Whole number. Integers and rational numbers. Real numbers. Integer positive numbers. Integer negative numbers

In order to do any job effectively, you need tools to dig, you need a shovel or an excavator; to think you need words. Numbers are tools that allow you to work with quantities.

It seems that we all know what a number is: 1, 2, 3… But let's talk about numbers as tools.

Let's take three objects: an apple, a balloon, the Earth (Fig. 1). What do they have in common? Shape is all balls.

Rice. 1. Illustration for example

Take three other items (Fig. 2). What do they have in common? Color - they are all blue.

Rice. 2. Illustration for example

Let us now take three sets: three cars, three apples, three pencils (Fig. 3). What do they have in common? The number is three.

Rice. 3. Illustration for example

We can put an apple on each car, and stick a pencil in each apple (Fig. 4). A common property of these sets is the number of elements.

Rice. 4. Comparison of sets

However, there are few natural numbers for solving problems, so negative, rational, irrational, etc. were also introduced. Mathematics (especially that part of it that is studied at school) is a kind of mechanism for processing signs.

Take, for example, two piles of sticks, one with seventeen pieces, and the other with twenty-five (Fig. 5). How to find out how many sticks are in both piles?

Rice. 5. Illustration for example

If there is no mechanism, then it is not clear: you can only put the sticks in one pile and count them.

But if the number of sticks is written in the decimal system familiar to us (and), then we can use the mechanisms for addition. For example, we can add numbers in a column (Fig. 6): .

Rice. 6. Stacking

Also, we will not be able to add the numbers written like this: three hundred and seventy-four plus four hundred and eighty-five. But if you write down the numbers in the decimal system, then for addition there is an algorithm - addition in a column (Fig. 7):.

Rice. 7. Stacking

If there is a car, then it is worth building a smooth road, together they are effective. Similarly: if there is a plane, then an airfield is needed. That is, the mechanism itself and the surrounding infrastructure are connected - individually they are much less effective.

In this case, there is a tool - numbers written in a positional system, and an infrastructure has been invented for them: algorithms for performing various actions, for example, addition in a column.

The numbers written in the decimal positional system replaced others (Roman, etc.) precisely because efficient and simple algorithms were invented to work with them.

Let's take a closer look at the decimal positional system. There are two main ideas that underlie it (thanks to which it got its name).

1. Decimal: we count in groups, namely tens.

2. positionality: The contribution of a digit to a number depends on its position. For example, , : the numbers are different, although they consist of the same digits.

These two ideas helped create a system that is easy to perform and write down numbers, since we have a limited set of characters (in this case, numbers) to write an infinite number of numbers.

Emphasize the importance technologies on such an example. Suppose you need to move a heavy load. If you use manual labor, then everything will depend on how strong a person is carrying the load: one will cope, the other will not.

The invention of technology (for example, a car that can carry this load) equalizes the possibilities of people: a fragile girl or a weightlifter can sit behind the wheel, but both of them will be able to cope with the task of moving the load equally effectively. That is, technology can be taught to anyone, not just a specialist.

Addition and multiplication in a column is also a technology. Working with numbers written in the Roman numeral system is a difficult task, only specially trained people could do it. Any fourth grader can add and multiply numbers in the decimal system.

As we have said, people have invented different numbers, and all of them are needed. The next (after natural) important invention are negative numbers. With the help of negative numbers, counting has become easier. How did it happen?

If we subtract the smaller from the larger, then there is no need for negative numbers: it is clear that the larger number contains the smaller. But it turned out that it is worth introducing negative numbers as a separate object. It cannot be seen, touched, but it is useful.

Consider this example: You can do the calculations in a different order: then there is no problem, we have enough natural numbers.

But sometimes there is a need to perform actions sequentially. If we run out of money in our account, we are given a loan. Let us have rubles, and we spent on conversations. There are not enough rubles on the account, it is convenient to write it down with a minus sign, since if we return them, then the account will have:. This idea underlies the invention of such a tool as negative numbers.

In life, we often work with concepts that cannot be touched: joy, friendship, etc. But this does not prevent us from understanding and analyzing them. We can say that these are just invented things. Indeed, they are, but they help people do something. Also, the car was invented by man, but it helps us move. Numbers are also invented by man, but they help to solve problems.

Let's take such an object as a clock (Fig. 8). If you pull out a part from there, it is not clear what it is and why it is needed. Without a watch, this part does not exist. So the negative number exists within mathematics.

Rice. 8. Clock

Often teachers try to indicate what a negative number is. They give an example of a negative temperature (Fig. 9).

Rice. 9. Negative temperature

But this is only a name, a designation, and not the number itself. It was possible to introduce another scale, where the same temperature would be, for example, positive. In particular, negative temperatures on the Celsius scale in the Kelvin scale are expressed as positive numbers: .

That is, there is no negative quantity in nature. However, numbers are not only used to express quantities. Recall the basic functions of the number.

So, we talked about natural and integer numbers. The number is a handy tool that can be used to solve various problems. Of course, for those working inside mathematics, numbers are objects. As for those who make pliers, they are also objects, not tools. We will consider numbers as a tool that allows us to think and work with quantities.

The information in this article forms a general idea of whole numbers. First, the definition of integers is given and examples are given. Next, the integers on the number line are considered, from which it becomes clear which numbers are called positive integers, and which are negative integers. After that, it is shown how changes in quantities are described using integers, and negative integers are considered in the sense of debt.

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Integers - definition and examples

Definition.

Whole numbers are natural numbers, the number zero, as well as numbers opposite to natural ones.

The definition of integers states that any of the numbers 1, 2, 3, …, the number 0, and also any of the numbers −1, −2, −3, … is an integer. Now we can easily bring integer examples. For example, the number 38 is an integer, the number 70040 is also an integer, zero is an integer (recall that zero is NOT a natural number, zero is an integer), the numbers −999 , −1 , −8 934 832 are also examples of integers numbers.

It is convenient to represent all integers as a sequence of integers, which has the following form: 0, ±1, ±2, ±3, … The sequence of integers can also be written as follows: …, −3, −2, −1, 0, 1, 2, 3, …

It follows from the definition of integers that the set of natural numbers is a subset of the set of integers. Therefore, every natural number is an integer, but not every integer is a natural number.

Integers on the coordinate line

Definition.

Integer positive numbers are integers that are greater than zero.

Definition.

Integer negative numbers are integers that are less than zero.

Integer positive and negative numbers can also be determined by their position on the coordinate line. On a horizontal coordinate line, points whose coordinates are positive integers lie to the right of the origin. In turn, points with negative integer coordinates are located to the left of the point O .

It is clear that the set of all positive integers is the set of natural numbers. In turn, the set of all negative integers is the set of all numbers opposite to natural numbers.

Separately, we draw your attention to the fact that we can safely call any natural number an integer, and we can NOT call any integer a natural number. We can call natural only any positive integer, since negative integers and zero are not natural.

Integer non-positive and integer non-negative numbers

Let us give definitions of nonpositive integers and nonnegative integers.

Definition.

All positive integers together with zero are called integer non-negative numbers.

Definition.

Integer non-positive numbers are all negative integers together with the number 0 .

In other words, a non-negative integer is an integer that is greater than or equal to zero, and a non-positive integer is an integer that is less than or equal to zero.

Examples of non-positive integers are numbers -511, -10 030, 0, -2, and as examples of non-negative integers, let's give numbers 45, 506, 0, 900 321.

Most often, the terms "non-positive integers" and "non-negative integers" are used for brevity. For example, instead of the phrase "the number a is an integer, and a is greater than zero or equal to zero", you can say "a is a non-negative integer".

Description of changing values ​​using integers

It's time to talk about what integers are for.

The main purpose of integers is that with their help it is convenient to describe the change in the number of any items. Let's deal with this with examples.

Suppose there is a certain amount of parts in stock. If, for example, 400 more parts are brought to the warehouse, then the number of parts in the warehouse will increase, and the number 400 expresses this change in the quantity in a positive direction (in the direction of increase). If, for example, 100 parts are taken from the warehouse, then the number of parts in the warehouse will decrease, and the number 100 will express the change in the quantity in a negative direction (in the direction of decrease). Parts will not be brought to the warehouse, and parts will not be taken away from the warehouse, then we can talk about the invariability of the number of parts (that is, we can talk about a zero change in quantity).

In the examples given, the change in the number of parts can be described using the integers 400 , −100 and 0, respectively. A positive integer 400 indicates a positive change in quantity (increase). The negative integer −100 expresses a negative change in quantity (decrease). The integer 0 indicates that the quantity has not changed.

The convenience of using integers compared to using natural numbers is that there is no need to explicitly indicate whether the quantity is increasing or decreasing - the integer determines the change quantitatively, and the sign of the integer indicates the direction of the change.

Integers can also express not only a change in quantity, but also a change in some value. Let's deal with this using the example of temperature change.

An increase in temperature by, say, 4 degrees is expressed as a positive integer 4 . A decrease in temperature, for example, by 12 degrees can be described by a negative integer −12. And the invariance of temperature is its change, determined by the integer 0.

Separately, it must be said about the interpretation of negative integers as the amount of debt. For example, if we have 3 apples, then the positive integer 3 represents the number of apples we own. On the other hand, if we have to give 5 apples to someone, and we do not have them available, then this situation can be described using a negative integer −5. In this case, we "own" −5 apples, the minus sign indicates debt, and the number 5 quantifies debt.

The understanding of a negative integer as a debt allows one, for example, to justify the rule for adding negative integers. Let's take an example. If someone owes 2 apples to one person and one apple to another, then the total debt is 2+1=3 apples, so −2+(−1)=−3 .

Bibliography.

• Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.

In this article, we will define a set of integers, consider which integers are called positive and which are negative. We will also show how integers are used to describe the change in some quantities. Let's start with the definition and examples of integers.

Whole numbers. Definition, examples

First, let's recall the 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.

It follows from the definition that any of the numbers 1 , 2 , 3 is an integer. . , the number 0 , as well as the numbers - 1 , - 2 , - 3 , . .

Accordingly, we give examples. The numbers 39 , - 589 , 10000000 , - 1596 , 0 are whole numbers.

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 straight line.

The reference point on the coordinate line corresponds to the number 0, and the points lying on both sides of zero correspond to positive and negative integers. Each point corresponds to a single integer.

Any point on a straight line whose coordinate is an integer can be reached by setting aside a certain number of unit segments from the origin.

Positive and negative integers

Of all integers, it is logical to distinguish between positive and negative integers. Let's 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, for which the number 0 is taken. 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 reverse is also true. The reciprocal 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.

Earlier we said that natural numbers are a subset of integers. Let's clarify this point. The set of natural numbers are 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. Answering the question whether negative numbers are natural, one must boldly say - no, they are not.

Non-positive and non-negative integers

Let's give 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 greater than or equal to zero, you can say: a is a non-negative integer.

Using Integers When Describing Changes in Values

What are integers used for? First of all, with their help it is convenient to describe and determine the change in the number of any objects. Let's take an example.

Let a certain number of crankshafts be stored in the warehouse. If another 500 crankshafts are brought to the warehouse, their number will increase. The number 500 just expresses the change (increase) in the number of parts. If then 200 parts are taken away from the warehouse, then this number will also characterize the change in the number of crankshafts. This time, in the direction of reduction.

If nothing is taken from the warehouse, and nothing is brought, then the number 0 will indicate the invariance of the number of parts.

The obvious convenience of using integers, in contrast to natural numbers, is that their sign clearly indicates the direction of change in magnitude (increase or decrease).

A decrease in temperature by 30 degrees can be characterized by a negative number - 30 , and an increase by 2 degrees - by a positive integer 2 .

Here is 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 amount of the debt, and the minus sign indicates that we must give back the coins.

If we owe 2 coins to one person and 3 to another, then the total debt (5 coins) can be calculated by the rule of adding negative numbers:

2 + (- 3) = - 5

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

1) I divide immediately by, since both numbers are 100% divisible by:

2) I will divide by the remaining large numbers (s), since they are divided by without a remainder (at the same time, I will not decompose - it is already a common divisor):

6 2 4 0 = 1 0 ⋅ 4 ⋅ 1 5 6

6 8 0 0 = 1 0 ⋅ 4 ⋅ 1 7 0

3) I will leave and alone and begin to consider the numbers and. Both numbers are exactly divisible by (end in even digits (in this case, we present as, but can be divided by)):

4) We work with numbers and. Do they have common divisors? It’s as easy as in the previous steps, and you can’t say, so then we’ll just decompose them into simple factors:

5) As we can see, we were right: and have no common divisors, and now we need to multiply.
GCD

Task number 2. Find GCD of numbers 345 and 324

I can’t quickly find at least one common divisor here, so I just decompose into prime factors (as few as possible):

Exactly, GCD, and I did not initially check the divisibility criterion for, and, perhaps, I would not have to do so many actions.

But you checked, right?

As you can see, it's quite easy.

Least common multiple (LCM) - saves time, helps to solve problems outside the box

Let's say you have two numbers - and. What is the smallest number that is divisible by without a trace(i.e. completely)? Hard to imagine? Here's a visual clue for you:

Do you remember what the letter means? That's right, just whole numbers. So what is the smallest number that fits x? :

In this case.

Several rules follow from this simple example.

Rules for quickly finding the NOC

Rule 1. If one of two natural numbers is divisible by another number, then the larger of these two numbers is their least common multiple.

Find the following numbers:

• NOC (7;21)
• NOC (6;12)
• NOC (5;15)
• NOC (3;33)

Of course, you easily coped with this task and you got the answers -, and.

Note that in the rule we are talking about TWO numbers, if there are more numbers, then the rule does not work.

For example, LCM (7;14;21) is not equal to 21, since it cannot be divided without a remainder by.

Rule 2. If two (or more than two) numbers are coprime, then the least common multiple is equal to their product.

find NOC for the following numbers:

• NOC (1;3;7)
• NOC (3;7;11)
• NOC (2;3;7)
• NOC (3;5;2)

Did you count? Here are the answers - , ; .

As you understand, it is not always so easy to take and pick up this same x, so for slightly more complex numbers there is the following algorithm:

Shall we practice?

Find the least common multiple - LCM (345; 234)

Let's break down each number:

Why did I just write?

Remember the signs of divisibility by: divisible by (the last digit is even) and the sum of the digits is divisible by.

Accordingly, we can immediately divide by, writing it as.

Now we write out the longest expansion in a line - the second:

Let's add to it the numbers from the first expansion, which are not in what we wrote out:

Note: we wrote out everything except for, since we already have it.

Now we need to multiply all these numbers!

Find the least common multiple (LCM) yourself

What answers did you get?

Here's what happened to me:

How long did it take you to find NOC? My time is 2 minutes, I really know one trick, which I suggest you open right now!

If you are very attentive, then you probably noticed that for the given numbers we have already searched for GCD and you could take the factorization of these numbers from that example, thereby simplifying your task, but this is far from all.

Look at the picture, maybe some other thoughts will come to you:

Well? I'll give you a hint: try to multiply NOC And GCD among themselves and write down all the factors that will be when multiplying. Did you manage? You should end up with a chain like this:

Take a closer look at it: compare the factors with how and are decomposed.

What conclusion can you draw from this? Right! If we multiply the values NOC And GCD between themselves, then we get the product of these numbers.

Accordingly, having numbers and meaning GCD(or NOC), we can find NOC(or GCD) in the following way:

1. Find the product of numbers:

2. We divide the resulting product by our GCD (6240; 6800) = 80:

That's all.

Let's write the rule in general form:

Try to find GCD if it is known that:

Did you manage? .

Negative numbers - "false numbers" and their recognition by mankind.

As you already understood, these are numbers opposite to natural ones, that is:

It would seem that they are so special?

But the fact is that negative numbers “won” their rightful place in mathematics right up to the 19th century (until that moment there was a huge amount of controversy whether they exist or not).

The negative number itself arose because of such an operation with natural numbers as "subtraction".

Indeed, subtract from - that's a negative number. That is why the set of negative numbers is often called "an extension of the set of natural numbers".

Negative numbers were not recognized by people for a long time.

So, Ancient Egypt, Babylon and Ancient Greece - the lights of their time, did not recognize negative numbers, and in the case of obtaining negative roots in the equation (for example, as we have), the roots were rejected as impossible.

For the first time negative numbers got their right to exist in China, and then in the 7th century in India.

What do you think about this confession?

That's right, negative numbers began to denote debts (otherwise - shortage).

It was believed that negative numbers are a temporary value, which as a result will change to positive (that is, the money will still be returned to the creditor). However, the Indian mathematician Brahmagupta already then considered negative numbers on an equal footing with positive ones.

In Europe, the usefulness of negative numbers, as well as the fact that they can denote debt, came much later, that is, a millennium.

The first mention was seen in 1202 in the "Book of the Abacus" by Leonard of Pisa (I say right away that the author of the book has nothing to do with the Leaning Tower of Pisa, but the Fibonacci numbers are his work (the nickname of Leonardo of Pisa is Fibonacci)).

So, in the XVII century, Pascal believed that.

How do you think he justified it?

That's right, "nothing can be less than NOTHING".

An echo of those times remains the fact that a negative number and the operation of subtraction are denoted by the same symbol - minus "-". And true: . Is the number " " positive, which is subtracted from, or negative, which is added to? ... Something from the series "which comes first: the chicken or the egg?" Here is such a kind of this mathematical philosophy.

Negative numbers secured their right to exist with the advent of analytic geometry, in other words, when mathematicians introduced such a thing as a real axis.

It was from this moment that equality came. However, there were still more questions than answers, for example:

proportion

This proportion is called the Arno paradox. Think about it, what is doubtful about it?

Let's talk together " " more than " " right? Thus, according to logic, the left side of the proportion should be greater than the right side, but they are equal ... Here it is the paradox.

As a result, mathematicians agreed that Karl Gauss (yes, yes, this is the one who considered the sum (or) of numbers) in 1831 put an end to it.

He said that negative numbers have the same rights as positive ones, and the fact that they do not apply to all things does not mean anything, since fractions do not apply to many things either (it does not happen that a digger digs a hole, you cannot buy a ticket to the cinema, etc.).

Mathematicians calmed down only in the 19th century, when the theory of negative numbers was created by William Hamilton and Hermann Grassmann.

That's how controversial they are, these negative numbers.

Emergence of "emptiness", or the biography of zero.

In mathematics, a special number.

At first glance, this is nothing: add, subtract - nothing will change, but you just have to attribute it to the right to "", and the resulting number will be many times greater than the original one.

By multiplying by zero, we turn everything into nothing, but we cannot divide by "nothing". In a word, the magic number)

The history of zero is long and complicated.

A trace of zero is found in the writings of the Chinese in 2000 AD. and even earlier with the Maya. The first use of the zero symbol, as it is today, was seen among the Greek astronomers.

There are many versions of why such a designation "nothing" was chosen.

Some historians are inclined to believe that this is an omicron, i.e. The first letter of the Greek word for nothing is ouden. According to another version, the word “obol” (a coin of almost no value) gave life to the symbol of zero.

Zero (or zero) as a mathematical symbol first appears among the Indians(note that negative numbers began to “develop” there).

The first reliable evidence of writing zero dates back to 876, and in them "" is a component of the number.

Zero also came to Europe belatedly - only in 1600, and just like negative numbers, it faced resistance (what can you do, they are Europeans).

“Zero was often hated, feared for a long time, and even banned”— writes the American mathematician Charles Seif.

So, the Turkish Sultan Abdul-Hamid II at the end of the 19th century. ordered his censors to delete the H2O water formula from all chemistry textbooks, taking the letter "O" for zero and not wanting his initials to be defamed by the proximity to the despicable zero.

On the Internet you can find the phrase: “Zero is the most powerful force in the Universe, it can do anything! Zero creates order in mathematics, and it also brings chaos into it. Absolutely correct point :)

Summary of the section and basic formulas

The set of integers consists of 3 parts:

• natural numbers (we will consider them in more detail below);
• numbers opposite to natural ones;
• zero - " "

The set of integers is denoted letter Z.

1. Natural numbers

Natural numbers are the numbers that we use to count objects.

The set of natural numbers is denoted letter N.

In operations with integers, you will need the ability to find GCD and LCM.

Greatest Common Divisor (GCD)

To find the NOD you need:

1. Decompose numbers into prime factors (into numbers that cannot be divided by anything other than itself or by, for example, etc.).
2. Write down the factors that are part of both numbers.
3. Multiply them.

Least common multiple (LCM)

To find the NOC you need:

1. Factorize numbers into prime factors (you already know how to do this very well).
2. Write out the factors included in the expansion of one of the numbers (it is better to take the longest chain).
3. Add to them the missing factors from the expansions of the remaining numbers.
4. Find the product of the resulting factors.

2. Negative numbers

These are numbers that are opposite to natural numbers, that is:

Now I want to hear from you...

I hope you appreciated the super-useful "tricks" of this section and understood how they will help you in the exam.

And more importantly, in life. I'm not talking about it, but believe me, this one is. The ability to count quickly and without errors saves in many life situations.

Now it's your turn!

Write, will you use grouping methods, divisibility criteria, GCD and LCM in calculations?

Maybe you have used them before? Where and how?

Perhaps you have questions. Or suggestions.

Write in the comments how you like the article.

And good luck with your exams!

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

The first ideas about the 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 a natural number.

Shot called a part (share) of a unit or several equal parts of it.

Designated: , where m,n- whole numbers;

Fractions with denominator 10 n, Where n is an integer, they are 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 already caused by the development of mathematics itself (algebra). Descartes in the 17th century introduces the concept negative number.

Numbers whole (positive and negative), fractional (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 variables, it turned out to be necessary to expand the concept of number - 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 (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(represented as a fraction) and irrational(not representable as a fraction ) form a set real (real) numbers.

Separately in mathematics, complex numbers are distinguished.

Complex numbers arise in connection with the problem of solving squares for the case D< 0 (здесь D is the discriminant of the quadratic equation). For a long time, these numbers did not find physical use, 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 as: z= a+ bi. Here a And b – real numbers, A i – imaginary unit.e. i 2 = -1. Number a called abscissa, a b-ordinate complex number a+ bi. Two complex numbers a+ bi And a-bi called conjugate complex numbers.

Properties:

1. Real number A can also be written as a complex number: 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. Recording 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. The 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 between two complex numbers a+ bi(reduced) and c+ di(subtracted) is called a complex number ( a-c) + (b-d)i. Thus, when subtracting two complex numbers, their abscissas and ordinates are subtracted separately.

Multiplication. The product of complex numbers a+ bi And c+ di is called a complex number.

(ac-bd) + (ad+ bc)i. This definition stems from two requirements:

1) numbers a+ bi And c+ di must multiply 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(divided) to another c+ di (divider) - means to find the third number e+ fi(chat), which, when multiplied by a divisor c+ di, which 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 doing all the transformations, we get:

Task 1: Add, subtract, multiply and divide z 1 to 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 whose second power 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 number -3, dot B is the number 2, and O-zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, 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 called the length of the vector OP, depicting a complex number on the coordinate ( comprehensive) plane. Complex number modulus a+ bi denoted by | 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; Rez

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 is called exact, the second - approximate. Most often it is convenient to use an approximate number instead of an exact number, especially since in many cases the exact number cannot be found at all.

So, if they say that there are 29 students in the class, then the number 29 is exact. If they say that the distance from Moscow to Kiev 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 some extent.

The result of operations with approximate numbers is also an approximate number. By performing some operations on exact numbers (dividing, extracting the root), you can also get approximate numbers.

The theory of approximate calculations allows:

1) knowing the degree of accuracy of the data, assess 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 approximate numbers is rounding. Round off both approximate and exact numbers.

Rounding a given number to some of its digits is the replacement of it with a new number, which is obtained from the given one by discarding all of 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 the greatest proximity of the rounded number to the rounded one, the following rules should be used: in order to round the 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. This takes into account the following:

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

2) if the first discarded digit is greater than 5 or equal to 5, then the last remaining digit is increased by one (rounding up).

Let's show this with examples. Round up:

a) up to tenths of 12.34;

b) up to hundredths of 3.2465; 1038.785;

c) up to thousandths of 3.4335.

d) up to 12375 thousand; 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 tenths, 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 quantity under consideration is unknown, but only approximate. Then the absolute error is also unknown. In these cases, indicate the limit that it does not exceed. This number is called the marginal absolute error. They say that the exact value of a number is equal to its approximate value with an error less than the boundary error. For example, the number 23.71 is the approximate value of the number 23.7125 with an accuracy of 0.01, since the absolute approximation error is 0.0025 and less than 0.01. Here the boundary absolute error is equal to 0.01 * .

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

x≈a(±Δ a)

should be understood as follows: the exact value of the quantity x is in between 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.

Conversely, if 7.3< X< 7,4, тоX≈ 7.35 (±0.05). Absolute or marginal absolute error does not characterize the quality of the measurement. The same absolute error can be considered significant and insignificant, depending on the number that expresses the measured value. 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, while at the same time, when measuring the distance between two houses on the same street, such accuracy will be unacceptable. Therefore, 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 measure of accuracy is the relative error.

Relative error is the ratio of the absolute error to the value of the approximate number. The ratio of the boundary absolute error to the approximate number is called the boundary relative error; denote it like this: Relative and boundary relative errors are usually expressed as a percentage. For example, if measurements show 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 an approximate value, i.e. their half-sum, then the boundary absolute error is equal to the half-difference of these numbers. In this case X≈ 12.5 (±0.2). Here, the boundary absolute error is 0.2 km, and the boundary relative