What is a rational number? What are rational numbers? What other ones are there?

Quarters

Orderliness. a And b there is a rule that allows one to uniquely identify one and only one of three relationships between them: “< », « >" 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, but b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
Adding Fractions
Addition operation. For any rational numbers a And b there is a so-called summation rule c. At the same time, the number itself c called amount numbers a And b and is denoted by , and the process of finding such a number is called summation. The summation rule has next view: .
Multiplication operation. For any rational numbers a And b there is a so-called multiplication rule, which assigns them some rational number c. At the same time, the number itself c called work numbers a And b and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule looks like this: .
Transitivity of the order relation. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
Associativity of addition. The order in which three rational numbers are added does not affect the result.
Presence of zero. There is a rational number 0 that preserves every other rational number when added.
The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
Commutativity of multiplication. Changing the places of rational factors does not change the product.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
Availability of unit. There is a rational number 1 that preserves every other rational number when multiplied.
Presence of reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
Distributivity of multiplication relative to addition. The multiplication operation is coordinated 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. /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 exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

All other properties inherent in rational numbers are not distinguished as basic ones, because, generally speaking, they are no longer based directly on the properties of integers, but can be proven based on the given basic properties or directly by the definition of some mathematical object. There are a lot of such additional properties. It makes sense to list only a few of them here.

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Countability of a set

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 is enough to give an algorithm that enumerates rational numbers, i.e., establishes a bijection between the sets of rational and natural numbers.

The simplest of these algorithms looks like this. An endless table is created ordinary fractions, on each i-th line in each j the th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are denoted by , where i- the number of the table row in which the cell is located, and j- column number.

The resulting table is traversed using a “snake” according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected based on the first match.

In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. Formal sign irreducibility is the equality of the greatest common divisor of the numerator and denominator of the fraction to one.

Following this algorithm, we 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 by simply 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 confusion, since at first glance it seems that it is much more extensive than the set of natural numbers. In fact, this is not so and there are enough natural numbers to enumerate all rational ones.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number

Rational numbers of the form 1 / n at large n arbitrarily small quantities can be measured. This fact creates the misleading impression that rational numbers can be used to measure any geometric distances. 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.: chapter. ed. physics and mathematics lit. ed. "Science", 1977
I. L. Khmelnitsky. Introduction to the theory of algebraic systems

Links

Wikimedia Foundation. 2010.

Number- an important mathematical concept that has changed over the centuries.

The first ideas about number arose from counting people, animals, fruits, various products, etc. The result is natural numbers: 1, 2, 3, 4, ...

Historically, the first extension of the concept of number is the addition of fractional numbers to the natural number.

Fraction a part (share) of a unit or several equal parts is called.

Designated by: , where m, n- integers;

Fractions with denominator 10 n, Where n- an integer, called decimal: .

Among decimals special place occupy periodic fractions: - pure periodic fraction, - mixed periodic fraction.

Further expansion of the concept of number is caused by the development of mathematics itself (algebra). Descartes in the 17th century. introduces the concept negative number.

The numbers integers (positive and negative), fractions (positive and negative), and zero are called rational numbers. Any rational number can be written as a finite and periodic fraction.

To study continuously changing variable quantities, it turned out that a new expansion of the concept of number was necessary - the introduction of real (real) numbers - by adding irrational numbers to rational numbers: irrational numbers are infinite decimal non-periodic fractions.

Irrational numbers appeared when measuring incommensurable segments (the side and diagonal of a square), in algebra - when extracting roots, an example of a transcendental, irrational number is π, e .

Numbers natural(1, 2, 3,...), whole(..., –3, –2, –1, 0, 1, 2, 3,...), rational(representable as a fraction) and irrational(not representable as a fraction ) form a set real (real) numbers.

Complex numbers are distinguished separately in mathematics.

Complex numbers arise in connection with the problem of solving squares for the case D< 0 (здесь D– discriminant of a quadratic equation). For a long time, these numbers did not find physical application, which is why they were called “imaginary” numbers. However, now they are very widely used in various fields of physics and technology: electrical engineering, hydro- and aerodynamics, elasticity theory, etc.

Complex numbers are written in the form: z= a+ bi. Here a And b – real numbers, A i – imaginary unit, i.e.e. i 2 = –1. Number a called abscissa,a b –ordinate complex number a+ bi. Two complex numbers a+ bi And a–bi are called conjugate complex numbers.

Properties:

1. Real number A can also be written in complex number form: a+ 0i or a – 0i. For example 5 + 0 i and 5 – 0 i mean the same number 5.

2. Complex number 0 + bi called purely imaginary number. Record bi means the same as 0 + bi.

3. Two complex numbers a+ bi And c+ di are considered equal if a= c And b= d. Otherwise complex numbers not equal.

Actions:

Addition. Sum of complex numbers a+ bi And c+ di is called a complex number ( a+ c) + (b+ d)i. Thus, When adding complex numbers, their abscissas and ordinates are added separately.

Subtraction. The difference of two complex numbers a+ bi(diminished) and c+ di(subtrahend) is called a complex number ( a–c) + (b–d)i. Thus, When subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. Product of complex numbers a+ bi And c+ di is called a complex number:

(ac–bd) + (ad+ bc)i. This definition follows from two requirements:

1) numbers a+ bi And c+ di must be multiplied like algebraic binomials,

2) number i has the main property: i 2 = –1.

EXAMPLE ( a+ bi)(a–bi)= a 2 +b 2 . Hence, workof two conjugate complex numbers is equal to a positive real number.

Division. Divide a complex number a+ bi(divisible) by another c+ di (divider) - means to find the third number e+ f i(chat), which when multiplied by a divisor c+ di, results in the dividend a+ bi. If the divisor is not zero, division is always possible.

EXAMPLE Find (8 + i) : (2 – 3i) .

Solution. Let's rewrite this ratio as a fraction:

Multiplying its numerator and denominator by 2 + 3 i and after performing all the transformations, we get:

Task 1: Add, subtract, multiply and divide z 1 on z 2

Extracting the square root: Solve the equation x 2 = -a. To solve this equation we are forced to use numbers of a new type - imaginary numbers . Thus, imaginary called the number the second power of which is a negative number. According to this definition of imaginary numbers we can define and imaginary unit:

Then for the equation x 2 = – 25 we get two imaginary root:

Task 2: Solve the equation:

1) x 2 = – 36; 2) x 2 = – 49; 3) x 2 = – 121

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point A means the number –3, dot B– number 2, and O-zero. In contrast, complex numbers are represented by points on the coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a+ bi will be represented by a dot P with abscissaA and ordinateb. This coordinate system is called complex plane .

Module complex number is the length of the vector OP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex number a+ bi denoted | a+ bi| or) letter r and is equal to:

Conjugate complex numbers have the same modulus.

The rules for drawing up a drawing are almost the same as for a drawing in a Cartesian coordinate system. Along the axes you need to set the dimension, note:

e
unit along the real axis; Re z

imaginary unit along the imaginary axis. Im z

Task 3. Construct the following complex numbers on the complex plane: , , , , , , ,

1. Numbers are exact and approximate. The numbers we encounter in practice are of two kinds. Some give the true value of the quantity, others only approximate. The first are called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact one, especially since in many cases exact number impossible to find at all.

So, if they say that there are 29 students in a class, then the number 29 is accurate. If they say that the distance from Moscow to Kyiv is 960 km, then here the number 960 is approximate, since, on the one hand, our measuring instruments are not absolutely accurate, on the other hand, the cities themselves have a certain extent.

The result of actions with approximate numbers is also an approximate number. By performing some operations on exact numbers (division, root extraction), you can also obtain approximate numbers.

The theory of approximate calculations allows:

1) knowing the degree of accuracy of the data, evaluate the degree of accuracy of the results;

2) take data with an appropriate degree of accuracy sufficient to ensure the required accuracy of the result;

3) rationalize the calculation process, freeing it from those calculations that will not affect the accuracy of the result.

2. Rounding. One source of obtaining approximate numbers is rounding. Both approximate and exact numbers are rounded.

Rounding a given number to a certain digit is called replacing it with a new number, which is obtained from the given one by discarding all its digits written to the right of the digit of this digit, or by replacing them with zeros. These zeros are usually underlined or written smaller. To ensure that the rounded number is as close as possible to the one being rounded, you should use the following rules: to round a number to one of a certain digit, you must discard all the digits after the digit of this digit, and replace them with zeros in the whole number. The following are taken into account:

1) if the first (on the left) of the discarded digits is less than 5, then the last remaining digit is not changed (rounding down);

2) if the first digit to be discarded is greater than 5 or equal to 5, then the last digit left is increased by one (rounding with excess).

Let's show this with examples. Round:

a) up to tenths 12.34;

b) to hundredths 3.2465; 1038.785;

c) up to thousandths 3.4335.

d) up to thousand 12375; 320729.

a) 12.34 ≈ 12.3;

b) 3.2465 ≈ 3.25; 1038.785 ≈ 1038.79;

c) 3.4335 ≈ 3.434.

d) 12375 ≈ 12,000; 320729 ≈ 321000.

3. Absolute and relative errors. The difference between the exact number and its approximate value is called the absolute error of the approximate number. For example, if the exact number 1.214 is rounded to the nearest tenth, we get an approximate number of 1.2. In this case absolute error the approximate number 1.2 is equal to 1.214 - 1.2, i.e. 0.014.

But in most cases exact value the quantity under consideration is unknown, but only approximate. Then the absolute error is unknown. In these cases, indicate the limit that it does not exceed. This number is called the limiting absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the marginal error. For example, the number 23.71 is an approximate value of the number 23.7125 with an accuracy of 0.01, since the absolute error of the approximation is 0.0025 and less than 0.01. Here the limiting absolute error is 0.01 *.

Boundary absolute error of the approximate number A denoted by the symbol Δ a. Record

x≈a(±Δ a)

should be understood as follows: the exact value of the quantity x is between the numbers A– Δ a And A+ Δ A, which are called the lower and upper bounds, respectively X and denote NG x VG X.

For example, if x≈ 2.3 (±0.1), then 2.2<x< 2,4.

Vice versa, if 7.3< X< 7,4, тоX≈ 7.35 (±0.05). The absolute or marginal absolute error does not characterize the quality of the measurement performed. The same absolute error can be considered significant and insignificant depending on the number with which the measured value is expressed. For example, if we measure the distance between two cities with an accuracy of one kilometer, then such accuracy is quite sufficient for this change, but at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable. Consequently, the accuracy of the approximate value of a quantity depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. Therefore, the relative error is a measure of accuracy.

Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the limiting absolute error to the approximate number is called the limiting relative error; they designate it like this: . Relative and marginal relative errors are usually expressed as percentages. For example, if measurements showed that the distance X between two points is more than 12.3 km, but less than 12.7 km, then the arithmetic mean of these two numbers is taken as its approximate value, i.e. their half-sum, then the marginal absolute error is equal to the half-difference of these numbers. In this case X≈ 12.5 (±0.2). Here the limiting absolute error is 0.2 km, and the limiting relative

) are numbers with a positive or negative sign (integers and fractions) and zero. A more precise concept of rational numbers sounds like this:

Rational number- a number that is represented as a common fraction m/n, where the numerator m are integers, and the denominator n- natural numbers, for example 2/3.

Infinite non-periodic fractions are NOT included in the set of rational numbers.

a/b, Where a∈ Z (a belongs to integers), b∈ N (b belongs to natural numbers).

Using rational numbers in real life.

In real life, the set of rational numbers is used to count the parts of some integer divisible objects, For example, cakes or other foods that are cut into pieces before consumption, or for roughly estimating the spatial relationships of extended objects.

Properties of rational numbers.

Basic properties of rational numbers.

1. Orderliness a And b there is a rule that allows you to unambiguously identify 1 and only one of 3 relations between them: “<», «>" or "=". This is the rule - ordering rule and formulate it like this:

2 positive numbers a=m a /n a And b=m b /n b are related by the same relationship as 2 integers m a⋅ n b And m b⋅ n a;
2 negative numbers a And b are related by the same ratio as 2 positive numbers |b| And |a|;
When a positive and b- negative, then a>b.

∀ a,b∈ Q(a ∨ a>b∨ a=b)

2. Addition operation. For all rational numbers a And b There is summation rule, which associates them with a certain rational number c. At the same time, the number itself c- This sum numbers a And b and it is denoted as (a+b) summation.

Summation Rule looks like this:

m a/n a + m b/n b =(m a⋅ n b + m b⋅ n a)/(n a⋅ n b).

∀ a,b∈ Q∃ !(a+b)∈ Q

3. Multiplication operation. For all rational numbers a And b There is multiplication rule, it associates them with a certain rational number c. The number c is called work numbers a And b and denote (a⋅b), and the process of finding this number is called multiplication.

Multiplication rule looks like this: m a n a⋅ m b n b =m a⋅ m b n a⋅ n b.

∀a,b∈Q ∃(a⋅b)∈Q

4. Transitivity of the order relation. For any three rational numbers a, b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c.

∀ a,b,c∈ Q(a ∧ b ⇒ a ∧ (a = b∧ b = c⇒ a = c)

5. Commutativity of addition. Changing the places of the rational terms does not change the sum.

∀ a,b∈ Q a+b=b+a

6. Addition associativity. The order in which 3 rational numbers are added does not affect the result.

∀ a,b,c∈ Q (a+b)+c=a+(b+c)

7. Presence of zero. There is a rational number 0, it preserves every other rational number when added.

∃ 0 ∈ Q∀ a∈ Q a+0=a

8. Presence of opposite numbers. Any rational number has an opposite rational number, and when they are added, the result is 0.

∀ a∈ Q∃ (−a)∈ Q a+(−a)=0

9. Commutativity of multiplication. Changing the places of rational factors does not change the product.

∀ a,b∈ Q a⋅ b=b⋅ a

10. Associativity of multiplication. The order in which 3 rational numbers are multiplied has no effect on the result.

∀ a,b,c∈ Q(a⋅ b)⋅ c=a⋅ (b⋅ c)

11. Unit availability. There is a rational number 1, it preserves every other rational number in the process of multiplication.

∃ 1 ∈ Q∀ a∈ Q a⋅ 1=a

12. Availability reciprocal numbers . Every rational number other than zero has an inverse rational number, multiplying by which we get 1 .

∀ a∈ Q∃ a−1∈ Q a⋅ a−1=1

13. Distributivity of multiplication relative to addition. The multiplication operation is related to addition using the distributive law:

∀ a,b,c∈ Q(a+b)⋅ c=a⋅ c+b⋅ c

14. Relationship between the order relation and the operation of addition. The same rational number is added to the left and right sides of a rational inequality.

∀ a,b,c∈ Q a ⇒ a+c

15. Relationship between the order relation and the multiplication operation. The left and right sides of a rational inequality can be multiplied by the same non-negative rational number.

∀ a,b,c∈ Q c>0∧ a ⇒ a⋅ c ⋅ c

16. Axiom of Archimedes. Whatever the rational number a, it is easy to take so many units that their sum will be greater a.

Rational numbers

Quarters

Orderliness. a And b there is a rule that allows one to uniquely identify one and only one of the three between them 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, but b- negative, then a > b. src="/pictures/wiki/files/57/94586b8b651318d46a00db5413cf6c15.png" border="0">
Adding Fractions
Addition operation. For any rational numbers a And b there is a so-called summation rule c. At the same time, the number itself c called amount numbers a And b 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 a And b there is a so-called multiplication rule, which assigns them some rational number c. At the same time, the number itself c called work numbers a And b and is denoted by , and the process of finding such a number is also called multiplication. The multiplication rule looks like this: .
Transitivity order relations. For any triple of rational numbers a , b And c If a less b And b less c, That a less c, and if a equals b And b equals c, That a equals c. 6435">Commutativity of addition. Changing the places of rational terms does not change the sum.
Associativity addition. The order in which three rational numbers are added does not affect the result.
Availability zero. There is a rational number 0 that preserves every other rational number when added.
The presence of opposite numbers. Any rational number has an opposite rational number, which when added to gives 0.
Commutativity of multiplication. Changing the places of rational factors does not change the product.
Associativity of multiplication. The order in which three rational numbers are multiplied does not affect the result.
Availability units. There is a rational number 1 that preserves every other rational number when multiplied.
Availability reciprocal numbers. Any rational number has an inverse rational number, which when multiplied by gives 1.
Distributivity multiplication relative to addition. The multiplication operation is coordinated 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. /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 exceeds a. src="/pictures/wiki/files/55/70c78823302483b6901ad39f68949086.png" border="0">

Additional properties

Src="/pictures/wiki/files/48/0caf9ffdbc8d6264bc14397db34e8d72.png" border="0">

Countability of a set

Numbering of rational numbers

To estimate the number of rational numbers, you need to find power there are many of them. It is easy to prove that the set of rational numbers countably. To do this, it is enough to give an algorithm that numbers rational numbers, i.e., establishes bijection between sets of rational and natural numbers.

The simplest of these algorithms looks like this. An endless table of ordinary fractions is compiled, on each i-th line in each j the th column of which the fraction is located. For definiteness, it is assumed that the rows and columns of this table are numbered starting from one. Table cells are denoted by , where i- the number of the table row in which the cell is located, and j- column number.

The resulting table is traversed using a “snake” according to the following formal algorithm.

These rules are searched from top to bottom and the next position is selected based on the first match.

In the process of such a traversal, each new rational number is associated with another natural number. That is, the fraction 1/1 is assigned to the number 1, the fraction 2/1 to the number 2, etc. It should be noted that only irreducible fractions are numbered. A formal sign of irreducibility is equality to one greatest common divisor numerator and denominator of the fraction.

Lack of rational numbers

The hypotenuse of such a triangle cannot be expressed by any rational number

Rational numbers of the form 1 / n at large n can be measured arbitrarily small quantities. This fact creates the misleading impression that rational numbers can measure any geometric distances. 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.: chapter. ed. physics and mathematics lit. ed. "Science", 1977
I. L. Khmelnitsky. Introduction to the theory of algebraic systems