TO whole numbers These are natural numbers, zero, as well as the numbers opposite to natural.

Integers - These are positive integers.

For example: 1, 3, 7, 19, 23, etc. We use such numbers to count (on the table there are 5 apples, the car has 4 wheels, etc.)

Latin letter \\ MathBB (N) - is indicated lots of natural numbers .

Negative numbers cannot be attributed to natural numbers (the chair cannot have a negative amount of legs) and fractional numbers (Ivan could not sell 3.5 bicycles).

Numbers opposite to natural, are negative integers: -8, -148, -981, ....

Arithmetic actions with integers

What can be done with integers? They can be multiplied, fold and deduct from each other. We will analyze every operation on a specific example.

Addition of integers

Two integers with the same signs are folded as follows: the modules of these numbers are made and the final sign is made before the amount received:

(+11) + (+9) = +20

Subtracting integers

Two integers with different signs It is folded as follows: from the Module of a larger, the module is subtracted by the smaller module and before the response received is a sign of a larger one by the number of numbers:

(-7) + (+8) = +1

Multiplication of integers

To multiply one integer on another, you need to multiply the modules of these numbers and put the "+" sign before the response received, if the initial numbers were with the same signs, and the sign "-", if the initial numbers were with different signs:

(-5) \\ CDOT (+3) \u003d -15

(-3) \\ CDOT (-4) \u003d +12

We should remember the following the rule of multiplication of integers:

+ \\ Cdot + \u003d +

+ \\ CDOT - \u003d -

- \\ Cdot + \u003d -

- \\ Cdot - \u003d +

There is a rule of multiplying multiple numbers. Let's remember it:

The mark of the work will be "+", if the number of multipliers with a negative sign is even and "-", if the number of multipliers with a negative sign is odd.

(-5) \\ Cdot (-4) \\ CDOT (+1) \\ CDOT (+6) \\ CDOT (+1) \u003d +120

Division of integers

The division of two integers is made as follows: the module of the same number is divided into the module of the other and if the characters of numbers are the same, then in front of the private sign "+" sign, and if the signs of the initial numbers are different, then the "-" sign is installed.

(-25) : (+5) = -5

Properties of addition and multiplication of integers

We will analyze the basic properties of addition and multiplication for any integers A, B and C:

1. a + b \u003d b + a - the remedy property of the addition;
2. (A + B) + C \u003d A + (B + C) - a combatant property of addition;
3. a \\ cdot b \u003d b \\ Cdot A - Multiplication Property;
4. (A \\ Cdot C) \\ Cdot B \u003d A \\ CDOT (B \\ CDOT C) - the combination of multiplication properties;
5. a \\ CDOT (B \\ Cdot C) \u003d A \\ CDOT B + A \\ CDOT C - distribution property of multiplication.

There are many varieties of numbers, one of them are integer numbers. Integers appeared in order to facilitate the bill not only in positive side, but also in negative.

Consider an example:
In the afternoon there was a temperature of 3 degrees. By evening, the temperature fell by 3 degrees.
3-3=0
On the street it became 0 degrees. At night, the temperature fell by 4 degrees and began to show on the thermometer -4 degree.
0-4=-4

A number of integers.

We will not be able to describe such a task with natural numbers, consider this task on the coordinate direct.

We got a number of numbers:
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

This number of numbers are called a number of integers.

Whole positive numbers. Whole negative numbers.

A number of integers consists of positive and negative numbers. To the right of zero are natural numbers or they are also called whole positive numbers. And to the left of zero go whole negative numbers.

Zero is neither positive negative number. It is the border between positive and negative numbers.

- It is a set of numbers consisting of natural numbers, as many negative numbers and zero.

A number of integers in positive and in negative side is an infinite set.

If we take two any integers, the numbers standing between these whole numbers will be called the final set.

For example:
Take the integers from -2 to 4. All numbers standing between these numbers are included in the final 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.
The integers are indicated by the Latin letter Z. All many natural numbers and integers can be ported in the figure.


Involatory integers In other words, these are negative integers.
Non-negative integers - These are positive integers.

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To make a lot of simplifying life when you need to calculate something to win the precious time on the OGE or EGE to make less stupid mistakes - read this section!

This is what you will learn:

• how faster, easier and more accurate to count usinggrouping numbers When adding and subtracting,
• how without errors, quickly multiply and divide using multiplication Rules and Signs of Destinations,
• how to significantly speed up calculations using the smallest common multiple (NOC) and the greatest common divisor (Node).

The possession of the receptions of this section can translate the scale of the scales in one direction or another ... You will enter the university of dreams or not, you will have to pay huge money for your training or your parents or you will do on the budget.

Let "S Dive Right in ... (drove!)

P.S. Last valuable advice ...

Lots of integers Consists of 3 parts:

1. integers (Consider them in more detail below);
2. natural numbers (everything will be in place as soon as you know what natural numbers are);
3. zero - " " (Where without him?)

letter Z.

Integers

"God created natural numbers, everything else is the work of human hands" (c) the German mathematician Kronkener.

Natural numbers are The numbers that we use for the account items and it is on this that their history of the emergence is based on - the need to count arrows, skins, etc.

1, 2, 3, 4 ... n

letter N.

Accordingly, it is not included in this definition (you can not count what is not?) And even more so do not enter negative values \u200b\u200b(is it apple?).

In addition, not included fractional numbers (We also can't say "I have a laptop", or "I sold the car")

Anyone natural number You can write with 10 digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Thus, 14 is not a digit. This is the number. What figures does it consist of? That's right, from numbers and.

Addition. Grouping when adding to quickly count and less wrong

What is interesting can you say about this procedure? Of course, you will reply now "from the permutation of the terms of the amount does not change." It would seem primitive, familiar with the first class a rule, however, when solving large examples it instantly forgotten!

Do not forget about it -use groupingTo facilitate the process of counting and reduce the likelihood of errors, because you will not have a calculator on the exam.

Watch yourself, what expression it is easier to fold?

• 4 + 5 + 3 + 6
• 4 + 6 + 5 + 3

Of course the second! Although the result is the same. But! Considering the second way you have less chances to make a mistake and you will do everything faster!

So, you think in your mind like this:

4 + 5 + 3 + 6 = 4 + 6 + 5 + 3 = 10 + 5 + 3 = 18

Subtraction. Grouping when subtracting to read faster and mistake

When subtracting, we can also group subtractable numbers, for example:

32 - 5 - 2 - 6 = (32 - 2) - 5 - 6 = 30 - 5 - 6 = 19

And what if subtraction alternates in the example with the addition? You can also group, you will answer, and that's right. Just ask, do not forget about signs in front of numbers, for example: 32 - 5 - 2 - 6 = (32 - 2) - (6 + 5) = 30 - 11 = 19

Remember: Incorrect signs will lead to an erroneous result.

Multiplication. How to multiply in mind

Obviously, from change places of multipliers The value of the work will not change:

2 ⋅ 4 ⋅ 6 ⋅ 5 = (2 ⋅ 5 ) ⋅ (4 ⋅ 6 ) = 1 0 ⋅ 2 4 = 2 4 0

I will not tell you "Use it when solving examples" (you yourself understood the hint, right?), And I'll tell you how to quickly multiply some numbers in the mind. So, look attentively at the table:

And a little more about multiplication. Of course, you remember two special occasions ... Guess what I mean? This is:

Oh yes, still consider signs of divisibility. There are only 7 rules on the signs of divisibility, of which the first 3 you already know exactly!

But the rest is not at all difficult to remember.

7 signs of the divisibility of numbers that will help you quickly read in the mind!

• The first three rules you, of course, know.
• Fourth and fifth easy to remember - when dividing on and we look, whether the amount of numbers constituting the number is divided into this.
• When dividing on we pay attention to the two last digits of the number - is it divided by the number they make on?
• When dividing the number should be simultaneously sharing on and on. That's all the wisdom.

You think now - "Why do I need all this"?

First, the exam goes without calculator And these rules will help you navigate in the examples.

And secondly, you heard the tasks about Node and Nok.? Familiar abbreviation? Let's start remembering and understand.

The greatest common divider (node) is needed to reduce fractions and fast computing

Suppose you have two numbers: and. On what the greatest number are both of these numbers? You, without thinking, answer, because you know that:

12 = 4 * 3 = 2 * 2 * 3

8 = 4 * 2 = 2 * 2 * 2

What are the numbers in the expansion? That's right, 2 * 2 \u003d 4. So your answer was. Holding this simple example in my head, you will not forget the algorithm how to find Node. Try to "build" him in my head. Happened?

To find a node need:

1. Ensure the numbers on simple factors (on such numbers that cannot be divided into anything except or on, for example, 3, 7, 11, 13, etc.).
2. Multiply them.

You understand why we needed signs of divisibility? So that you looked at the number and could start dividing without a residue.

For example, find the nodes of 290 and 485

First number - .

Looking at him, you can immediately say that it is divided into, write down:

it is impossible to divide anything else, but you can - and, we get:

290 = 29 * 5 * 2

Take another number - 485.

According to the signs of divisibility, it must be divided into, as it ends. We divide:

We analyze the original number.

• It cannot be divided into it (the last digit is odd),
• - not divided by, then the number is also not divided into,
• on and on it is also not divided (the amount of numbers included in the number is not divided into and on)
• it is also not divided, because it is not divided on and,
• it is also not divided, because it is not divided on and.
• it is impossible to divide on target,

So the number can be decomposed only on and.

And now we find Node These numbers (s). What is this number? Right, .

Practice?

Task number 1. Find Nodes Numbers 6240 and 6800

1) I divide at once, since both numbers are 100% divided into:

Task number 2. Find Nodes Numbers 345 and 324

Here I can not quickly find at least one general divisorSo simply lay out on simple factors (as little as possible):

The smallest total multiple (NOC) - saves time, helps to solve the tasks of non-standard

Suppose you have two numbers - and. What is the smallest number that is divided and without residue (i.e., a focus)? Hard to imagine? Here you have a visual tip:

Do you remember what is indicated by the letter? Right just whole numbers. So what the smallest number is suitable in place x? :

In this case.

From this simple example It follows several rules.

Rules for quick finding NOK

Rule 1. If one of two natural numbers is divided into another number, then more of these two numbers is their smallest multiple.

Find the following numbers:

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

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

Note, we are talking about two numbers in the rule, if the numbers are larger, the rule does not work.

For example, NOC (7; 14; 21) is not equal to 21, as it is not divided without residue.

Rule 2. If two (or more than two) numbers are mutually simple, then the smallest common multiple is equal to their work.

Find Nok. In the following numbers:

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

Calculated? Here are the answers - ,; .

As you understand, it is not always possible to take it so easily and pick up this very x, so there is a next algorithm for a little more difficult numbers:

Practice?

We find the lowest total multiple - NOC (345; 234)

Find the smallest total multiple (NOK) yourself

What answers did you get?

That's what happened to me:

How much time did you spend on finding Nok.? My time is 2 minutes, truth I know one trickI suggest you open right now!

If you are very attentive, then you probably noticed that for the specified numbers we already searched Node And the decomposition of the factors of these numbers you could take from that example, thereby simplifying the task, but this is not all.

Look at the picture, you may come to you some more thoughts:

Well? I'll make a hint: try multiply Nok. and Node Between themselves and write down all the factors that will be with multiplies. Cope? You should get this chain:

Look towards her closer: Compare multipliers with how they unfold and.

What conclusion can you make this? Right! If we change the values Nok. and Node Mean, then we will get the work of these numbers.

Accordingly, having numbers and value Node (or Nok.) we can find Nok. (or Node) According to such a scheme:

1. Find a product of numbers:

2. Delim the resulting work on our Node (6240; 6800) = 80:

That's all.

We write a rule in general form:

Try to find NodeIf it is known:

Cope? .

Negative numbers - "lzhenchul" and their recognition by humanity.

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

Negative numbers can be folded, deduct, multiply and divide - everything is like in natural. It would seem that they are so special about them? And the fact is that the negative numbers "dismantled" their rightful place in mathematics already until the XIX century (before that moment was great amount Disputes, they exist or not).

The negative number itself occurred due to such an operation with natural numbers as "subtraction". Indeed, from the subtraction - here is a negative number. That is why, many negative numbers are often called "the expansion of the set natural numbers».

Negative numbers have not been admitted for a long time. So, Ancient Egypt, Babylon I. Ancient Greece - Sights of their time, did not recognize negative numbers, and in 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 received their right to exist in China, and then in the VII century in India. What do you think, what is the reason for this recognition? That's right, negative numbers began to denote debts (otherwise there is a shortage). It was believed that negative numbers are a temporary value, which as a result will change to a positive (that is, the creditor will be returned by the creditor). However, the Indian brahmagupta mathematician has already considered negative numbers on a par with positive.

In Europe, the usefulness of negative numbers, as well as to the fact that they can denote the debts, they came significantly later, both of the Millennium. The first mention was noticed in 1202 in the "Book of Abaka" Leonard Pisansky (I immediately speak - to the Pisa Tower The author of the book relationship does not have anything, but the number of Fibonacci is his hands (nickname Leonardo Pisansky - Fibonacci)). Further, the Europeans came to the fact that negative numbers may indicate not only debts, but also a shortage of anything, however, it is not all recognized.

So, in the XVII century Pascal believed that. What do you think, what did he justify it? True, "nothing can be less than nothing." The echoes of those times remain the fact that the negative number and the subtraction operation is indicated by the same symbol - the minus "-". And truth:. The number "" is positive, which is deducted from, or negative, which is summed up to? ... something from the series "What is the first: chicken or egg?" Here is such a kind of mathematical philosophy.

Negative numbers secured their right to exist with the advent of analytical geometry, in other words, when mathematics introduced such a concept as a numerical axis.

From now on, equality has come. However, any equal questions were more than answers, for example:

proportion

This proportion is called "Arno Paradox". Think what is dubious in it?

Let's talk together "" more than "" right? Thus, according to logic, the left part of the proportion should be greater than the right, but they are equal ... So he and the paradox.

As a result, mathematics agreed before Karl Gauss (yes, yes, this is the one who considered the amount (or) numbers) in 1831 put the point - he said that negative numbers have the same rights as positive, and The fact that they apply not to all things does not mean anything, since the fraraty is also not applicable to many things (there is no way that the pit is digging the farmer, it is impossible to buy a ticket to the movies, etc.).

Mathematics calmed down only in the XIX century, when William Hamilton and German Grassman was created the theory of negative numbers.

These are these controversial, these negative numbers.

The emergence of "emptiness", or a scratch biography.

In mathematics - a special number. At first glance, this is nothing: add, take away - nothing will change, but it is only worth it to the right to "", and the obtained number will be more initial. We all turn into a zero to zero in nothing, but divided into "nothing", that is, we cannot. In short, the magical number)

The history of zero is long and confusing. Zero trail found in the compositions of the Chinese in 2 thousand AD. And even earlier by Maya. The first use of the zero symbol, which is what it is today, was noticed from Greek astronomers.

There are many versions why it was chosen exactly the designation "nothing". Some historians tend to the fact that this is an ohomikron, i.e. The first letter of the Greek word nothing - Ouden. According to another version, the life of the zero symbol gave the word "Obol" (a coin, almost no values).

Zero (or zero) as a mathematical symbol for the first time appears in Indians (notice, negative numbers began to "develop" there. The first reliable evidence of the recording of zero belongs to 876, and in them "- the number of numbers.

In Europe, Zero also came with the intake - only in 1600g., And as well as negative numbers, came across resistance (what can you do, they are, Europeans).

"Zero often hated, they were afraid that they were afraid, but forbidden," Charles's American mathematician writes safe. So, the Turkish Sultan Abdul-Hamid II at the end of the XIX. He ordered his censors to strike out all the textbooks of chemistry the Water formula H2O, taking the letter "O" for zero and not wanting his initials to be broken by the neighborhood with a contemptible zero. "

On the Internet, you can meet the phrase: "Zero is the most powerful force in the universe, he can all! Zero creates order in mathematics, and it also contributes to the chaos. " Absolutely correctly noticed :)

Summary of section and basic formulas

Many integers consist of 3 parts:

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

Many integers are indicated letter Z.

1. Natural numbers

Natural numbers are the numbers that we use items to account.

Many natural numbers are indicated letter N.

In operations with integers, you need the ability to find NOD and NOC.

The greatest common divider (node)

To find a node need:

1. Dismix numbers on simple factors (on such numbers that cannot be divided into anything, except or on, for example, etc.).
2. To write down the multipliers that are part of both numbers.
3. Multiply them.

The smallest total multiple (NOK)

To find the NOC need:

1. Dismix numbers on simple factors (you can already do it perfectly).
2. To write down the factors included in the decomposition of one of the numbers (it is better to take the longest chain).
3. Add missing multipliers to them from the expansions of the other numbers.
4. Find a product of the resulting multipliers.

2. Negative numbers

these are the numbers opposite to natural, that is:

Now I want to hear you ...

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

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

Now your move!

Write, will you apply grouping methods, signs of divisibility, nodes and noks in the calculations?

Maybe you used them earlier? Where and how?

Perhaps you have questions. Or suggestions.

Write in the comments as your article.

And good luck on the exams!

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For the successful passing of the USE, for admission to the Institute on the budget and, most importantly, for life.

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But, think myself ...

What you need to be sure to be better than others on the exam and be ultimately ... happier?

Fill a hand by solving tasks on this topic.

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And if you did not solve them (a lot!), You definitely be a foolishly mistaken or just do not have time.

It's like in sport - you need to repeat many times to win for sure.

Find where you want a collection, mandatory with solutions, detailed analysis And decide, decide, decide!

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Information of this article forms general view about whole numbers. First, the definition of integers is given and examples are given. Next, integers are considered on a numeric line, where it becomes clear what numbers are called integer positive numbers, and which are integer negative. After that, it is shown how with the help of integers, changes are described, and all negative numbers are considered in the sense of debt.

Navigating page.

Integers - definition and examples

Definition.

Whole numbers - These are natural numbers, the number of zero, as well as the numbers opposed to natural.

The definition of integers argues that any of the numbers 1, 2, 3, ..., the number 0, as well as any of the numbers -1, -2, -3, ... is whole. Now we can easily bring examples of integers. For example, the number 38 is an integer, the number 70 040 is also an integer, zero is an integer (we recall that zero is not a natural number, zero is an integer), the number -999, -1, -8 934 832 - also are examples of integers numbers.

All integers are conveniently represented as a sequence of integers, which has the following form: 0, ± 1, ± 2, ± 3, ... The sequence of integers can be recorded and so: …, −3, −2, −1, 0, 1, 2, 3, …

From the definition of integers it follows that the set of natural numbers is a subset of many integers. Therefore, any natural number is integer, but not any integer is natural.

Integers on the coordinate direct

Definition.

Whole positive numbers - These are integers that are more zero.

Definition.

Whole negative numbers - These are integers that are less than zero.

Calibly positive and negative numbers can also be determined by their position on the coordinate direct. On the horizontal coordinate direct point, whose coordinates are whole positive numbers, lie to the right of reference. In turn, the points with the whole negative coordinates are located to the left of the point O.

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

Separately, we will draw your attention to the fact that any natural number we can boldly be called the whole, and any integer we can call natural. Natural we can name only any integer positive number, as the whole negative numbers and zero are not natural.

Interestable and whole non-negative numbers

Let us give the definition of integer inseparable numbers and integer non-negative numbers.

Definition.

All the whole positive numbers along with the number of zero called whole non-negative numbers.

Definition.

Interesting numbers - These are all whole negative numbers along with a number of 0.

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

Examples of integer non-quantities are the numbers -511, -10 030, 0, -2, and as examples of integer non-negative numbers, we give numbers 45, 506, 0, 900 321.

Most often, the terms "whole inhabitants" and "whole non-negative numbers" are used for shortness of presentation. For example, instead of the phrase "Number A is a whole, and a more zero or equal to zero," one can say "a - a non-negative number".

Description of changes in values \u200b\u200busing integers

It's time to talk about what the whole numbers are needed.

The main purpose of integers is that with their help it is convenient to describe the change in the number of any items. Tell me on the examples.

Let there be a number of details in the warehouse. If the warehouse is also brought to the warehouse, for example, 400 parts, the number of parts in the warehouse will increase, and the number 400 expresses this change in the amount in a positive side (upwards). If it is taken from the warehouse, for example, 100 parts, then the number of parts in the warehouse will decrease, and the number 100 will express the change in the amount in the negative side (up to the reduction). There will be no details on the warehouse, and they will not take part from the warehouse, then we can talk about the number of parts (that is, it can be about zero change in quantity).

In the examples given, the change in the number of parts can be described using integers 400, -100 and 0, respectively. A positive integer 400 shows a change in the number in a positive side (increase). A negative integer -100 expresses a change in quantity in the negative side (decrease). An integer 0 shows that the amount remains unchanged.

Ease of use of integers compared to the use of natural numbers is that it is not necessary to explicitly indicate the number of or decreases, - an integer determines the change in quantitatively, and the value of an integer indicates the direction of change.

The integers can also express not only the change in the quantity, but also a change in any value. We will deal with this on the example of a change in temperature.

Increased temperature, let's say, 4 degrees are expressed by a positive integer number 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 an integer 0.

Separately, you need to say about the interpretation of negative integers as the amount of debt. For example, if we have 3 apples, then a positive number 3 shows the number of apples we own. On the other hand, if we need to give 5 apples to anyone, and we do not have them in stock, then this situation can be described using a negative integer -5. In this case, we "possess" -5 apples, a minus sign indicates a debt, and the number 5 determines the debt quantitatively.

Understanding a negative integer as debt allows, for example, justify the rule of addition of negative integers. Let us give an example. If someone has 2 apples to one person and one apple - another, then the total debt is 2 + 1 \u003d 3 apples, so -2 + (- 1) \u003d - 3.

Bibliography.

• Vilenkin N.Ya. and others. mathematics. Grade 6: Textbook for general educational institutions.

Phrase " numeric sets"Quite often occurs in mathematics textbooks. There it is very often possible to meet phrases of such a plan:

"Bla-blah blah, where belongs to the set of natural numbers."

Completely, instead of the end of the phrase, you can see this record. It means the same as the text is slightly higher - the number belongs to the set of natural numbers. Many often do not attach attention to what a variety is defined by one or another variable. As a result, completely incorrect methods are used in solving the problem or proof of the theorem. This is due to the fact that the properties of the numbers belonging to various sets may have differences.

Numeric sets are not so much. Below you can see the definitions of various numeric sets.

The set of natural numbers includes all integers more than zero - positive integers.

For example: 1, 3, 20, 3057. The set does not include a figure of 0.

This numerical set includes all the integers more and less than zero, as well as zero.

For example: -15, 0, 139.

The rational numbers, generally speaking, are many fractions that are not reduced (if the fraction is reduced, it will already be an integer, and for this case it is not necessary to introduce another numerical set).

An example of the numbers included in the rational set: 3/5, 9/7, 1/2.

,

where - the final sequence of numbers of the integer part of the number belonging to the set of real numbers. This sequence is the ultimate, that is, the number of numbers into the cell phones of the real number of the final number.

- Infinite sequence of numbers in the fractional part of the real number. It turns out that in the fractional part there is an infinite number of numbers.

Such numbers cannot be submitted as a fraction. IN otherwise, a similar number could be attributed to a set of rational numbers.

Examples of real numbers:

Let's look at the value of the root of two more carefully. In the integer part there is only one digit - 1, so we can write:

In the fractional part (after the point), the numbers 4, 1, 4, 2 and so on. Therefore, for the first four digits can be written:

I dare to hope that now the recording of the definition of a set of real numbers has become clearer.

Conclusion

It should be remembered that the same function can exhibit completely different properties depending on which the set will belong to the variable. So remember the foundations - they will use you.

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