impression on the scientific community, since it proved the existence of many numbers without constructing a single specific example, but only based on general considerations. No specific example of a transcendental number can be extracted from this proof; a proof of this type is said to be unconstructive .

It's important to note that for a long time mathematicians dealt only with algebraic numbers. It took considerable effort to find even a few transcendental numbers. For the first time it was possible French mathematician Liouville in 1844, who proved a set of theorems allowing one to construct specific examples such numbers. For example, a transcendental number is the number 0,..., in which after the first unit there is one zero, after the second - two, after the third - 6, after the nth, respectively, n! zeros.

It has been proven that transcendental is decimal logarithm any integer except the numbers 10 n. Also the set of transcendental numbers includes sin α, cos α and tg α for any non-zero algebraic number α . The most striking representatives of transcendental numbers are usually considered to be numbers π And e. By the way, the proof of the transcendence of number π , carried out by the German mathematician Karl Linderman in 1882, was large scientific event, because it implied the impossibility of squaring a circle. The history of finding the squaring of a circle lasted four millennia, and the term itself became synonymous with insoluble problems.

Unit of measurement" href="/text/category/edinitca_izmereniya/" rel="bookmark">unit of measurement radius of a circle and designate x the length of the side of the required square, then the problem reduces to solving the equation: x 2 = π, from where: . As you know, with the help of a compass and ruler you can do all 4 arithmetic operations and extraction square root. This means that squaring the circle is possible if and only if, using a finite number of such actions, it is possible to construct a segment of length π. Thus, the unsolvability of this problem follows from the non-algebraic nature (transcendence) of the number π. Actually, the problem of squaring a circle is reduced to the problem of constructing a triangle with base πr and height r. An equal square can then be easily constructed for it.

In the previously mentioned list of 23 cardinal problems Mathematics number 7 was the problem concerning the transcendence of numbers formed in a certain way.

Hilbert's seventh problem. Let a --- positive algebraic number, not equal to 1, b --- irrational algebraic number. Prove that ab is a transcendental number.

In 1934 Soviet mathematician Gelfond and a little later the German mathematician Schneider proved the validity of this statement, and thus this problem was solved.

Two more are associated with the principle of dividing numbers into rational and irrational interesting facts, not immediately perceived as true.

T.2.4.(5) Theorem

Between any two different rational numbers there is always a set of irrational numbers of continuum power.

Proof

Let there be two rational numbers, a And b. Let us construct a linear, and therefore one-to-one, function f(x) = (x - a) / (b - a). Because f(a) = 0 and f(b) = 1, then f(x) maps the segment [ a; b] into the segment , while maintaining the rationality of the numbers. Therefore, the powers of the sets [ a; b] and real numbers are equal, and, as has been proven, the power of the segment is equal to the power of the continuum. Selecting only irrational numbers from the resulting set, we obtain that between any two rational numbers there is always a continuum of irrational numbers, Q.E.D.

Generally this theorem intuitively seems quite logical. The following, at first glance, is perceived with skepticism.

T. 2.4.(6) Theorem

Between any two different irrational numbers there is always a countable set of rational numbers.

Proof

Let there be two irrational numbers a And b, we write their corresponding digits as a 1a 2a 3... and b 1b 2b 2..., where ai, bi- decimal numbers. Let a < b, then there is N such that a N< b N. Let's construct a new number c, why let's put ci = ai = bi For i= 1, …, N-1. Let cN = bN-1. It's obvious that c < b. Since all digits of the number a after the Nth cannot be nines (then it will be a periodic fraction, i.e. a rational number), then we denote by M >= N such a digit of the number a, What a M< 9. Положим cj = aj, at N< j < M, и c M = 9. In this case c > a. So we got one rational number c, such that a < c < b. Adding to decimal notation numbers c any final number numbers behind we can get as many rational numbers as we want between a And b. By assigning each such number to its serial number, we obtain a one-to-one correspondence between the set of these numbers and the set of natural numbers, so the resulting set will be countable, Q.E.D.

At this stage, the proof of the following theorem becomes interesting and important, the meaning of which before the introduction of the scale of transfinite numbers was generally obvious, and with the appearance of such specific arithmetic requires rigorous proof.

T.2 Cantor's theorem

For any cardinal number α, α<2α.

Proof

1. Let us prove that at least α≤2α

As is known, the cardinality of the Boolean set M is equal to 2|M|. Let the set M = (m1, m2, m3, ...). The Boolean set M (the set of all its subsets) also includes sets each consisting of a single element, for example (m1), (m2), (m3), .... Only this type of subset will be |M|, and in addition to them, the Boolean also includes other subsets, which means that in any case |M| ≤ 2|M|

2. Let us prove the strictness of the inequality α<2α

Taking into account what was proven in paragraph 1. it is enough to show that a situation in which α=2α. Let us assume the opposite, let α=2α, i.e. |M| = 2|M|. This means that M is equivalent to P(M), which means there is a mapping of the set M onto its Boolean P(M). That. Each element m of the set M has a one-to-one correspondence with some subset Mm belonging to P(M). This means that any element m either belongs to the corresponding subset Mm or does not belong. Let us construct a set M* formed from all elements of the second kind (i.e. those m that do not belong to their corresponding subsets Mm)

By construction it is clear that if any element m belongs to M*, then it automatically does not belong to Mm. This, in turn, means that for any m the situation M*=Мm is impossible. This means that the set M* is different from all sets Mm and for it there is no one-to-one element m from the set M. This in turn means that the equality |M|= 2|M| wrong. That. it has been proven that |M| < 2|M| or α<2α , Q.E.D.

When applied to the consideration of infinite sets, this convincingly proves that the set of all subsets of natural numbers (and this, in fact, is the set of complexes of infinite length) is NOT equivalent to the set of natural numbers themselves. That is, À0 ≠ 2À0. And this means, by analogy, it is possible to construct an even more extensive set, for example, based on real numbers. In other words, the question regarding other types of infinite sets is: is there a set of cardinality greater than the cardinality of the set of real numbers? If such a question is answered positively, the next one immediately arises: is there a set of even greater power? Then even more. And finally, a logical global question: is there a set of the greatest cardinality?

T.2 Theorem

For any set A there is a set B whose cardinality is greater than A.

Proof

Consider the set IN all functions defined on the set A and taking values ​​0 and 1. Each point A sets A let us associate the function fa(x), which takes the value 1 at this point and the value 0 at other points. It is clear that different functions correspond to different points. It follows that the cardinality of the set IN no less than the power of the set A (|B|≥|A|).

Let us assume that there are many powers A And IN equal to each other. In this case, there is a one-to-one correspondence between the elements of the sets A And IN. Let us denote the function corresponding to the element A from many A, through fa(x). All functions of the family fa(x) take the value either 0 or 1. Let's construct a new function φ(x)=1- fх(x). Thus, to find the value of the function φ(x) at some point A, belonging to the set A, we must first find the corresponding function fa( A) and then subtract from unity the value of this function at the point A. From the construction it is clear that the function φ(x) is also defined on the set A and takes values ​​0 and 1. Therefore, φ(x) is an element of the set IN. Then there is a number b in the set A such that φ(x) = fb(x). Taking into account the previously introduced definition of the function φ(x)=1- fх(x), we obtain that for all x belonging to the set A, true 1 - fх(x)= fb(x). Let x = b. Then 1 - fb(b) = fb(b) and that means fb(b)=1/2. This result clearly contradicts the fact that the values ​​of the function fb(x) are equal to zero or one. Consequently, the accepted assumption is incorrect, which means there is no one-to-one correspondence between the elements of the sets A And IN (| A| ≠| B| ). Because | A| ≠|B| and at the same time | B| ≥| A| , Means | B| >|A| . This means that for any set A you can build a set IN more power. From this we can conclude that there is no set of the greatest cardinality, Q.E.D.

There is a fairly close connection between the constructed set of functions and the Boolean set A(the set of all subsets A). Consider the set IN all subsets of the set A. Let WITH– some subset in A. Let's take the function f(x) , which takes the value 1 if X belongs WITH, and the value is 0 otherwise. Thus, different subsets WITH correspond to various functions. On the contrary, each function f(x) , taking two values ​​0 and 1, corresponds to a subset in A, consisting of those elements X, in which the function takes the value 1. Thus, a one-to-one correspondence has been established between the set of functions defined on the set A and taking values ​​0 and 1, and the set of all subsets in A.

§ 2.5. Sets with cardinality greater than the cardinality of the continuum

So, there is no set of the greatest cardinality. The first two transfinite numbers had sets in nature that formed them (the set of natural numbers and the set of real numbers). If we start from the set of the continuum, then we can construct the set of all subsets of the continuum, we will obtain its Boolean, let’s call this set BR. By definition, the power of the set BR is equal to 2А. According to Cantor's theorem 2À≠À. It is obvious that the set BR is infinite, therefore, its cardinal number is a transfinite number and it cannot coincide with any of the two transfinite numbers considered earlier. This means it’s time to introduce the third transfinite number into our scale.

Aleph One ( À 1 ) – third transfinite number. By definition, this is the cardinality of the set of all subsets of the continuum. The same number corresponds to the cardinality of many other sets, for example:

· Sets of all linear functions that take any real values ​​(a linear function is a real function of one or more variables). Essentially, these are sets of all possible curves in a countably-dimensional space, where the number of dimensions n is any finite number or even À0.

· Sets of figures on the plane, i.e., sets of all subsets of points on the plane or sets of all subsets of pairs of real numbers.

· Sets of bodies in ordinary three-dimensional space, and also, generally speaking, in any countable-dimensional space, where the number of dimensions n is any finite number or even À0.

Since the number À1 is introduced as the cardinality of the Boolean set with cardinality À, we obtain the statement that À1 =2À.

§ 2.6. Paradoxes of set theory

A reasonable question arises: what next? What happens if we construct the set of all subsets of the set BR. What will its cardinal number be equal to (of course, by analogy we can assume that it is 2À1) and, most importantly, what real-life set will this correspond to? Are there infinite sets larger than BR and how many are there?

Although we have shown that the largest transfinite number does not exist, as research shows, it is unsafe to ascend further and further to new large cardinal numbers - this leads to antinomy (paradoxes). Indeed, whatever the set of cardinal numbers, it is always possible to find a cardinal number that is greater than all the numbers in a given set and, therefore, not included in it. That. no such set contains all the cardinal numbers, and the set of all cardinal numbers is unthinkable.

It is quite natural that every mathematician wants to deal with a consistent theory, that is, one in which it is impossible to simultaneously prove two theorems that clearly deny each other. Is Cantor's theory consistent? To what extent can we expand the range of sets under consideration? Unfortunately, not everything is so rosy. If we introduce such a seemingly harmless concept as “the set of all sets U,” a number of interesting points arise.

https://pandia.ru/text/78/390/images/image009_32.gif" width="81" height="75 src="> T.2.6.(2) Russell's paradox

Let B be the set of all sets that do not contain themselves as their own elements. Then two theorems can be proven.

Theorem 2.6.(2).1.

B belongs to V.

Proof

Let's assume the opposite, i.e. IN doesn't belong IN. By definition, this means that IN belongs IN. We get a contradiction - therefore, the original assumption is incorrect and IN belongs IN, Q.E.D.

Theorem 2.6.(2).2.

B does not belong to V.

Proof

Let's assume the opposite, i.e. IN belongs IN. By definition of a set IN any element of it cannot have itself as its own element, therefore, IN doesn't belong IN. A contradiction - therefore, the original assumption is incorrect and IN doesn't belong IN, Q.E.D.

It is easy to see that Theorems 2.6.(2).1. and 2.6.(2).2. exclude each other.

Unfortunately, even excluding all superextensive sets from consideration does not save Cantor's theory. In essence, Russell's paradox affects logic, that is, the methods of reasoning by which new concepts are formed when moving from one true statement to another.

Already when deriving a paradox, the logical law of excluded middle is used, which is one of the integral methods of reasoning in classical mathematics (i.e., if the statement not-A is true, then A is false). If you think about the essence of things, then you can generally move away from set theory and mathematics in general.

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It is one of the basic undefined concepts of mathematics. A set is understood as a collection (collection, class, family...) of some objects united by some characteristic. So we can talk about the many students at the institute, about the many fish in the Black Sea, about the many roots of the equation x 2 + 2x + 2 = 0, about many all natural numbers, etc.

The objects that make up a set are called its elements. Sets are usually denoted in capital letters Latin alphabet A, B,..., X, Y,..., and their elements - in small letters a, b,... ..., x, y,...

If an element x belongs to the set X, then write x О X; record xÏ X or x Î X means that element x does not belong to set X.

For example, the notation A=(1,3,15) means that set A consists of three numbers 1, 3 and 15; the notation A=(x:0≤x≤2) means that the set A consists of all real (unless otherwise stated) numbers satisfying the inequality 0 ≤ x ≤ 2.

Many A is called a subset of the set B if every element of the set A is an element of the set B. Symbolically this is denoted as AÌ B (“A is included in B”) or BÉ A (“the set B includes the set A”).

They say that sets A and B are equal or the same, and write A=B if AÌ B and BÌ A. In other words, sets, consisting of the same elements, are called equal.

Association(or the sum) of sets A and B is a set consisting of elements, each of which belongs to at least one of these sets. The union (sum) of sets is denoted by AUB (or A+B). Briefly, you can write АУВ = (x: xєA or xєB).

The intersection (or product) of sets A and B is a set consisting of elements, each of which belongs to set A and set B. The intersection (product) of sets is denoted A∩B (or A*B). Briefly we can write A∩B=(x:xєA and xєB)

In the future, we will use some simple logical symbols to shorten records:

ΑÞ ß - means “from the sentence α follows the sentence ß”;

ΑÛ ß - “the propositions α and ß are equivalent,” that is, from α follows ß and from ß follows α;

" - means “for anyone”, “for everyone”;

$ - “exists”, “will be found”;

: - “takes place”, “such that”;

→ - “compliance”.

For example:
1) the entry “xО А:α means: “for every element xО А the proposition α holds”;
2) (х єA U В)<==>(x є A or x є B); this entry defines the union of sets A and B.

13.2. Numerical sets. Set of real numbers

Sets whose elements are numbers are called numerical. Examples of number sets are:

N=(1; 2; 3; ...; n; ... ) - set of natural numbers;

Zo=(0; 1; 2; ...; n; ... ) - set of non-negative integers;

Z=(0; ±1; ±2; ...; ±n; ...) - set of integers;

Q=(m/n: mО Z,nО N) - set of rational numbers.

R-set of real numbers.

There is a relationship between these sets

NÌ ZoÌ ZÌ QÌ R.

Many R contains rational and irrational numbers. Every rational number is expressed either as a finite decimal fraction or as an infinite periodic fraction. So, 1/2= 0.5 (= 0.500...), 1/3=0.333... are rational numbers.

Real numbers that are not rational are called irrational.

Theorem 13.1.

There is no rational number whose square equal to the number 2.

▼Suppose that there is a rational number, represented by an irreducible fraction m/n, whose square is equal to 2. Then we have:

(m/n) 2 =2, i.e. m 2 =2n 2.

It follows that m 2 (and therefore m) - even number, i.e. m=2k. Substituting m=2k into the equality m 2 =2n 2, we get 4k 2 = 2n 2, i.e. 2k 2 =n 2,

It follows that the number is n-even, i.e. n=2l. But then the fraction m/n=2k/2l is reducible. This contradicts the assumption that m/n is an irreducible fraction. Therefore, there is no rational number whose square is equal to the number 2. ▲

An irrational number is expressed as an infinite non-periodic fraction. So, √2=1.4142356... are irrational numbers. We can say: the set of real numbers is the set of all infinite decimal fractions. And write it down

R=(x: x=α,α 1 α 2 α 3 ...), where aєZ, and i є(0,1,...,9).

Many R real numbers have the following properties.

1. It is ordered: for any two different numbersα and b, one of two relations holds: a

2. Many R is dense: between any two distinct numbers a and b there is an infinite set of real numbers x, that is, numbers satisfying the inequality a<х

So, if a

(a

3. Many R continuous. Let the set R be divided into two non-empty classes A and B such that each real number is contained in only one class and for each pair of numbers aєA and bєB the inequality a

The property of continuity allows us to establish a one-to-one correspondence between many of all real numbers and the set of all points on a line. This means that each number xєR corresponds to a certain (single) point on the numerical axis and, conversely, each point on the axis corresponds to a certain (single) real number. Therefore, instead of the word “number” they often say “dot”.