INSTRUCTIONS AND PROPHECIES OF THE Blessed MOTHER ALIPIA GOLOSEEVSKY, Kyiv...
quarters
- Orderliness. a and b there is a rule that allows you to uniquely identify between them one and only one of the three relations: “<
», « >' or ' = '. This rule is called ordering rule and is formulated as follows: two non-negative numbers and are related by the same relation as two integers and ; two non-positive numbers a and b are related by the same relation as two non-negative numbers and ; if suddenly a non-negative, and b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
summation of fractions
- addition operation. For any rational numbers a and b there is a so-called summation rule c. However, the number itself c called sum numbers a and b and is denoted , and the process of finding such a number is called summation. The summation rule has the following form:
.
- multiplication operation. For any rational numbers a and b there is a so-called multiplication rule, which puts them in correspondence with some rational number c. However, the number itself c called work numbers a and b and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule is as follows:
.
- Transitivity of the order relation. For any triple of rational numbers a , b and c if a less b and b less c, then a less c, what if a equals b and b equals c, then a equals c. 6435">Commutativity of addition. The sum does not change from changing the places of rational terms.
- Associativity of addition. The order in which three rational numbers are added does not affect the result.
- The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
- The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
- Commutativity of multiplication. By changing the places of rational factors, the product does not change.
- Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
- The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
- The presence of reciprocals. Any rational number has an inverse rational number, which, when multiplied, gives 1.
- Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
- Connection of the order relation with the operation of addition. to the left and right parts rational inequality, you can add the same rational number. /pictures/wiki/files/51/358b88fcdff63378040f8d9ab9ba5048.png" border="0">
- Axiom of Archimedes. Whatever the rational number a, you can take so many units that their sum will exceed a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">
Additional properties
All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they no longer rely directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.
Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">
Set countability
Numbering of rational numbers
To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers.
The simplest of these algorithms is as follows. An infinite table of ordinary fractions is compiled, on each i-th line in each j th column of which is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted , where i- the row number of the table in which the cell is located, and j- column number.
The resulting table is managed by a "snake" according to the following formal algorithm.
These rules are searched from top to bottom and the next position is selected by the first match.
In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions 1 / 1 are assigned the number 1, fractions 2 / 1 - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.
Following this algorithm, one can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers, simply by assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.
The statement about the countability of the set of rational numbers may cause some bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.
Insufficiency of rational numbers
The hypotenuse of such a triangle is not expressed by any rational number
Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates a deceptive impression that rational numbers can measure any geometric distances in general. It is easy to show that this is not true.
Notes
Literature
- I. Kushnir. Handbook of mathematics for schoolchildren. - Kyiv: ASTARTA, 1998. - 520 p.
- P. S. Alexandrov. Introduction to set theory and general topology. - M.: head. ed. Phys.-Math. lit. ed. "Science", 1977
- I. L. Khmelnitsky. Introduction to the theory of algebraic systems
Links
Wikimedia Foundation. 2010 .
In this lesson, we will get acquainted with the set of rational numbers. Let's analyze the basic properties of rational numbers, learn how to translate decimals to ordinary and vice versa.
We have already talked about the sets of natural and integer numbers. The set of natural numbers is a subset of integers.
Now we have learned what fractions are, we have learned how to work with them. A fraction, for example, is not an integer. This means that it is necessary to describe a new set of numbers, which will include all fractions, and this set needs a name, a clear definition and designation.
Let's start with the title. The Latin word ratio is translated into Russian as ratio, fraction. The name of the new set "rational numbers" comes from this word. That is, "rational numbers" can be translated as "fractional numbers".
Let's figure out what numbers this set consists of. It can be assumed that it consists of all fractions. For example, such -. But such a definition would not be entirely correct. A fraction is not a number itself, but a form of writing a number. In the example below, two different fractions represent the same number:
Then it will be more accurate to say that rational numbers are those numbers that can be represented as a fraction. And this is actually almost the same definition that is used in mathematics.
This set is denoted by the letter . And how are the sets of natural and integer numbers connected with the new set of rational numbers? A natural number can be written as a fraction in an infinite number of ways. And since it can be represented as a fraction, it is also rational.
The situation is similar with negative integers. Any negative integer can be expressed as a fraction 
. Can zero be represented as a fraction? Of course you can, also in an infinite number of ways. 
.
Thus, all natural numbers and all integers are also rational numbers. The sets of natural and integer numbers are subsets of the set of rational numbers ().
Closure of sets with respect to arithmetic operations
The need to introduce new numbers - integers, then rational ones - can be explained not only by problems from real life. The arithmetic operations themselves tell us this. Let's add two natural numbers: . We get a natural number again.
They say that the set of natural numbers is closed under the operation of addition (closed under addition). Think for yourself whether the set of natural numbers is closed under multiplication.
As soon as we try to subtract from a number equal to it or greater, then we do not have enough natural numbers. Introducing zero and negative integers fixes the situation:
The set of integers is closed under subtraction. We can add and subtract any whole numbers without fear that we won't have a number to write down the result (closed under addition and subtraction).
Is the set of integers closed under multiplication? Yes, the product of any two integers results in an integer (closed under addition, subtraction, and multiplication).
There is one more action left - division. Is the set of integers closed under division? The answer is obvious: no. Let's divide by . Among the integers there is no one to write down the answer: .
But using a fractional number, we can almost always write down the result of dividing one whole number by another. Why almost? Recall that, by definition, you can't divide by zero.
Thus, the set of rational numbers (which arises from the introduction of fractions) claims to be a set that is closed under all four arithmetic operations.
Let's check.
That is, the set of rational numbers is closed under addition, subtraction, multiplication, and division, excluding division by zero. In this sense, we can say that the set of rational numbers is arranged "better" than the previous sets of natural and integer numbers. Does this mean that the rational numbers are the last set of numbers we study? No. Subsequently, we will have other numbers that cannot be written as fractions, for example, irrational ones.
Numbers as a tool
Numbers are a tool that man created as needed.
Rice. 1. Use of natural numbers
Further, when it was necessary to lead cash settlements, they began to put plus or minus signs in front of the number, showing whether it is necessary to increase or decrease the original value. So there were negative and positive numbers. The new set was called the set of integers ().
Rice. 2. Use of fractional numbers
Therefore, it appears new tool, new numbers are fractions. We write them in different equivalent ways: ordinary and decimal fractions ( 
).
All numbers - "old" (integer) and "new" (fractional) - were combined into one set and called it the set of rational numbers ( - rational numbers)
So, a rational number is a number that can be represented as an ordinary fraction. But this definition in mathematics is still a little more precise. Any rational number can be represented as a fraction with a positive denominator, that is, the ratio of an integer to a natural number: 
.
Then we get the definition: a number is called rational if it can be represented as a fraction with an integer numerator and a natural denominator ( 
).
In addition to ordinary fractions, we also use decimals. Let's see how they are related to the set of rational numbers.
There are three types of decimal fractions: finite, periodic, and non-periodic.
Infinite non-periodic fractions: such fractions also have an infinite number of digits after the decimal point, but there is no period. An example is the decimal notation for the number PI: 
Any finite decimal fraction is, by definition, an ordinary fraction with a denominator, and so on.
We read the decimal fraction aloud and write it in the form of an ordinary:,.
In the reverse transition from writing in the form of an ordinary fraction to a decimal one, final decimal fractions or infinite periodic fractions can be obtained.
Change from fraction to decimal
The simplest case is when the denominator of a fraction is a power of ten: and so on. Then we use the definition of a decimal fraction:
There are fractions in which the denominator is easily reduced to this form: . It is possible to go to such a notation if only twos and fives are included in the expansion of the denominator.
The denominator consists of three twos and one five. Each one forms a ten. So we are missing two. Multiply by both the numerator and the denominator:
It could have been done differently. Divide by a column by (see Fig. 1).
Rice. 2. Long division
In the case of c, the denominator cannot be turned into or another bit number, since its expansion includes a triple. There is only one way left - to divide into a column (see Fig. 2).
Such a division at each step will give the remainder and the quotient. This process is endless. That is, we got an infinite periodic fraction with a period
Let's practice. Convert ordinary fractions to decimals.
In all these examples, we got the final decimal fraction, since there were only twos and fives in the expansion of the denominator.
(let's check ourselves by dividing into a table - see Fig. 3).
Rice. 3. Long division
Rice. 4. Long division
(see fig. 4)
The expansion of the denominator includes a triple, which means to bring the denominator to the form , etc. will not work. We divide by into a column. The situation will repeat itself. There will be an infinite number of triples in the result record. In this way, .
(see fig. 5)
Rice. 5. Long division
So, any rational number can be represented as an ordinary fraction. This is his definition.
And any ordinary fraction can be represented as a finite or infinite periodic decimal fraction.
Types of writing fractions:
writing a decimal fraction in the form of an ordinary: ; 
;
writing an ordinary fraction as a decimal: (final fraction); (infinite periodic).
That is, any rational number can be written as a finite or periodic decimal fraction. In this case, the final fraction can also be considered periodic with a period of zero.
Sometimes a rational number is given just such a definition: a rational number is a number that can be written as a periodic decimal fraction.
Periodic Fraction Transformation
Consider first a fraction whose period consists of one digit and has no preperiod. Let's denote this number as . The method is to get another number with the same period:
This can be done by multiplying the original number by . So the number has the same period. Subtract from the number itself:
To make sure that we did everything right, let's now make the transition in the opposite direction, in the way we already know - by dividing into a column by (see Fig. 1).
In fact, we get a number in its original form with a period of .
Consider a number with a preperiod and a longer period: . The method remains exactly the same as in the previous example. You need to get a new number with the same period and a pre-period of the same length. To do this, you need the comma to move to the right by the length of the period, i.e. for two characters. Multiply the original number by:
Subtract the original expression from the resulting expression:
So, what is the translation algorithm. A periodic fraction must be multiplied by a number of the form, etc., in which there are as many zeros as there are digits in the period of the decimal fraction. We get a new periodic. For example:
We subtract another from one periodic fraction, we get the final decimal fraction:
It remains to express the original periodic fraction in the form of an ordinary.
To practice on your own, write down a few periodic fractions. Using this algorithm, bring them to the form of an ordinary fraction. To check on a calculator, divide the numerator by the denominator. If everything is correct, then you get the original periodic fraction
So, we can write any finite or infinite periodic fraction as an ordinary fraction, as a ratio of natural and integer numbers. Those. all such fractions are rational numbers.
What about non-periodic fractions? It turns out that non-periodic fractions cannot be represented as ordinary fractions (we will accept this fact without proof). So they are not rational numbers. They are called irrational.
Infinite non-periodic fractions
As we have already said, a rational number in decimal notation is either a finite or a periodic fraction. So, if we can build an infinite non-periodic fraction, then we will get a non-rational, that is, an irrational number.
Here is one way to do this: The fractional part of this number consists only of zeros and ones. The number of zeros between ones increases by . It is impossible to single out a repeating part here. That is, the fraction is not periodic.
Practice constructing non-recurring decimal fractions, that is, irrational numbers, on your own
An example of an irrational number known to us is the number pi ( ). There is no period in this entry. But, besides pi, there are infinitely many other irrational numbers. We will talk more about irrational numbers later.
- Mathematics 5th grade. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I., 31st ed., ster. - M: Mnemosyne, 2013.
- Mathematics 5th grade. Erina T.M.. Workbook to the textbook Vilenkina N.Ya., M .: Exam, 2013.
- Mathematics 5th grade. Merzlyak A.G., Polonsky V.B., Yakir M.S., M.: Ventana - Graf, 2013.
- Math-prosto.ru ().
- Cleverstudents.ru ().
- Mathematics-repetition.com().
Homework
Definition of rational numbers:
A rational number is a number that can be represented as a fraction. The numerator of such a fraction belongs to the set of integers, and the denominator belongs to the set of natural numbers.
Why are numbers called rational?
In Latin "ratio" (ratio) means ratio. Rational numbers can be represented as a ratio, i.e. in other words, as a fraction.
Rational number example
The number 2/3 is a rational number. Why? This number is represented as a fraction, the numerator of which belongs to the set of integers, and the denominator belongs to the set of natural numbers.
For more examples of rational numbers, see the article.
Equal rational numbers
Different fractions can represent the same rational number.
Consider the rational number 3/5. This rational number is equal to
Let's reduce the numerator and denominator by common factor 2:
| 6 | = | 2 * 3 | = | 3 |
|---|---|---|---|---|
| 10 | 2 * 5 | 5 |
We got the fraction 3/5, which means that
Set of rational numbers
The set of rational numbers is denoted and can be written as follows:
It turns out that different entries can represent the same fraction, for example, and , (all fractions that can be obtained from each other by multiplying or dividing by the same natural number represent the same rational number). Since by dividing the numerator and denominator of a fraction by their greatest common divisor, one can obtain the only irreducible representation of a rational number, one can speak of their set as a set irreducible fractions with coprime integer numerator and natural denominator:
Here is the greatest common divisor of numbers and .
The set of rational numbers is a natural generalization of the set of integers. It is easy to see that if a rational number has a denominator , then it is an integer. The set of rational numbers is located everywhere densely on the number axis: between any two different rational numbers there is at least one rational number (and, therefore, an infinite set of rational numbers). However, it turns out that the set of rational numbers has a countable cardinality (that is, all its elements can be renumbered). Note, by the way, that even the ancient Greeks were convinced of the existence of numbers that cannot be represented as a fraction (for example, they proved that there is no rational number whose square is 2).
Terminology
Formal definition
Formally, rational numbers are defined as the set of equivalence classes of pairs with respect to the equivalence relation if . In this case, the operations of addition and multiplication are defined as follows:
Related definitions
Proper, improper and mixed fractions
correct A fraction is called if the modulus of the numerator is less than the modulus of the denominator. Proper fractions represent rational numbers, modulo less than one. A fraction that is not proper is called wrong and represents a rational number greater than or equal to one modulo.
An improper fraction can be represented as the sum of a whole number and a proper fraction called mixed fraction . For example, . A similar notation (with a missing addition sign), although used in elementary arithmetic, is avoided in strict mathematical literature due to the similarity of the notation mixed fraction with the notation of the product of an integer and a fraction.
Shot Height
Height of a common fraction is the sum of the modulus of the numerator and denominator of this fraction. Height of a rational number is the sum of the modulus of the numerator and denominator of the irreducible ordinary fraction corresponding to this number.
For example, the height of a fraction is . The height of the corresponding rational number is , since the fraction is reduced by .
Comment
Term fractional number (fraction) sometimes [ clarify] is used as a synonym for the term rational number, and sometimes a synonym for any non-integer number. In the latter case, fractional and rational numbers are different things, since then non-integer rational numbers are just a special case of fractional ones.
Properties
Basic properties
The set of rational numbers satisfy sixteen basic properties that can be easily obtained from the properties of integers.
- Orderliness. For any rational numbers, there is a rule that allows you to uniquely identify between them one and only one of the three relations: "", "" or "". This rule is called ordering rule and is formulated as follows: two positive numbers and are related by the same relation as two integers and ; two non-positive numbers and are related by the same relation as two non-negative numbers and ; if suddenly non-negative, but - negative, then .
summation of fractions
- addition operation. summation rule sum numbers and and is denoted by , and the process of finding such a number is called summation. The summation rule has the following form: .
- multiplication operation. For any rational numbers and there is a so-called multiplication rule, which puts them in correspondence with some rational number . The number itself is called work numbers and and is denoted , and the process of finding such a number is also called multiplication. The multiplication rule has the following form: .
- Transitivity of the order relation. For any triple of rational numbers , and if less than and less than , then less than , and if equal to and equal to , then equal to .
- Commutativity of addition. From a change in the places of rational terms, the sum does not change.
- Associativity of addition. The order in which three rational numbers are added does not affect the result.
- The presence of zero. There is a rational number 0 that preserves every other rational number when summed.
- The presence of opposite numbers. Any rational number has an opposite rational number, which, when summed, gives 0.
- Commutativity of multiplication. By changing the places of rational factors, the product does not change.
- Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
- The presence of a unit. There is a rational number 1 that preserves every other rational number when multiplied.
- The presence of reciprocals. Any non-zero rational number has an inverse rational number, multiplication by which gives 1.
- Distributivity of multiplication with respect to addition. The multiplication operation is consistent with the addition operation through the distribution law:
- Connection of the order relation with the operation of addition. The same rational number can be added to the left and right sides of a rational inequality.
- Connection of the order relation with the operation of multiplication. The left and right sides of a rational inequality can be multiplied by the same positive rational number.
- Axiom of Archimedes. Whatever the rational number , you can take so many units that their sum will exceed .
Additional properties
All other properties inherent in rational numbers are not singled out as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proved on the basis of the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense here to cite just a few of them.
Set countability
To estimate the number of rational numbers, you need to find the cardinality of their set. It is easy to prove that the set of rational numbers is countable. To do this, it suffices to give an algorithm that enumerates rational numbers, that is, establishes a bijection between the sets of rational and natural numbers. The following simple algorithm can serve as an example of such a construction. An infinite table of ordinary fractions is compiled, on each -th row in each -th column of which there is a fraction. For definiteness, it is assumed that the rows and columns of this table are numbered from one. Table cells are denoted by , where is the row number of the table in which the cell is located, and is the column number.
The resulting table is managed by a "snake" according to the following formal algorithm.
These rules are searched from top to bottom and the next position is selected by the first match.
In the process of such a bypass, each new rational number is assigned to the next natural number. That is, fractions are assigned the number 1, fractions - the number 2, etc. It should be noted that only irreducible fractions are numbered. The formal sign of irreducibility is the equality to unity of the greatest common divisor of the numerator and denominator of the fraction.
Following this algorithm, one can enumerate all positive rational numbers. This means that the set of positive rational numbers is countable. It is easy to establish a bijection between the sets of positive and negative rational numbers, simply by assigning to each rational number its opposite. That. the set of negative rational numbers is also countable. Their union is also countable by the property of countable sets. The set of rational numbers is also countable as the union of a countable set with a finite one.
Of course, there are other ways to enumerate the rational numbers. For example, for this you can use structures such as the Calkin - Wilf tree, the Stern - Brokaw tree or the Farey series.
The statement about the countability of the set of rational numbers may cause some bewilderment, since at first glance one gets the impression that it is much larger than the set of natural numbers. In fact, this is not the case, and there are enough natural numbers to enumerate all rational ones.
Insufficiency of rational numbers
see also
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| Rational numbers | |||||
Notes
Literature
- I. Kushnir. Handbook of mathematics for schoolchildren. - Kyiv: ASTARTA, 1998. - 520 p.
- P. S. Alexandrov. Introduction to set theory and general topology. - M.: head. ed. Phys.-Math. lit. ed. "Science", 1977
- I. L. Khmelnitsky. Introduction to the theory of algebraic systems
Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. Here are the numbers:
This is a natural series of numbers.
Zero is a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite set of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It cannot be specified, because there is an infinite set of natural numbers.
The sum of natural numbers is a natural number. So, the addition of natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers There is not always a natural number. If for natural numbers a and b
where c is a natural number, it means that a is evenly divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is the natural number by which the first number is evenly divisible.
Every natural number is divisible by 1 and itself.
Simple natural numbers are only divisible by 1 and themselves. Here we mean divided completely. Example, numbers 2; 3; 5; 7 is only divisible by 1 and itself. These are simple natural numbers.
One is not considered a prime number.
The numbers that more than one and which are not simple are called composite. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers is one, prime numbers and composite numbers.
The set of natural numbers is denoted by the Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab)c = a(bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are natural numbers, zero and the opposite of natural numbers.
Numbers that are opposite to natural numbers are integers. negative numbers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are integers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
It can be seen from the examples that any integer is a periodic fraction with a period of zero.
Any rational number can be represented as a fraction m/n, where m integer,n natural number. Let's represent the number 3,(6) from the previous example as such a fraction.
