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 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 total debt(5 coins) can be calculated by the rule of adding negative numbers:
2 + (- 3) = - 5
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If we add the number 0 to the left of a series of natural numbers, we get a series of positive integers:
0, 1, 2, 3, 4, 5, 6, 7, ...
Integer negative numbers
Let's consider a small example. The figure on the left shows a thermometer that shows a temperature of 7 °C heat. If the temperature drops by 4°C, the thermometer will show 3°C of heat. A decrease in temperature corresponds to a subtraction action:
Note: all degrees are written with the letter C (Celsius), the sign of the degree is separated from the number by a space. For example, 7 °C.
If the temperature drops by 7 °C, the thermometer will show 0 °C. A decrease in temperature corresponds to a subtraction action:
If the temperature drops by 8 °C, then the thermometer will show -1 °C (1 °C of frost). But the result of subtracting 7 - 8 cannot be written using natural numbers and zero.
Let's illustrate subtraction on a series of positive integers:
1) We count 4 numbers to the left from the number 7 and get 3:
2) We count 7 numbers to the left from the number 7 and get 0:
It is impossible to count 8 numbers in a series of positive integers from the number 7 to the left. To make action 7 - 8 feasible, we expand the series of positive integers. To do this, to the left of zero, we write (from right to left) in order all natural numbers, adding to each of them a - sign, showing that this number is to the left of zero.
The entries -1, -2, -3, ... read minus 1 , minus 2 , minus 3 , etc.:
5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
The resulting series of numbers is called next to whole numbers. The dots on the left and right in this entry mean that the series can be continued indefinitely to the right and left.
To the right of the number 0 in this row are the numbers that are called natural or whole positive(briefly - positive).
To the left of the number 0 in this row are the numbers that are called whole negative(briefly - negative).
The number 0 is an integer, but is neither positive nor negative. It separates positive and negative numbers.
Hence, a series of integers consists of negative integers, zero, and positive integers.
Integer Comparison
Compare two integers- means to find out which of them is greater, which is less, or to determine that the numbers are equal.
You can compare integers using a row of integers, since the numbers in it are arranged from smallest to largest if you move along the row from left to right. Therefore, in a series of integers, you can replace commas with a less than sign:
5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < ...
Hence, Of two integers, the one on the right is the greater, and the one on the left is the smaller., Means:
1) Any positive number is greater than zero and greater than any negative number:
1 > 0; 15 > -16
2) Any negative number less than zero:
7 < 0; -357 < 0
3) Of the two negative numbers, the one that is to the right in the series of integers is greater.
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.
Definition: natural numbers are called numbers that are used to count objects (1, 2, 3, 4, 5, ...) [The number 0 is not natural. It also has its own separate history in the history of mathematics and appeared much later than the natural numbers.]
The set of all natural numbers (1, 2, 3, 4, 5, ...) is denoted by the letter N.
Whole numbers
Having learned to count, the next thing we do is learn to perform arithmetic operations on numbers. Usually, first (on counting sticks) they learn to perform addition and subtraction.With addition, everything is clear: adding any two natural numbers, as a result we always get the same natural number. But in subtraction, we find that we cannot subtract the larger from the smaller so that the result is a natural number. (3 − 5 = what?) This is where the idea of negative numbers comes in. (Negative numbers are no longer natural)
At the stage of occurrence of negative numbers (and they appeared later than fractional ones) there were also their opponents who considered them nonsense. (Three objects can be shown on the fingers, ten can be shown, a thousand objects can be represented by analogy. And what is "minus three bags"? - At that time, although numbers were already used on their own, in isolation from specific objects, the number of which they designate, were still in the minds of people much closer to these specific subjects than today.) But, like the objections, the main argument in favor of negative numbers came from practice: negative numbers made it possible to conveniently keep track of debts. 3 - 5 = -2 - I had 3 coins, I spent 5. So, I not only ran out of coins, but I also owe 2 coins to someone. If I return one, the debt will change to −2+1=−1, but can also be represented as a negative number.
As a result, negative numbers appeared in mathematics, and now we have an infinite number of natural numbers (1, 2, 3, 4, ...) and there are the same number of their opposites (−1, −2, −3, −4 , ...). Let's add another 0 to them. And the set of all these numbers will be called integers.
Definition: The natural numbers, their opposites, and zero make up the set of integers. It is denoted by the letter Z.
Any two integers can be subtracted from each other or added to get an integer as a result.
The idea of integer addition already suggests the possibility of multiplication as simply a faster way to perform addition. If we have 7 bags of 6 kilograms each, we can add 6 + 6 + 6 + 6 + 6 + 6 + 6 (add 6 to the current sum seven times), or we can simply remember that such an operation will always result in 42. Like the addition of six sevens, 7+7+7+7+7+7 will also always give 42.
The results of the addition operation certain numbers with itself certain the number of times for all pairs of numbers from 2 to 9 are written out and make up the multiplication table. To multiply integers greater than 9, a multiplication rule is invented in a column. (Which also applies to decimals, and which will be covered in one of the following articles.) Any two integers multiplied by each other will always result in an integer.
Rational numbers
Now division. By analogy with how subtraction is the inverse of addition, we come to the idea of division as the inverse of multiplication.When we had 7 bags of 6 kilograms, using multiplication, we easily calculated that the total weight of the contents of the bags is 42 kilograms. Imagine that we poured all the contents of all the bags into one common pile weighing 42 kilograms. And then they changed their minds and wanted to distribute the contents back to 7 bags. How many kilograms will fall into one bag if we distribute equally? - Obviously 6.
And if we want to distribute 42 kilograms into 6 bags? Here we think about what the same total 42 kilograms could be if we poured 6 bags of 7 kilograms into a pile. And that means when dividing 42 kilograms into 6 bags equally, we get 7 kilograms in one bag.
And if you divide 42 kilograms equally into 3 bags? And here, too, we begin to select a number that, when multiplied by 3, would give 42. For "table" values, as in the case of 6 7=42 => 42:6=7, we perform the division operation, simply remembering the multiplication table. For more complex cases, division into a column is used, which will be discussed in one of the following articles. In the case of 3 and 42, one can recall by "selection" that 3 · 14 = 42. Hence, 42:3=14. Each bag will contain 14 kilograms.
Now let's try to divide 42 kilograms equally into 5 bags. 42:5=?
We notice that 5 8=40 (small), and 5 9=45 (many). That is, neither 8 kilograms in a bag, nor 9 kilograms, out of 5 bags we will not get 42 kilograms in any way. At the same time, it is clear that in reality nothing prevents us from dividing any amount (cereals, for example) into 5 equal parts.
The operation of dividing integers by each other does not necessarily result in an integer. So we came to the concept of a fraction. 42:5 \u003d 42/5 \u003d 8 whole 2/5 (if counted in ordinary fractions) or 42:5 \u003d 8.4 (if counted in decimal fractions).
Common and decimal fractions
We can say that any ordinary fraction m / n (m is any integer, n is any natural) is just a special form of writing the result of dividing the number m by the number n. (m is called the numerator of the fraction, n is the denominator) The result of dividing, for example, the number 25 by the number 5 can also be written as an ordinary fraction 25/5. But this is not necessary, since the result of dividing 25 by 5 can be written simply as the integer 5. (And 25/5 = 5). But the result of dividing the number 25 by the number 3 can no longer be represented as an integer, so here it becomes necessary to use a fraction, 25:3=25/3. (You can select the integer part 25/3= 8 whole 1/3. In more detail ordinary fractions and operations with ordinary fractions will be discussed in the following articles.)Ordinary fractions are good because in order to represent the result of dividing any two integers as such a fraction, you just need to write the dividend into the numerator of the fraction, and the divisor into the denominator. (123:11=123/11, 67:89=67/89, 127:53=127/53, …) Then, if possible, reduce the fraction and / or highlight the integer part (these operations with ordinary fractions will be discussed in detail in the following articles ). The problem is that performing arithmetic operations (addition, subtraction) with ordinary fractions is no longer as convenient as with integers.
For the convenience of writing (in one line) and for the convenience of calculations (with the possibility of calculations in a column, as for ordinary integers), in addition to ordinary fractions, decimal fractions were also invented. A decimal fraction is an ordinary fraction written in a special way with a denominator of 10, 100, 1000, etc. For example, the common fraction 7/10 is the same as the decimal fraction 0.7. (8/100 = 0.08; 2 integers 3/10=2.3; 7 integers 1/1000 = 7.001). A separate article will be devoted to the conversion of ordinary fractions to decimals and vice versa. Operations with decimal fractions - other articles.
Any whole number can be represented as a common fraction with denominator 1. (5=5/1; −765=−765/1).
Definition: All numbers that can be represented as a common fraction are called rational numbers. The set of rational numbers is denoted by the letter Q.
When dividing any two integers by each other (except when dividing by 0), we always get a rational number as a result. For ordinary fractions, there are rules for addition, subtraction, multiplication and division, which allow you to perform the corresponding operation with any two fractions and also get a rational number (fraction or integer) as a result.
The set of rational numbers is the first of the sets we have considered, in which you can add, subtract, multiply, and divide (except for dividing by 0) without ever going beyond this set (that is, always getting a rational number as a result) .
It would seem that there are no other numbers, all numbers are rational. But this is not so either.
Real numbers
There are numbers that cannot be represented as a fraction m / n (where m is an integer, n is a natural number).What are these numbers? We have not yet considered the exponentiation operation. For example, 4 2 \u003d 4 4 \u003d 16. 5 3 \u003d 5 5 5 \u003d 125. Just as multiplication is a more convenient form of notation and calculation of addition, so exponentiation is a form of notation for multiplying the same number by itself a certain number of times.
But now consider the operation, the inverse of raising to a power - extracting the root. The square root of 16 is the number that squared is 16, which is 4. The square root of 9 is 3. But the square root of 5 or 2, for example, cannot be represented by a rational number. (The proof of this statement, other examples of irrational numbers and their history can be found, for example, on Wikipedia)
In the GIA in grade 9, there is a task to determine whether a number containing a root in its entry is rational or irrational. The task is to try to convert this number to a form that does not contain a root (using the properties of the roots). If the root cannot be eliminated, then the number is irrational.
Another example of an irrational number is the number π, familiar to everyone from geometry and trigonometry.
Definition: Rational and irrational numbers together are called real (or real) numbers. The set of all real numbers is denoted by the letter R.
In real numbers, unlike rational numbers, we can express the distance between any two points on a line or plane.
If you draw a straight line and choose two arbitrary points on it, or choose two arbitrary points on a plane, then it may turn out that the exact distance between these points cannot be expressed by a rational number. (Example - the hypotenuse of a right triangle with legs 1 and 1, according to the Pythagorean theorem, will be equal to the root of two - that is, an irrational number. This also includes the exact length of the diagonal of a tetrad cell (the length of the diagonal of any ideal square with integer sides).)
And in the set of real numbers, any distances on a straight line, in a plane or in space can be expressed by the corresponding real number.
Negative numbers were first used in ancient China and in India, in Europe, they were introduced into mathematical use by Nicolas Shuquet (1484) and Michael Stiefel (1544).
Algebraic properties
is not closed under division of two integers (for example, 1/2). The following table illustrates several basic properties of addition and multiplication for any integers. a, b And c.
| addition | multiplication | |
| closure : | a + b- whole | a × b- whole |
| associativity : | a + (b + c) = (a + b) + c | a × ( b × c) = (a × b) × c |
| commutativity: | a + b = b + a | a × b = b × a |
| the existence of a neutral element: | a + 0 = a | a× 1 = a |
| the existence of an opposite element: | a + (−a) = 0 | a≠ ±1 ⇒ 1/ a is not whole |
| distributivity of multiplication with respect to addition: | a × ( b + c) = (a × b) + (a × c) | |
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|list5=Cardinal numbers - You should definitely transfer to the bed, it will not be possible here ...
The patient was so surrounded by doctors, princesses and servants that Pierre no longer saw that red-yellow head with a gray mane, which, despite the fact that he saw other faces, did not go out of sight for a moment during the entire service. Pierre guessed from the cautious movement of the people surrounding the chair that the dying man was being lifted and carried.
“Hold on to my hand, you’ll drop it like that,” he heard the frightened whisper of one of the servants, “from below ... another one,” voices said, and the heavy breathing and stepping of people’s feet became more hasty, as if the burden they were carrying was beyond their strength. .
The bearers, among whom was Anna Mikhailovna, drew level with the young man, and for a moment, from behind the backs and backs of the people’s heads, a high, fat, open chest, the fat shoulders of the patient, raised upwards by the people holding him under the armpits, and a gray-haired curly, lion head. This head, with an unusually wide forehead and cheekbones, a beautiful sensual mouth and a majestic cold look, was not disfigured by the proximity of death. She was the same as Pierre knew her three months ago, when the count let him go to Petersburg. But this head swayed helplessly from the uneven steps of the bearers, and the cold, indifferent look did not know where to stop.
A few minutes of fuss passed by the high bed; the people carrying the sick man dispersed. Anna Mikhailovna touched Pierre's hand and said to him: "Venez." [Go.] Pierre, together with her, went up to the bed, on which, in a festive pose, apparently related to the sacrament that had just been performed, the sick man was laid. He lay with his head propped high on the pillows. His hands were symmetrically laid out on a green silk blanket, palms down. When Pierre approached, the count looked directly at him, but looked with that look, the meaning and meaning of which cannot be understood by a person. Either this glance said absolutely nothing, only that, as long as there are eyes, one must look somewhere, or it said too much. Pierre stopped, not knowing what to do, and looked inquiringly at his leader, Anna Mikhailovna. Anna Mikhailovna made a hurried gesture to him with her eyes, pointing to the patient's hand and kissing it with her lips. Pierre, diligently stretching his neck so as not to catch on the blanket, carried out her advice and kissed her broad-boned and fleshy hand. Not a hand, not a single muscle of the count's face trembled. Pierre again looked inquiringly at Anna Mikhailovna, now asking what he should do. Anna Mikhaylovna pointed out to him with her eyes a chair that stood beside the bed. Pierre obediently began to sit down on an armchair, continuing to ask with his eyes whether he had done what was needed. Anna Mikhailovna nodded her head approvingly. Pierre again assumed the symmetrically naive position of the Egyptian statue, apparently condoling that his clumsy and fat body occupied such a large space, and using all his mental strength to seem as small as possible. He looked at the count. The count looked at the place where Pierre's face was, while he stood. Anna Mikhailovna, in her position, showed the touching importance of this last minute of meeting between father and son. This lasted two minutes, which seemed to Pierre an hour. Suddenly a shudder appeared in the large muscles and wrinkles of the count's face. The shudder intensified, the beautiful mouth twisted (it was only then that Pierre realized to what extent his father was close to death), an indistinct hoarse sound was heard from the twisted mouth. Anna Mikhailovna diligently looked into the patient's eyes and, trying to guess what he needed, she pointed either to Pierre, then to the drink, then in a whisper she called Prince Vasily inquiringly, then she pointed to the blanket. The patient's eyes and face showed impatience. He made an effort to look at the servant, who was standing at the head of the bed without leaving.
“They want to roll over onto the other side,” the servant whispered and rose to turn the count’s heavy body facing the wall.
Pierre got up to help the servant.
While the count was being turned over, one of his arms fell back helplessly, and he made a vain effort to drag it. Did the count notice that look of horror with which Pierre looked at this lifeless hand, or what other thought flashed through his dying head at that moment, but he looked at the disobedient hand, at the expression of horror in Pierre's face, again at the hand, and on the face he had a weak, suffering smile that did not suit his features, expressing, as it were, mockery at his own impotence. Suddenly, at the sight of this smile, Pierre felt a shudder in his chest, a pinching in his nose, and tears clouded his vision. The patient was turned over on his side against the wall. He sighed.
- Il est assoupi, [He dozed off,] - said Anna Mikhailovna, noticing the princess who came to replace. - Allons. [Let's go to.]
Pierre left.
