Decimal number system, classes and digits of natural numbers.

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9 categories of pronouns are allocated.
1. Personal pronouns: Pronoun 1 l. unit i points to the speaker, pronoun 2 l. unit you - to the interlocutor, addressee of the speech, pronoun 1 l. plural we - on the speaker and the interlocutor or several persons, including the speaker.
Pronoun 2 l. plural you indicate several persons, including the interlocutor and excluding the speaker, pronouns 3 l. unit he, she, it and 3 l. plural they
The pronouns I, you and the pronouns we, you do not correspond in number, i.e. pronouns we, you, I, you, as they differ in meaning: we are not a set of I, you are not a set of you
The pronoun we The pronoun you can be used as a form of polite address to refer to one person, the interlocutor.
The reflexive self, indicating the relationship of the subject to himself, can refer to any person: I bought myself a book. You bought yourself a book. She bought herself a book.
The pronoun itself can play the role of a particle, indicating the independence, independence of the subject's actions: And he does his own thing and does not pay attention to anyone.
Recently, many linguists have also singled out the mutually reflexive pronoun of each other. This pronoun does not have a nominative case form, and in indirect cases only the second component of addition changes - each other, each other, each other, etc. When declining this pronoun, simple prepositions are used, which are put into interposition between the components of each other's pronoun (for each other, with each other). Derived prepositions can stand both in interposition and in preposition before the whole pronoun (opposite each other, relative to each other and opposite each other, relative to each other).
Possessive (my, yours, ours, yours, his, hers, they denote that the subject belongs to any person: Can I take your book? Our children go to the same class. His composition is better than mine. They agree with nouns, speaking with them in definition roles.
The pronoun your can refer to the 1st, 2nd and 3rd person: I brought my books. You brought your books. He brought his books.
The pronouns his, her, them are a frozen form of the genitive case of personal pronouns he / it, she, they indicate belonging or relation to a person, object (his room, her hand, their views).
Demonstrative (this, that, such, such, such, such, so much (obsolete this, this They agree with nouns, acting as definitions with them. The pronoun such acts as a predicate in the sentence (The task is such that it will take a lot of time for it fulfillment
Interrogatives (who, what, what, what, which, whose, how many serve to express the question of the subject, quality, belonging, quantity: Who learned the poem? What problem could you not solve? How much does a ticket cost?
The pronoun who refers to an animate object. The verb-predicate with it is put in the masculine gender, even if the question refers to a female person (Which of the students completed the task?). The pronoun that refers to an inanimate object or an abstract concept. The verb-predicate with it is put in the middle gender (What happened?).
Relative - these are the same interrogative pronouns that are used not for a question, but for connecting the subordinate clause with the main one in a complex sentence. In the structure of the subordinate clause, relative pronouns act as allied words and perform the function of both main and secondary members of the sentence. For example: I saw a hut that stood on the edge of the forest. I have never seen it the way the House I grew up in was built by my grandfather.
Attributive (all, itself, indicate a generalized attribute of the subject and perform the function of agreed definitions in the sentence: All relatives came to him. Every year they rest in Sochi.
pronoun whole
pronoun self
Negative pronouns (no one, nothing, none, no one, not at all, no way, never, nowhere, nowhere, nowhere indicate the absence of an object, attribute or quality: No one could break the world record. I never saw a kangaroo. He won’t go anywhere today. Negative pronouns are formed in a prefixal way from interrogative pronouns.
Indefinite (someone, someone, anyone, something, anyone, anything, someone, something, something, anything, anything, someone, something, some, some, some, some, some, someone, someone, someone, some, some, some, some, some, somewhere, sometime, sometime, for some reason, some, someone's, somewhere, ever, from some, some, some where, some when indicate unknown or insufficiently known persons, objects, uncertain signs, qualities or Quantity: Suddenly someone came into the room He heard someone's footsteps You've already lost a few books
Indefinite pronouns are formed from interrogative prefixes (with the help of prefixes (prefixes) non-, something- and postfixal ways (with the help of postfixes - something, - or.

Our first lesson was called numbers. We have covered only a small part of this topic. In fact, the topic of numbers is quite extensive. It has a lot of subtleties and nuances, a lot of tricks and interesting chips.

Today we will continue the topic of numbers, but again we will not consider it all, so as not to complicate learning with unnecessary information, which at first is not really needed. We'll talk about grades.

Lesson content

What is a rank?

In simple terms, a digit is the position of a digit in a number or the place where the digit is located. Let's take the number 635 as an example. This number consists of three digits: 6, 3 and 5.

The position where the number 5 is located is called unit digit

The position where the number 3 is located is called tens digit

The position where the number 6 is located is called hundreds digit

Each of us heard from school such things as "ones", "tens", "hundreds". The digits, in addition to playing the role of the position of a digit in a number, tell us some information about the number itself. In particular, the digits tell us the weight of a number. They tell you how many ones, how many tens, and how many hundreds.

Let's return to our number 635. Five is in the category of units. What does it say? And this says that the discharge of units contains five units. It looks like this:

Three is in the tens place. This indicates that the tens digit contains three tens. It looks like this:

There is a six in the hundreds place. This means that there are six hundreds in the hundreds place. It looks like this:

If we add the number of resulting units, the number of tens and the number of hundreds, we get our original number 635

There are also higher digits such as the thousands digit, the tens of thousands digit, the hundreds of thousands digit, the millions digit, and so on. We will rarely consider such large numbers, but nevertheless it is also desirable to know about them.

For example, in the number 1,645,832, the ones place contains 2 units, the tens place contains 3 tens, the hundreds place contains 8 hundreds, the thousands place contains 5 thousand, the tens of thousands place contains 4 tens of thousands, the hundreds of thousands place contains 6 hundreds of thousands, the millions place contains 1 million.

At the first stages of studying the digits, it is desirable to understand how many units, tens, hundreds contains a particular number. For example, the number 9 contains 9 units. The number 12 contains two ones and one ten. The number 123 contains three ones, two tens and one hundred.

Grouping items

After counting some items, the digits can be used to group these items. For example, if we counted 35 bricks in the yard, then we can use discharges to group these bricks. In the case of grouping objects, the digits can be read from left to right. So, the number 3 in the number 35 will indicate that the number 35 contains three tens. And this means that 35 bricks can be grouped three times in ten pieces.

So, let's group the bricks three times ten pieces:

It turned out thirty bricks. But there are still five units of bricks left. We will call them as "five units"

It turned out three dozen and five units of bricks.

And if we did not begin to group the bricks into tens and ones, then we could say that the number 35 contains thirty-five units. This grouping would also be acceptable:

The same can be said about other numbers. For example, about the number 123. Earlier we said that this number contains three units, two tens and one hundred. But you can also say that this number contains 123 units. Moreover, you can group this number in another way, saying that it contains 12 tens and 3 units.

The words units, dozens, hundreds, replace the multiplicands 1, 10 and 100. For example, the number 3 is located in the units digit of the number 123. Using the multiplier 1, we can write that this unit is contained in the units digit three times:

100 x 1 = 100

If we add the results of 3, 20 and 100, we get the number 123

3 + 20 + 100 = 123

The same will happen if we say that the number 123 contains 12 tens and 3 units. In other words, the tens will be grouped 12 times:

10 x 12 = 120

And units three times:

1 x 3 = 3

This can be understood with the following example. If there are 123 apples, then you can group the first 120 apples 12 times in 10 pieces:

It turned out one hundred and twenty apples. But there are still three apples left. We will call them as "three units"

If we add the results 120 and 3, we again get the number 123

120 + 3 = 123

You can also group 123 apples into one hundred, two tens and three units.

Let's group a hundred:

Let's group two tens:

Let's group the three units:

If we add the results of 100, 20 and 3, we again get the number 123

100 + 20 + 3 = 123

And finally, consider the last possible grouping, where the apples will not be distributed into tens and hundreds, but will be collected together. In this case, the number 123 will be read as one hundred and twenty three units . This grouping would also be valid:

1 x 123 = 123

The number 523 can be read as 3 units, 2 tens and 5 hundreds:

1 × 3 = 3 (three ones)

10 × 2 = 20 (two tens)

100 × 5 = 500 (five hundred)

3 + 20 + 500 = 523

You can also read how 3 units 52 tens:

1 × 3 = 3 (three ones)

10 × 52 = 520 (fifty two tens)

3 + 520 = 523

Another number 523 can be read as 523 units:

1 × 523 = 523 (five hundred and twenty three units)

Where to apply ranks?

Bits greatly facilitate some calculations. Imagine that you are at the blackboard and solve a problem. You have almost finished the task, it remains only to evaluate the last expression and get the answer. The expression to be evaluated looks like this:

I don’t have a calculator at hand, but I want to quickly write down the answer and surprise everyone with the speed of my calculations. Everything is simple, if you separately add units, separately tens and separately hundreds. You need to start with the discharge of units. First of all, after the equal sign (=), you must mentally put three dots. Instead of these points, a new number will be located (our answer):

Now let's start adding. The units digit of 632 is the number 2, and the units digit of 264 is the number 4. This means that the units digit of 632 contains two ones, and the units digit of 264 contains four ones. We add 2 and 4 units - we get 6 units. We write the number 6 in the units place of the new number (our answer):

Next add up the tens. The tens place of the number 632 is the number 3, and the tens place of the number 264 is the number 6. This means that the tens place of the number 632 contains three tens, and the tens place of the number 264 contains six tens. We add 3 and 6 tens - we get 9 tens. We write the number 9 in the tens place of the new number (our answer):

Well, in the end, we add hundreds separately. The hundreds place of 632 is a 6, and the hundreds place of 264 is a 2. This means that the hundreds place of 632 contains six hundreds, and the hundreds place of 264 contains two hundreds. Adding 6 and 2 hundreds, we get 8 hundreds. We write the number 8 in the hundreds place of the new number (our answer):

Thus, if you add 264 to the number 632, you get 896. Of course, you will calculate such an expression faster and others will start to be surprised at your abilities. They will think that you are quickly calculating large numbers, when in fact you were calculating small ones. Agree that small numbers are easier to calculate than large ones.

discharge overflow

A digit is characterized by a single digit from 0 to 9. But sometimes, when calculating a numeric expression in the middle of a solution, a digit overflow may occur.

For example, adding the numbers 32 and 14 does not overflow. Adding the units of these numbers will give 6 units in the new number. And adding tens of these numbers will give 4 tens in the new number. The answer will be 46 or six ones and four tens .

But when adding the numbers 29 and 13, an overflow will occur. Adding units of these numbers gives 12 units, and adding tens gives 3 tens. If in the new number in the units place we write the received 12 units, and in the tens place we write the received 3 tens, then we get an error:

The value of the expression 29 + 13 is 42 , not 312 . What should be done in case of overflow? In our case, the overflow happened in the ones place of the new number. When nine and three units are added together, we get 12 units. And only numbers in the range from 0 to 9 can be written in the units place.

The fact is that 12 units is not easy "twelve units" . Otherwise, this number can be read as "two ones and one ten" . The units digit is for units only. There is no room for dozens. This is where our mistake lies. Having added 9 units and 3 units, we got 12 units, which in another way can be called two units and one ten. By writing two units and one ten in one place, we made a mistake, which eventually led to the wrong answer.

To correct the situation, two units must be written in the units digit of the new number, and the remaining ten should be transferred to the next tens digit. After adding the tens in the example 29 + 13, we will add to the result the ten that remained when adding the units.

So, out of 12 units, we write two units in the unit category of the new number, and transfer one ten to the next bit

As you can see in the figure, we presented 12 ones as 1 ten and 2 ones. We have written two ones in the units place of the new number. And one ten was transferred to the ranks of tens. We will add this ten to the result of adding the tens of the numbers 29 and 13. In order not to forget about it, we inscribed it above the tens of the number 29.

Now add up the tens. Two tens plus one tens is three tens, plus one tens left over from the previous addition. As a result, in the tens place we get four tens:

Example 2. Add the numbers 862 and 372 by digits.

Let's start with units. The units digit of 862 contains the number 2, and the units digit of 372 also contains the number 2. This means that the units digit of 862 contains two ones, and the units digit of 372 also contains two ones. We add 2 units plus 2 units - we get 4 units. We write the number 4 in the units place of the new number:

Next add up the tens. The tens place of 862 is the number 6, and the tens place of 372 is the number 7. This means that the tens place of 862 contains six tens, and the tens place of 372 contains seven tens. Adding 6 tens and 7 tens equals 13 tens. An overflow has occurred. 13 tens is a ten repeated 13 times. And if you repeat the ten 13 times, you get the number 130

10 x 13 = 130

The number 130 consists of three tens and one hundred. We will write three tens in the tens place of the new number, and send one hundred to the next place:

As you can see in the figure, we represented 13 tens (number 130) as 1 hundred and 3 tens. We wrote three tens in the tens place of the new number. And one hundred was transferred to the ranks of hundreds. We will add this hundred to the result of adding hundreds of numbers 862 and 372. In order not to forget about it, we inscribed it over hundreds of numbers 862.

Now add hundreds. Eight hundred plus three hundred is eleven hundred plus one hundred left over from the previous addition. The result is twelve hundred in the hundreds place:

There is also a hundreds place overflow here, but this does not result in an error since the solution is complete. If desired, with 12 hundreds, you can carry out the same actions that we performed with 13 tens.

12 hundreds is a hundred repeated 12 times. And if you repeat a hundred 12 times, you get 1200

100 x 12 = 1200

There are two hundred and one thousand in 1200. Two hundred are written into the hundreds place of the new number, and one thousand has moved to the thousands place.

Now let's look at subtraction examples. First, let's remember what subtraction is. This is an operation that allows you to subtract another from one number. Subtraction consists of three parameters: minuend, subtrahend and difference. You also need to subtract by digits.

Example 3. Subtract 12 from 65.

Let's start with units. The units place of the number 65 contains the number 5, and the units place of the number 12 contains the number 2. This means that the units place of the number 65 contains five ones, and the units place of the number 12 contains two ones. Subtract two units from five units, we get three units. We write the number 3 in the units place of the new number:

Now subtract the tens. The tens place of the number 65 is the number 6, and the tens place of the number 12 is the number 1. This means that the tens place of the number 65 contains six tens, and the tens place of the number 12 contains one tens. Subtract one ten from six tens, we get five tens. We write the number 5 in the tens place of the new number:

Example 4. Subtract 15 from 32

The ones place of 32 contains two ones, and the ones place of 15 contains five ones. Five units cannot be subtracted from two units, because two units is less than five units.

Let's group 32 apples so that the first group has three dozen apples, and the second has the remaining two units of apples:

So, we need to subtract 15 apples from these 32 apples, that is, subtract five units and one dozen apples. And subtract by ranks.

Five units of apples cannot be subtracted from two units of apples. To perform a subtraction, two 1's must take a few apples from the adjacent group (the tens digit). But you can’t take as much as you want, since dozens are strictly ordered in ten pieces. The tens digit can give two units only one whole ten.

So, we take one ten from the category of tens and give it to two units:

Two units of apples are now joined by one dozen apples. It turns out 12 units of apples. And from twelve you can subtract five, you get seven. We write the number 7 in the units place of the new number:

Now subtract the tens. Since the tens place gave one ten to the units, now it has not three, but two tens. Therefore, subtract one ten from two tens. Only ten remain. We write the number 1 in the tens place of the new number:

In order not to forget that one ten (or a hundred or a thousand) was taken in some category, it is customary to put a dot over this category.

Example 5. Subtract 286 from 653

The ones place of 653 contains three ones, and the ones place of 286 contains six ones. Six units cannot be subtracted from three units, so we take one ten at the tens place. We put a dot over the tens discharge to remember that we took one ten from there:

Taken one ten and three units together form thirteen units. From thirteen units, you can subtract six units, you get seven units. We write the number 7 in the units place of the new number:

Now subtract the tens. Previously, the tens place of 653 contained five tens, but we took one ten from it, and now the tens place contains four tens. Eight tens cannot be subtracted from four tens, so we take one hundred at the hundreds place. We put a dot over the hundreds place to remember that we took one hundred from there:

Taken one hundred and four tens together form fourteen tens. Eight tens can be subtracted from fourteen tens to get six tens. We write the number 6 in the tens place of the new number:

Now subtract hundreds. The hundreds place of 653 used to contain six hundred, but we took one hundred from it, and now the hundreds place contains five hundred. You can subtract two hundred from five hundred to get three hundred. We write the number 3 in the hundreds place of the new number:

It is much more difficult to subtract from numbers like 100, 200, 300, 1000, 10000. That is, numbers with zeros at the end. To perform a subtraction, each digit has to borrow tens/hundreds/thousands from the next digit. Let's see how it goes.

Example 6

The ones place of 200 contains zero ones, and the ones place of 84 contains four ones. Four units cannot be subtracted from zero, so we take one ten at the tens place. We put a dot over the tens discharge to remember that we took one ten from there:

But there are no tens in the tens place that we could take, since there is also a zero. In order for the tens place to be able to give us one ten, we must take one hundred from the hundreds place for it. We put a dot over the hundreds place to remember that we took one hundred from there for the tens place:

Taken one hundred is ten tens. From these ten tens we take one ten and give it to units. This taken one ten and the previous zero ones together form ten ones. From ten units, you can subtract four units, you get six units. We write the number 6 in the units place of the new number:

Now subtract the tens. To subtract the units, we turned to the tens place for one ten, but at that time this place was empty. So that the tens place can give us one tens, we took one hundred from the hundreds place. We named this one hundred "ten tens" . We gave one dozen to units. So at the moment, the tens place contains not ten, but nine tens. Eight tens can be subtracted from nine tens to get one tens. We write the number 1 in the tens place of the new number:

Now subtract hundreds. For the tens digit, we took one hundred from the hundreds digit. So now the hundreds place contains not two hundred, but one. Since there is no hundreds place in the subtrahend, we transfer this one hundred to the hundreds place of the new number:

Naturally, subtracting with such a traditional method is quite difficult, especially at first. Having understood the principle of subtraction, you can use non-standard methods.

The first way is to decrease the number that has zeros on the end by one unit. Next, subtract the subtracted from the result obtained and add the unit to the resulting difference, which was originally subtracted from the reduced. Let's solve the previous example in this way:

The number being reduced here is 200. Let's decrease this number by one. If you subtract 1 from 200, you get 199. Now, in the example 200 - 84, instead of the number 200, we write the number 199 and solve the example 199 - 84. And the solution to this example is not difficult. We subtract units from units, tens from tens, and simply transfer a hundred to a new number, since there are no hundreds in the number 84:

We got the answer 115. Now we add the unit to this answer, which we initially subtracted from the number 200

Got the final answer 116.

Example 7. Subtract 91899 from 100000

Subtract one from 100000, we get 99999

Now subtract 91899 from 99999

To the result of 8100 we add the unit that we subtracted from 100000

Received final response 8101.

The second way to subtract is to consider the digit in the digit as an independent number. Let's solve some examples in this way.

Example 8. Subtract 36 from 75

So, in the units place of the number 75 there is the number 5, and in the units place of the number 36 there is the number 6. Six cannot be subtracted from five, so we take one unit from the next number in the tens place.

The number 7 is located in the tens place. We take one unit from this number and mentally add it to the left of the number 5

And since one unit is taken from the number 7, this number will decrease by one unit and turn into the number 6

Now, in the units place of the number 75, there is the number 15, and in the units place of the number 36, the number is 6. You can subtract 6 from 15, you get 9. We write the number 9 in the units place of the new number:

Move on to the next number in the tens place. Previously, the number 7 was located there, but we took one unit from this number, so now the number 6 is located there. And in the tens place of the number 36 is the number 3. You can subtract 3 from 6, you get 3. We write the number 3 in the tens place of the new number:

Example 9. Subtract 84 from 200

So, in the units place of the number 200 there is a zero, and in the units place of the number 84 there is a four. Four cannot be subtracted from zero, so we take one unit from the next number in the tens place. But the tens place is also zero. Zero cannot give us one. In this case, we take the number 20 as the next.

We take one unit from the number 20 and mentally add it to the left of zero, which is located in the category of units. And since one unit is taken from the number 20, this number will turn into the number 19

The units place is now 10. Ten minus four equals six. We write the number 6 in the ones place of the new number:

Move on to the next number in the tens place. Previously, there was a zero, but this zero, together with the next number 2, formed the number 20, from which we took one unit. As a result, the number 20 turned into the number 19. It turns out that now the number 9 is in the tens place of the number 200, and the number 8 is in the tens place of the number 84. Nine minus eight equals one. We write the number 1 in the tens place of our answer:

We move on to the next number, which is in the hundreds place. Previously, the number 2 was located there, but we took this number, together with the number 0, for the number 20, from which we took one unit. As a result, the number 20 turned into the number 19. It turns out that now the number 1 is located in the hundreds place of the number 200, and the hundreds place is empty in the number 84, so we transfer this unit to the new number:

This method at first seems complicated and meaningless, but in fact it is the easiest. Basically, we will use it when adding and subtracting numbers in a column.

Stacking

Column addition is a school operation that many people remember, but it doesn’t hurt to remember it again. Addition in a column occurs by digits - units are added to units, tens to tens, hundreds to hundreds, thousands to thousands.

Let's look at a few examples.

Example 1. Add 61 and 23.

First, we write down the first number, and under it the second number so that the units and tens of the second number are under the units and tens of the first number. We connect all this with an addition sign (+) vertically:

Now we add the units of the first number with the units of the second number, and add the tens of the first number with the tens of the second number:

Got 61 + 23 = 84.

Example 2 Add 108 and 60

Now we add the units of the first number with the units of the second number, the tens of the first number with the tens of the second number, the hundreds of the first number with the hundreds of the second number. But only the first number 108 has a hundred. In this case, the number 1 from the hundreds place is added to the new number (our answer). As they said at school, "demolishes":

It can be seen that we demolished the number 1 to our answer.

When it comes to addition, there is no difference in which order the numbers are written. Our example could have been written like this:

The first entry, where the number 108 was at the top, is more convenient to calculate. A person has the right to choose any record, but it must be remembered that units must be written strictly under units, tens under tens, hundreds under hundreds. In other words, the following entries will be incorrect:

If suddenly, when adding the corresponding digits, a number is obtained that does not fit into the digit of a new number, then it is necessary to write down one digit from the least significant digit, and transfer the rest to the next digit.

In this case, we are talking about the discharge overflow, which we talked about earlier. For example, adding 26 and 98 results in 124. Let's see how it turned out.

We write the numbers in a column. Units under units, tens under tens:

We add the units of the first number with the units of the second number: 6+8=14. We got the number 14, which will not fit into the category of units of our answer. In such cases, we first pull out from 14 the digit in the ones place and write it in the units place of our answer. In the units digit of the number 14 is the number 4. We write this figure in the units digit of our answer:

And where to put the number 1 out of 14? This is where things get interesting. We carry this unit to the next digit. It will be added to the tens place of our answer.

Adding tens to tens. 2 plus 9 equals 11, plus we add the unit that we got from the number 14. Adding our unit to 11, we get the number 12, which we write in the tens place of our answer. Since this is the end of the solution, there is no longer a question of whether the answer received will fit in the tens place. 12 we write down in full, forming the final answer.

Got the answer 124.

Using the traditional addition method, when adding 6 and 8 units, you get 14 units. 14 units is 4 units and 1 ten. We wrote down four units in the category of units, and sent one ten to the next category (to the digits of tens). Then, adding 2 tens and 9 tens, we got 11 tens, plus we added 1 tens, which remained after adding the units. The result was 12 tens. These twelve tens we wrote down in their entirety, forming the final answer 124.

This simple example demonstrates a school situation in which they say "Four write, one in the mind" . If you solve examples and after adding the digits you still have a number that you need to keep in mind, write it down above the digit where it will be added later. This will keep you from forgetting about her:

Example 2. Add the numbers 784 and 548

We write the numbers in a column. Units under units, tens under tens, hundreds under hundreds:

We add the units of the first number with the units of the second number: 4+8=12. The number 12 does not fit into the units category of our answer, so we take the number 2 out of 12 from the units category and write it into the units category of our answer. And the number 1 is transferred to the next digit:

Now add up the tens. We add 8 and 4 plus the unit that remains from the previous operation (the unit remains from 12, in the figure it is highlighted in blue). We add 8+4+1=13. The number 13 will not fit in the tens place of our answer, so we will write the number 3 in the tens place, and transfer the unit to the next place:

Now add hundreds. We add 7 and 5 plus the one left over from the previous operation: 7+5+1=13. We write the number 13 in the hundreds place:

Column subtraction

Example 1. Subtract 53 from 69.

Let's write the numbers in a column. Units under units, tens under tens. Then subtract by digits. Subtract the units of the second number from the units of the first number. Subtract the tens of the second number from the tens of the first number:

Received the answer 16.

Example 2 Find the value of the expression 95 − 26

The ones digit of 95 contains 5 ones, and the ones digit of 26 contains 6 ones. Six units cannot be subtracted from five units, so we take one ten at the tens place. This ten and the existing five units together make 15 units. From 15 units, you can subtract 6 units, you get 9 units. We write the number 9 in the category of units of our answer:

Now subtract the tens. The tens place of the number 95 used to contain 9 tens, but we took one tens from this place, and now it contains 8 tens. And the tens place of the number 26 contains 2 tens. Two tens can be subtracted from eight tens to get six tens. We write the number 6 in the tens place of our answer:

Let's use in which each digit included in the number is considered as a separate number. When subtracting large numbers in a column, this method is very convenient.

The number 5 is located in the unit category of the minuend. And the number 6 is in the unit category of the subtrahend. Do not subtract the six from the five. Therefore, we take one unit from the number 9. The taken unit is mentally added to the left of the five. And since we took one unit from the number 9, this number will decrease by one unit:

As a result, the five turns into the number 15. Now you can subtract 6 from 15. It turns out 9. We write the number 9 in the units of our answer:

Let's move on to the tens. Previously, the number 9 was located there, but since we took one unit from it, it turned into the number 8. The number 2 is located in the tens place of the second number. Eight minus two will be six. We write the number 6 in the tens place of our answer:

Example 3 Find the value of the expression 2412 − 2317

We write this expression in a column:

In the units place of the number 2412 there is the number 2, and in the units place of the number 2317 there is the number 7. You can’t subtract the seven from the two, so we take the unit from the next number 1. We mentally add the taken unit to the left of the two:

As a result, the two turns into the number 12. Now you can subtract 7 from 12. It turns out 5. We write the number 5 in the category of units of our answer:

Let's move on to tens. In the tens place of the number 2412, the number 1 was previously located, but since we took one unit from it, it turned into 0. And in the tens place of the number 2317, the number 1 is located. One cannot be subtracted from zero. Therefore, we take one unit from the next number 4. We mentally add the taken unit to the left of zero. And since we took one unit from the number 4, this number will decrease by one unit:

As a result, zero turns into the number 10. Now you can subtract 1 from 10. It turns out 9. We write the number 9 in the tens place of our answer:

The hundreds place of 2412 used to be a 4, but now it is a 3. The hundreds place of 2317 is also a 3. Three minus three is zero. The same goes for the thousands digits in both numbers. Two minus two equals zero. And if the difference between the leading digits is zero, then this zero is not recorded. Therefore, the final answer will be the number 95.

Example 4. Find the value of the expression 600 − 8

The units place of 600 is zero, and the units place of 8 is the number itself. From zero, do not subtract the eight, so we take the unit from the next number. But the next number is also zero. Then we take the number 60 for the next number. We take one unit from this number and mentally add it to the left of zero. And since we took one unit from the number 60, this number will decrease by one unit:

Now the number 10 is in the units place. You can subtract 8 from 10, you get 2. We write the number 2 in the units place of the new number:

Move on to the next number in the tens place. The tens place used to have a zero, but now there is a 9, and there is no tens place in the second number. Therefore, the number 9 is transferred to a new number:

Move on to the next number in the hundreds place. The hundreds place used to have the number 6, but now it has the number 5, and there is no hundreds place in the second number. Therefore, the number 5 is transferred to a new number:

Example 5 Find the value of the expression 10000 − 999

Let's write this expression in a column:

In the units place of the number 10000 there is a 0, and in the units place of the number 999 there is the number 9. You can’t subtract nine from zero, so we take one unit from the next number in the tens place. But the next digit is also zero. Then we take 1000 for the next number and take one from this number:

The next number in this case was 1000. Taking a unit from it, we turned it into the number 999. And the taken unit was added to the left of zero.

Further calculation was not difficult. Ten minus nine equals one. Subtracting numbers in the tens place of both numbers gave zero. Subtracting numbers in the hundreds place of both numbers also gave zero. And nine from the category of thousands was transferred to a new number:

Example 6. Find the value of the expression 12301 − 9046

Let's write this expression in a column:

In the units place of the number 12301 there is the number 1, and in the units place of the number 9046 there is the number 6. Six cannot be subtracted from the unit, so we take one unit from the next number in the tens place. But the next bit is zero. Zero can't give us anything. Then we take 1230 for the next number and take one from this number:

Numerals are an independent part of speech, indicating the number of objects, their serial number, as well as the total number. Depending on the purpose, numerals are divided into three broad groups, each of which is described in detail in this article. Also here are grammatical signs of numerals and illustrative examples.

Numerals are studied in the 6th grade. This part of speech can denote the number of some objects, the number of the object in order, as well as the quantity as a whole. Based on this value, they are divided into 3 large groups.

Numerals can be quantitative, ordinal and collective. Each of these groups of words has its own characteristics.

Table"Lexico-grammatical categories of numerals in Russian" with examples

Changing cardinal numbers

numeral	How is it changing	Examples
1	By gender, numbers and cases	One - one, one, one
2,3, 4	By case and gender	Two, two, two, three
5-20,30	By cases, as a noun of the 3rd cl.	Wed: tablecloth - ten, eleven
50-80, 200-900	By cases, while both parts change	Fifty, three hundred
40, 90, 100	By cases, they have only 2 forms	forty - forty One hundred - one hundred ninety - ninety
1000	By cases as a noun. 1st sk.	Wed: Candles - a thousand
1000000,	By cases as a noun. 2nd sk.	Wed: home - a million - a billion

Ordinal numbers: grammatical features

The numerals of this category are grammatically similar to adjectives. They can also have forms of different cases, genders and numbers.

Ordinal numbers can also have a different structure. When declining the composite numbers of this category, only the last word changes in them. For example.

Name the main forms and classes of words in which zero endings are distinguished.

Compose phrases using different cases, using the nickname of the literary character Richard the Lionheart

15. What is the ending in the genitive plural of the names of paired items (boots, trousers)? How to say correctly: no socks or no socks?

16. Which of the forms relate to the official business, and which to the colloquial style of speech: grams, pendants / grams, pendant? tomatoes, oranges / tomato, orange?

17. How do they form the genitive case of nouns that are used only in the plural: frost, twilight, nursery? (frost, twilight, nursery)

18. What nouns in Russian are classified as animate / inanimate?

The category of animation of nouns manifests itself in the coincidence of the form of V. p. and R. p. in the singular and plural in masculine nouns (I see a person, a deer, friends, bears) and only in the plural of feminine and neuter nouns (I see women, animals).

The main forms and classes of words in which zero endings are distinguished:

1. im.p. unit nouns m.r. - garden, snow;

2. im.p. unit nouns f.r. - joy, mouse;

3. im.p. unit m.r. short adjectives and participles: sad, offended, withdrawn;

4. im.p. some numerals: twelve, six, one;

5. rod.p. plural some nouns: stocking_ (stockings), family_, (se [m "y a]);

6. im.p. unit m.r. possessive adjectives: Zeus_ (cf. Zeusova), sister_ (sister), fish_ (fish [b "y a]).

7. unit m.r. past verbs temp. and conditional mood: walked_, talked_, was_, would come_, would get lost_.

20. Name the most difficult cases of using a name
numeral.

21. What should be remembered when using compound cardinal numbers in speech?

When using compound quantitative numbers in speech, it should be remembered that 1) they must be in the same case as the noun denoting the object; 2) by cases, all the components forming the numeral change.

22. How are the numbers 40, 90, 100 different from other numbers?

Numerals forty, ninety and one hundred differ from others in that in all indirect cases, except for the accusative, they have the ending -a.

23. What is the peculiarity of the use of fractional numbers?

When using numerals denoting fractional numbers, the noun always stands in the genitive case of the singular, and all parts of the numeral change in cases: im. - three whole and six tenths of a percent, genus. - three point six percent etc.

24. How do ordinal numbers change in cases?

In compound ordinal numbers, only the last word changes in cases, and all previous words are used in the initial form (nominative singular).

25. How to use the collective numerals two, three correctly?

Numerals two, three, four, five, six, seven, as well as infrequent eight, nine, ten can only be used in certain cases:

1) with nouns husband. and general gender denoting males (three friends, two orphans);

2) with personal pronouns we you they (there are four of us, there are two of them, there are two of us, there were seven of them);

3) with nouns that have only plural forms. h. (two sledges, three scissors, four collars, three jeans, two days);

4) with nouns denoting baby animals (two cubs, seven kids);

5) with nouns denoting paired objects (two boots, two mittens, three mittens) this form is colloquial, it is preferable to use a pair of boots, a pair of mittens;

6) with nouns children, guys, people, with a noun face in the meaning of "human" two children, three boys, three young people, four strangers);

7) with nouns that are formed by transition from adjectives or participles and name a group of people (four acquaintances, five vacationers).

26. Can "three wolves" eat one teddy bear? Can it be said three friends, three girlfriends, two scissors, two bears, three students, three students?

three friends, three girlfriends, two scissors, two bears, three students, three students?

27. Tell us about the influence of the category of animateness / inanimateness in combinations of numerals with nouns

Collective numerals in combination with animate nouns are used in all cases ( tro e students, tro them students, three them students etc.). In combination with inanimate nouns, these numerals are used only in the nominative and accusative cases; in other cases, quantitative numbers should be used: three days have passed(them.), waited three days(vin.). Combinations are not allowed: * about three days(genus.), * thanks to three days(dat.), * after three days(tv.), * about three days(proposition). It should say: near three days(genus.), thanks to three days(dat.), after three for days(tv.), about three days(proposition).

28. What form does the predicate take in the numeral?

With compound numbers ending in one , the predicate, as a rule, is put in the form singular, for example: Twenty-one delegates arrived at the meeting.

2. If words act as the subject or part of it majority, minority, part, few , then the predicate is used in the form singular, for example: Part of the group remained at the scientific conference. Several students signed up for an elective seminar.

3. With words thousand, million, billion the form singular predicate preferred, and the form plural is allowed, for example: A thousand students came admissible came to the rally. A million inhabitants of the country did not participate ( admissible did not participate) in the voting.

· quality(answer the question "what?");

o form forms of quality degrees ( red - red, rubella, reddish);

o have degree of comparison: comparative (kind - kinder) and excellent degree ( kinder - kindest, stronger - strongest and etc.).

Consumption compound comparative degree, which is formed by a combination of auxiliary words more or less. The correct form is more, less + adjective in a positive degree. For example: more merciful, less rude. But never with words more, less the forms of a simple comparative and a simple superlative degree of comparison cannot be combined, i.e. combinations such as better, more beautiful, less beautiful in Russian literary language unacceptable a pleonastic combination occurs;

appear in full and short form (cheerful - cheerful). Not all quality adjectives can have short forms. There are no such forms:

a) for adjectives formed by the transition to qualitative from
other categories or participles: golden character, the lion's share,
outstanding musician;

b) most adjectives denoting color, as well as
calling the colors of animals: pink scarf, black horse, brown
boots.

c) for adjectives with suffixes - ck,- ov,- ev:friendly advice,
draft version, comrade.

d) for adjectives with the suffix -l, the stem of which coincides with the stem of the past tense verb: experienced person, backward views,
Old man,(however, if the final consonant -l of the stem is not
suffix, the formation of a short form is possible: dim look - dim,
round table- round).

e) for adjectives with prefixes pre-, times- and race in the sense of a greater degree of quality: a gracious person, a cheerful student,
messed up girl.(However, when the specified meanings of prefixes are not
are understood, the formation of a short form is permissible: beautiful - beautiful
reasonable - reasonable);

o form adverbs, ending in -о/-е and -и ( cheerful - fun);

o it is possible to form nouns from quality adjectives ( red - redness);

o most quality adjectives are combined with the adverb "very" ( very big, very tasty);

· relative(answering the question “which one?”)

o relative adjectives do not have degrees; designate the material from which the object is made, the spatial and temporal features of the object: wood - wood, January - January, yesterday - yesterday's;

o most relative adjectives do not go well with "very";

· possessive- answer the question "whose?" and denote belonging to something animal or person ( paternal, sisters, fox);

o do not have degrees;

o have neither synonyms nor antonyms, although the very combination of a possessive adjective with a noun is synonymous with a construction with the meaning of possession.

In order to attribute an adjective to any category, it is enough to find at least one sign of this category in the adjective.