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- whole numbers;
Fractions with denominator 10 n, Where n- an integer, called decimal: .
Among decimal fractions, a special place is occupied by 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, the complex numbers are 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 the number is called 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 it is impossible to find an exact number 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, the absolute error of the approximate number 1.2 is 1.214 - 1.2, i.e. 0.014.
But in most cases, the exact value of the value under consideration is unknown, but only an approximate one. 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
There are many types of numbers, one of them is integers. Integers appeared in order to facilitate counting not only in the positive direction, but also in the negative direction.
Let's look at an example:
During the day the temperature outside was 3 degrees. By evening the temperature dropped by 3 degrees.
3-3=0
It became 0 degrees outside. And at night the temperature dropped by 4 degrees and the thermometer began to show -4 degrees.
0-4=-4
A series of integers.
We cannot describe such a problem using natural numbers; we will consider this problem on a coordinate line.
We got a series of numbers:
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
This series of numbers is called series of integers.
Positive integers. Negative integers.
The series of integers consists of positive and negative numbers. To the right of zero are the natural numbers, or they are also called positive integers. And to the left of zero they go negative integers.
Zero is neither a positive nor a negative number. It is the boundary between positive and negative numbers.
is a set of numbers consisting of natural numbers, negative integers and zero.
A series of integers in a positive and negative direction is an infinite number.
If we take any two integers, then the numbers between these integers will be called finite set.
For example:
Let's take integers from -2 to 4. All numbers between these numbers are included in the finite set. Our final set of numbers looks like this:
-2, -1, 0, 1, 2, 3, 4.
Natural numbers are denoted by the Latin letter N.
Integers are denoted by the Latin letter Z. The entire set of natural numbers and integers can be depicted in a picture. 
Non-positive integers in other words, they are negative integers.
Non-negative integers are positive integers.
In this article we will define the set of integers, consider which integers are called positive and which are negative. We will also show how integers are used to describe changes in certain quantities. Let's start with the definition and examples of integers.
Whole numbers. Definition, examples
First, let's remember about natural numbers ℕ. The name itself suggests that these are numbers that have naturally been used for counting since time immemorial. In order to cover the concept of integers, we need to expand the definition of natural numbers.
Definition 1. Integers
Integers are the natural numbers, their opposites, and the number zero.
The set of integers is denoted by the letter ℤ.
The set of natural numbers ℕ is a subset of the integers ℤ. Every natural number is an integer, but not every integer is a natural number.
From the definition it follows that any of the numbers 1, 2, 3 is an integer. . , the number 0, as well as the numbers - 1, - 2, - 3, . .
In accordance with this, we will give examples. The numbers 39, - 589, 10000000, - 1596, 0 are integers.
Let the coordinate line be drawn horizontally and directed to the right. Let's take a look at it to visualize the location of integers on a line.
The origin on the coordinate line corresponds to the number 0, and points lying on both sides of zero correspond to positive and negative integers. Each point corresponds to a single integer.
You can get to any point on a line whose coordinate is an integer by setting aside a certain number of unit segments from the origin.
Positive and negative integers
Of all the integers, it is logical to distinguish positive and negative integers. Let us give their definitions.
Definition 2: Positive integers
Positive integers are integers with a plus sign.
For example, the number 7 is an integer with a plus sign, that is, a positive integer. On the coordinate line, this number lies to the right of the reference point, which is taken to be the number 0. Other examples of positive integers: 12, 502, 42, 33, 100500.
Definition 3: Negative integers
Negative integers are integers with a minus sign.
Examples of negative integers: - 528, - 2568, - 1.
The number 0 separates positive and negative integers and is itself neither positive nor negative.
Any number that is the opposite of a positive integer is, by definition, a negative integer. The opposite is also true. The inverse of any negative integer is a positive integer.
It is possible to give other formulations of the definitions of negative and positive integers using their comparison with zero.
Definition 4: Positive integers
Positive integers are integers that are greater than zero.
Definition 5: Negative integers
Negative integers are integers that are less than zero.
Accordingly, positive numbers lie to the right of the origin on the coordinate line, and negative integers lie to the left of zero.
We said earlier that natural numbers are a subset of integers. Let's clarify this point. The set of natural numbers consists of positive integers. In turn, the set of negative integers is the set of numbers opposite to the natural ones.
Important!
Any natural number can be called an integer, but any integer cannot be called a natural number. When answering the question whether negative numbers are natural numbers, we must boldly say - no, they are not.
Non-positive and non-negative integers
Let's give some definitions.
Definition 6. Non-negative integers
Non-negative integers are positive integers and the number zero.
Definition 7. Non-positive integers
Non-positive integers are negative integers and the number zero.
As you can see, the number zero is neither positive nor negative.
Examples of non-negative integers: 52, 128, 0.
Examples of non-positive integers: - 52, - 128, 0.
A non-negative number is a number greater than or equal to zero. Accordingly, a non-positive integer is a number less than or equal to zero.
The terms "non-positive number" and "non-negative number" are used for brevity. For example, instead of saying that the number a is an integer that is greater than or equal to zero, you can say: a is a non-negative integer.
Using integers to describe changes in quantities
What are integers used for? First of all, with their help it is convenient to describe and determine changes in the quantity of any objects. Let's give an example.
Let a certain number of crankshafts be stored in a warehouse. If 500 more crankshafts are brought to the warehouse, their number will increase. The number 500 precisely expresses the change (increase) in the number of parts. If 200 parts are then taken from the warehouse, then this number will also characterize the change in the number of crankshafts. This time, downward.
If nothing is taken from the warehouse and nothing is delivered, then the number 0 will indicate that the number of parts remains unchanged.
The obvious convenience of using integers, as opposed to natural numbers, is that their sign clearly indicates the direction of change in the value (increase or decrease).
A decrease in temperature by 30 degrees can be characterized by a negative integer - 30, and an increase by 2 degrees - by a positive integer 2.
Let's give another example using integers. This time, let's imagine that we have to give 5 coins to someone. Then, we can say that we have - 5 coins. The number 5 describes the size of the debt, and the minus sign indicates that we must give away the coins.
If we owe 2 coins to one person and 3 to another, then the total debt (5 coins) can be calculated using the rule of adding negative numbers:
2 + (- 3) = - 5
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TO integers include natural numbers, zero, and numbers opposite to natural numbers.
Integers are positive integers.
For example: 1, 3, 7, 19, 23, etc. We use such numbers for counting (there are 5 apples on the table, a car has 4 wheels, etc.)
Latin letter \mathbb(N) - denoted set of natural numbers.
Natural numbers cannot include negative numbers (a chair cannot have a negative number of legs) and fractional numbers (Ivan could not sell 3.5 bicycles).
The opposite of natural numbers are negative integers: −8, −148, −981, ….
Arithmetic operations with integers
What can you do with integers? They can be multiplied, added and subtracted from each other. Let's look at each operation using a specific example.
Addition of integers
Two integers with the same signs are added as follows: the modules of these numbers are added and the resulting sum is preceded by a final sign:
(+11) + (+9) = +20
Subtracting Integers
Two integers with different signs are added as follows: from the modulus of the larger number, the modulus of the smaller one is subtracted and the sign of the larger modulo number is placed in front of the resulting answer:
(-7) + (+8) = +1
Multiplying Integers
To multiply one integer by another, you need to multiply the moduli of these numbers and put a “+” sign in front of the resulting answer if the original numbers had the same signs, and a “−” sign if the original numbers had different signs:
(-5)\cdot (+3) = -15
(-3)\cdot (-4) = +12
The following should be remembered rule for multiplying integers:
+ \cdot + = +
+ \cdot - = -
- \cdot + = -
- \cdot - = +
There is a rule for multiplying multiple integers. Let's remember it:
The sign of the product will be “+” if the number of factors with a negative sign is even and “−” if the number of factors with a negative sign is odd.
(-5) \cdot (-4) \cdot (+1) \cdot (+6) \cdot (+1) = +120
Integer division
The division of two integers is carried out as follows: the modulus of one number is divided by the modulus of the other, and if the signs of the numbers are the same, then the sign “+” is placed in front of the resulting quotient, and if the signs of the original numbers are different, then the sign “−” is placed.
(-25) : (+5) = -5
Properties of addition and multiplication of integers
Let's look at the basic properties of addition and multiplication for any integers a, b and c:
- a + b = b + a - commutative property of addition;
- (a + b) + c = a + (b + c) - combinative property of addition;
- a \cdot b = b \cdot a - commutative property of multiplication;
- (a \cdot c) \cdot b = a \cdot (b \cdot c)- associative properties of multiplication;
- a \cdot (b \cdot c) = a \cdot b + a \cdot c- distributive property of multiplication.
These are the numbers that are used when counting: 1, 2, 3... etc.
Zero is not natural.
Natural numbers are usually denoted by the symbol N.
Whole numbers. Positive and negative numbers
Two numbers that differ from each other only by sign are called opposite, for example, +1 and -1, +5 and -5. The "+" sign is usually not written, but it is assumed that there is a "+" in front of the number. Such numbers are called positive. Numbers preceded by a "-" sign are called negative.
The natural numbers, their opposites and zero are called integers. The set of integers is denoted by the symbol Z.
Rational numbers
These are finite fractions and infinite periodic fractions. For example,
The set of rational numbers is denoted Q. All integers are rational.
Irrational numbers
An infinite non-periodic fraction is called an irrational number. For example:
The set of irrational numbers is denoted J.
Real numbers
The set of all rational and all irrational numbers is called set of real (real) numbers.
Real numbers are represented by the symbol R.
Rounding numbers
Consider the number 8,759123... . Rounding to the nearest whole number means writing down only the part of the number that is before the decimal point. Rounding to tenths means writing down the whole part and one digit after the decimal point; round to the nearest hundredth - two digits after the decimal point; up to thousandths - three digits, etc.
