Column division(you can also find the name division corner) is a standard procedure inarithmetic, designed to divide simple or complex multi-digit numbers by breakingdivided into a number of simpler steps. As with all division problems, one number, calleddivisible, is divided into another, calleddivider, producing a result calledprivate.

The column can be used to divide natural numbers without a remainder, as well as to divide natural numbers with the remainder.

Rules for writing when dividing by a column.

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results whendividing natural numbers in a column. Let’s say right away that writing long division isIt is most convenient on paper with a checkered line - this way there is less chance of straying from the desired row and column.

First, the dividend and the divisor are written in one line from left to right, after which between the writtennumbers represent a symbol of the form.

For example, if the dividend is 6105 and the divisor is 55, then their correct notation when dividing inthe column will be like this:

Look at the following diagram illustrating places to write dividend, divisor, quotient,remainder and intermediate calculations when dividing by a column:

From the above diagram it is clear that the required quotient (or incomplete quotient when divided with a remainder) will bewritten below the divisor under the horizontal bar. And intermediate calculations will be carried out belowdivisible, and you need to take care in advance about the availability of space on the page. In this case, one should be guidedrule: the greater the difference in the number of characters in the entries of the dividend and the divisor, the greaterspace will be required.

Division of a natural number by a single-digit natural number, column division algorithm.

How to do long division is best explained with an example.Calculate:

512:8=?

First, let's write down the dividend and divisor in a column. It will look like this:

We will write their quotient (result) under the divisor. For us this is number 8.

1. Define an incomplete quotient. First we look at the first digit on the left in the dividend notation.If the number defined by this figure is greater than the divisor, then in the next paragraph we have to workwith this number. If this number is less than the divisor, then we need to add the following to considerationon the left the figure in the notation of the dividend, and work further with the number determined by the two consideredin numbers. For convenience, we highlight in our notation the number with which we will work.

2. Take 5. The number 5 is less than 8, which means you need to take one more number from the dividend. 51 is greater than 8. So.this is an incomplete quotient. We put a dot in the quotient (under the corner of the divisor).

After 51 there is only one number 2. This means we add one more point to the result.

3. Now, remembering multiplication table by 8, find the product closest to 51 → 6 x 8 = 48→ write the number 6 into the quotient:

We write 48 under 51 (if we multiply 6 from the quotient by 8 from the divisor, we get 48).

Attention! When writing under an incomplete quotient, the rightmost digit of the incomplete quotient should be aboverightmost digit works.

4. Between 51 and 48 on the left we put “-” (minus). Subtract according to the rules of subtraction in column 48 and below the lineLet's write down the result.

However, if the result of the subtraction is zero, then it does not need to be written (unless the subtraction is inthis point is not the very last action that completely completes the division process column).

The remainder is 3. Let's compare the remainder with the divisor. 3 is less than 8.

Attention!If the remainder is greater than the divisor, then we made a mistake in the calculation and the product iscloser than the one we took.

5. Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we do notbegan to write down zero) we write down the number located in the same column in the record of the dividend. If inThere are no numbers in the dividend entry in this column, then division by column ends here.

The number 32 is greater than 8. And again, using the multiplication table by 8, we find the nearest product → 8 x 4 = 32:

The remainder was zero. This means that the numbers are completely divided (without remainder). If after the lastsubtraction results in zero, and there are no more digits left, then this is the remainder. We add it to the quotient inparentheses (eg 64(2)).

Column division of multi-digit natural numbers.

Division by a multi-digit natural number is done in a similar way. At the same time, in the firstThe “intermediate” dividend includes so many high-order digits that it becomes larger than the divisor.

For example, 1976 divided by 26.

• The number 1 in the most significant digit is less than 26, so consider a number made up of two digits senior ranks - 19.
• The number 19 is also less than 26, so consider a number made up of the digits of the three highest digits - 197.
• The number 197 is greater than 26, divide 197 tens by 26: 197: 26 = 7 (15 tens left).
• Convert 15 tens to units, add 6 units from the units digit, we get 156.
• Divide 156 by 26 to get 6.

So 1976: 26 = 76.

If at some division step the “intermediate” dividend turns out to be less than the divisor, then in the quotient0 is written, and the number from this digit is transferred to the next, lower digit.

Division with decimal fraction in quotient.

Decimals online. Converting decimals to fractions and fractions to decimals.

If the natural number is not divisible by a single digit natural number, you can continuebitwise division and get a decimal fraction in the quotient.

For example, divide 64 by 5.

• Divide 6 tens by 5, we get 1 ten and 1 ten as a remainder.
• We convert the remaining ten into units, add 4 from the ones category, and get 14.
• We divide 14 units by 5, we get 2 units and a remainder of 4 units.
• We convert 4 units to tenths, we get 40 tenths.
• Divide 40 tenths by 5 to get 8 tenths.

So 64:5 = 12.8

Thus, if, when dividing a natural number by a natural single-digit or multi-digit numberthe remainder is obtained, then you can put a comma in the quotient, convert the remainder into units of the following,smaller digit and continue dividing.

The division of natural numbers, especially multi-digit ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also find the name corner division. Let us immediately note that the column can be used to both divide natural numbers without a remainder and divide natural numbers with a remainder.

In this article we will look at how long division is performed. Here we will talk about recording rules and all intermediate calculations. First, let's focus on dividing a multi-digit natural number by a single-digit number with a column. After this, we will focus on cases when both the dividend and the divisor are multi-valued natural numbers. The entire theory of this article is provided with typical examples of division by a column of natural numbers with detailed explanations of the solution process and illustrations.

Page navigation.

Rules for recording when dividing by a column

Let's start by studying the rules for writing the dividend, divisor, all intermediate calculations and results when dividing natural numbers by a column. Let's say right away that it is most convenient to do column division in writing on paper with a checkered line - this way there is less chance of straying from the desired row and column.

First, the dividend and divisor are written in one line from left to right, after which a symbol of the form is drawn between the written numbers. For example, if the dividend is the number 6 105 and the divisor is 5 5, then their correct recording when dividing into a column will be as follows:

Look at the following diagram to illustrate where to write the dividend, divisor, quotient, remainder, and intermediate calculations in long division.

From the above diagram it is clear that the required quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal line. And intermediate calculations will be carried out below the dividend, and you need to take care in advance about the availability of space on the page. In this case, you should be guided by the rule: the greater the difference in the number of characters in the entries of the dividend and divisor, the more space will be required. For example, when dividing by a column the natural number 614,808 by 51,234 (614,808 is a six-digit number, 51,234 is a five-digit number, the difference in the number of characters in the records is 6−5 = 1), intermediate calculations will require less space than when dividing the numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3). To confirm our words, we present complete records of division by a column of these natural numbers:

Now you can proceed directly to the process of dividing natural numbers by a column.

Column division of a natural number by a single-digit natural number, column division algorithm

It is clear that dividing one single-digit natural number by another is quite simple, and there is no reason to divide these numbers into a column. However, it will be helpful to practice your initial long division skills with these simple examples.

Example.

Let us need to divide with a column of 8 by 2.

Solution.

Of course, we can perform division using the multiplication table, and immediately write down the answer 8:2=4.

But we are interested in how to divide these numbers with a column.

First, we write down the dividend 8 and the divisor 2 as required by the method:

Now we begin to find out how many times the divisor is contained in the dividend. To do this, we sequentially multiply the divisor by the numbers 0, 1, 2, 3, ... until the result is a number equal to the dividend (or a number greater than the dividend, if there is a division with a remainder). If we get a number equal to the dividend, then we immediately write it under the dividend, and in the place of the quotient we write the number by which we multiplied the divisor. If we get a number greater than the dividend, then under the divisor we write the number calculated at the penultimate step, and in place of the incomplete quotient we write the number by which the divisor was multiplied at the penultimate step.

Let's go: 2·0=0 ; 2·1=2 ; 2·2=4 ; 2·3=6 ; 2·4=8. We have received a number equal to the dividend, so we write it under the dividend, and in place of the quotient we write the number 4. In this case, the record will take the following form:

The final stage of dividing single-digit natural numbers with a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract the numbers above this line in the same way as is done when subtracting natural numbers in a column. The number resulting from the subtraction will be the remainder of the division. If it is equal to zero, then the original numbers are divided without a remainder.

In our example we get

Now we have before us a completed recording of the column division of the number 8 by 2. We see that the quotient of 8:2 is 4 (and the remainder is 0).

Answer:

8:2=4 .

Now let's look at how a column divides single-digit natural numbers with a remainder.

Example.

Divide 7 by 3 using a column.

Solution.

At the initial stage, the entry looks like this:

We begin to find out how many times the dividend contains the divisor. We will multiply 3 by 0, 1, 2, 3, etc. until we get a number equal to or greater than the dividend 7. We get 3·0=0<7 ; 3·1=3<7 ; 3·2=6<7 ; 3·3=9>7 (if necessary, refer to the article comparing natural numbers). Under the dividend we write the number 6 (it was obtained at the penultimate step), and in place of the incomplete quotient we write the number 2 (the multiplication was carried out by it at the penultimate step).

It remains to carry out the subtraction, and the division by a column of single-digit natural numbers 7 and 3 will be completed.

Thus, the partial quotient is 2 and the remainder is 1.

Answer:

7:3=2 (rest. 1) .

Now you can move on to dividing multi-digit natural numbers by columns into single-digit natural numbers.

Now we'll figure it out long division algorithm. At each stage, we will present the results obtained by dividing the multi-digit natural number 140,288 by the single-digit natural number 4. This example was not chosen by chance, since when solving it we will encounter all possible nuances and will be able to analyze them in detail.

First we look at the first digit on the left in the dividend notation. If the number defined by this figure is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend, and continue to work with the number determined by the two digits under consideration. For convenience, we highlight in our notation the number with which we will work.

The first digit from the left in the notation of the dividend 140288 is the digit 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the notation of the dividend. At the same time, we see the number 14, with which we have to work further. We highlight this number in the notation of the dividend.

The following steps from the second to the fourth are repeated cyclically until the division of natural numbers by a column is completed.

Now we need to determine how many times the divisor is contained in the number we are working with (for convenience, let's denote this number as x). To do this, we sequentially multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When the number x is obtained, we write it under the highlighted number according to the recording rules used when subtracting natural numbers in a column. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (in subsequent passes of 2-4 points of the algorithm, this number is written to the right of the numbers already there). When a number is obtained that is greater than the number x, then under the highlighted number we write the number obtained at the penultimate step, and in place of the quotient (or to the right of the numbers already there) we write the number by which the multiplication was carried out at the penultimate step. (We carried out similar actions in the two examples discussed above).

Multiply the divisor 4 by the numbers 0, 1, 2, ... until we get a number that is equal to 14 or greater than 14. We have 4·0=0<14 , 4·1=4<14 , 4·2=8<14 , 4·3=12<14 , 4·4=16>14 . Since at the last step we received the number 16, which is greater than 14, then under the highlighted number we write the number 12, which was obtained at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate point the multiplication was carried out precisely by it.


At this stage, from the selected number, subtract the number located under it using a column. The result of the subtraction is written under the horizontal line. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at that point is the very last action that completely completes the process of long division). Here, for your own control, it would not be amiss to compare the result of the subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake was made somewhere.

We need to subtract the number 12 from the number 14 with a column (for the correctness of the recording, we must remember to put a minus sign to the left of the numbers being subtracted). After completing this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with the divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next point.


Now, under the horizontal line to the right of the numbers located there (or to the right of the place where we did not write down the zero), we write down the number located in the same column in the notation of the dividend. If there are no numbers in the record of the dividend in this column, then the division by column ends there. After this, we select the number formed under the horizontal line, accept it as a working number, and repeat points 2 to 4 of the algorithm with it.

Under the horizontal line to the right of the number 2 already there, we write down the number 0, since it is the number 0 that is in the record of the dividend 140,288 in this column. Thus, the number 20 is formed under the horizontal line.


We select this number 20, take it as a working number, and repeat with it the actions of the second, third and fourth points of the algorithm.

Multiply the divisor 4 by 0, 1, 2, ... until we get the number 20 or a number that is greater than 20. We have 4·0=0<20 , 4·1=4<20 , 4·2=8<20 , 4·3=12<20 , 4·4=16<20 , 4·5=20 . Так как мы получили число, равное числу 20 , то записываем его под отмеченным числом, а на месте частного, справа от уже имеющегося там числа 3 записываем число 5 (на него производилось умножение).


We carry out the subtraction in a column. Since we are subtracting equal natural numbers, then by virtue of the property of subtracting equal natural numbers, the result is zero. We do not write down the zero (since this is not the final stage of division with a column), but we remember the place where we could write it (for convenience, we will mark this place with a black rectangle).


Under the horizontal line to the right of the remembered place we write down the number 2, since it is precisely it that is in the record of the dividend 140,288 in this column. Thus, under the horizontal line we have the number 2.


We take the number 2 as the working number, mark it, and we will once again have to perform the actions of 2-4 points of the algorithm.

We multiply the divisor by 0, 1, 2, and so on, and compare the resulting numbers with the marked number 2. We have 4·0=0<2 , 4·1=4>2. Therefore, under the marked number we write the number 0 (it was obtained at the penultimate step), and in the place of the quotient to the right of the number already there we write the number 0 (we multiplied by 0 at the penultimate step).


We perform the subtraction in a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with the divisor 4. Since 2<4 , то можно спокойно двигаться дальше.


Under the horizontal line to the right of the number 2, add the number 8 (since it is in this column in the entry for the dividend 140 288). Thus, the number 28 appears under the horizontal line.


We take this number as a working number, mark it, and repeat steps 2-4.

There shouldn't be any problems here if you have been careful up to now. Having completed all the necessary steps, the following result is obtained.

All that remains is to carry out the steps from points 2, 3, 4 one last time (we leave this to you), after which you will get a complete picture of dividing the natural numbers 140,288 and 4 into a column:

Please note that the number 0 is written in the very bottom line. If this was not the last step of division by a column (that is, if in the record of the dividend there were numbers left in the columns on the right), then we would not write this zero.

Thus, looking at the completed record of dividing the multi-digit natural number 140,288 by the single-digit natural number 4, we see that the quotient is the number 35,072 (and the remainder of the division is zero, it is in the very bottom line).

Of course, when dividing natural numbers by a column, you will not describe all your actions in such detail. Your solutions will look something like the following examples.

Example.

Perform long division if the dividend is 7 136 and the divisor is a single-digit natural number 9.

Solution.

At the first step of the algorithm for dividing natural numbers by columns, we get a record of the form

After performing the actions from the second, third and fourth points of the algorithm, the column division record will take the form

Repeating the cycle, we will have

One more pass will give us a complete picture of the column division of the natural numbers 7,136 and 9

Thus, the partial quotient is 792, and the remainder is 8.

Answer:

7 136:9=792 (rest. 8) .

And this example demonstrates what long division should look like.

Example.

Divide the natural number 7,042,035 by the single-digit natural number 7.

Solution.

The most convenient way to do division is by column.

Answer:

7 042 035:7=1 006 005 .

Column division of multi-digit natural numbers

We hasten to please you: if you have thoroughly mastered the column division algorithm from the previous paragraph of this article, then you almost already know how to perform column division of multi-digit natural numbers. This is true, since stages 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first point.

At the first stage of dividing multi-digit natural numbers into a column, you need to look not at the first digit on the left in the notation of the dividend, but at the number of them equal to the number of digits contained in the notation of the divisor. If the number defined by these numbers is greater than the divisor, then in the next paragraph we have to work with this number. If this number is less than the divisor, then we need to add to the consideration the next digit on the left in the notation of the dividend. After this, the actions specified in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

All that remains is to see the application of the column division algorithm for multi-valued natural numbers in practice when solving examples.

Example.

Let's perform column division of multi-digit natural numbers 5,562 and 206.

Solution.

Since the divisor 206 contains 3 digits, we look at the first 3 digits on the left in the dividend 5,562. These numbers correspond to the number 556. Since 556 is greater than the divisor 206, we take the number 556 as a working number, select it, and move on to the next stage of the algorithm.

Now we multiply the divisor 206 by the numbers 0, 1, 2, 3, ... until we get a number that is either equal to 556 or greater than 556. We have (if multiplication is difficult, then it is better to multiply natural numbers in a column): 206 0 = 0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556. Since we received a number that is greater than the number 556, then under the highlighted number we write the number 412 (it was obtained at the penultimate step), and in place of the quotient we write the number 2 (since we multiplied by it at the penultimate step). The column division entry takes the following form:

We perform column subtraction. We get the difference 144, this number is less than the divisor, so you can safely continue performing the required actions.

Under the horizontal line to the right of the number there we write the number 2, since it is in the record of the dividend 5562 in this column:

Now we work with the number 1,442, select it, and go through steps two through four again.

Multiply the divisor 206 by 0, 1, 2, 3, ... until you get the number 1442 or a number that is greater than 1442. Let's go: 206·0=0<1 442 , 206·1=206<1 442 , 206·2=412<1 332 , 206·3=618<1 442 , 206·4=824<1 442 , 206·5=1 030<1 442 , 206·6=1 236<1 442 , 206·7=1 442 . Таким образом, под отмеченным числом записываем 1 442 , а на месте частного правее уже имеющегося там числа записываем 7 :

We carry out the subtraction in a column, we get zero, but we don’t write it down right away, we just remember its position, because we don’t know whether the division ends here, or whether we’ll have to repeat the steps of the algorithm again:

Now we see that we cannot write any number under the horizontal line to the right of the remembered position, since there are no digits in the record of the dividend in this column. Therefore, this completes the division by column, and we complete the entry:

• Mathematics. Any textbooks for 1st, 2nd, 3rd, 4th grades of general education institutions.
• Mathematics. Any textbooks for 5th grade of general education institutions.

Let's first look at simple cases of division, when the quotient results in a single-digit number.

Let's find the value of the quotient numbers 265 and 53.

To make it easier to choose the quotient number, let's divide 265 not by 53, but by 50. To do this, divide 265 by 10, the result will be 26 (the remainder is 5). And if we divide 26 by 5, it will be 5. The number 5 cannot be immediately written down in the quotient, since it is a trial number. First you need to check if it fits. Let's multiply. We see that the number 5 has come up. And now we can write it down privately.

The value of the quotient of the numbers 265 and 53 is 5. Sometimes, when dividing, the test digit of the quotient does not fit, and then it needs to be changed.

Let's find the value of the quotient numbers 184 and 23.

The quotient will be a single digit number.

To make it easier to choose the quotient number, let's divide 184 not by 23, but by 20. To do this, divide 184 by 10, the result will be 18 (remainder 4). And we divide 18 by 2, it becomes 9. 9 is a test number, we won’t write it in the quotient right away, but we’ll check if it fits. Let's multiply. And 207 is greater than 184. We see that the number 9 is not suitable. The quotient will be less than 9. Let's try to see if the number 8 is suitable. Let's multiply. We see that the number 8 is suitable. We can write it down privately.

The value of the quotient of 184 and 23 is 8.

Let's consider more complex cases of division. Let's find the value of the quotient of 768 and 24.

The first incomplete dividend is 76 tens. This means that the quotient will have 2 digits.

Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to choose the quotient number, let's divide 76 not by 24, but by 20. That is, you need to divide 76 by 10, there will be 7 (the remainder is 6). And divide 7 by 2, you get 3 (remainder 1). 3 is the test digit of the quotient. First let's check if it fits. Let's multiply. . The remainder is less than the divisor. This means that the number 3 is suitable and now we can write it in place of the tens of the quotient.

Let's continue the division. The next partial dividend is 48 units. Let's divide 48 by 24. To make it easier to choose the quotient, let's divide 48 not by 24, but by 20. That is, if we divide 48 by 10, there will be 4 (the remainder is 8). And we divide 4 by 2, it becomes 2. This is the test digit of the quotient. We must first check if it will fit. Let's multiply. We see that the number 2 fits and, therefore, we can write it in place of the units of the quotient.

The meaning of the quotient of 768 and 24 is 32.

Let's find the value of the quotient numbers 15,344 and 56.

The first incomplete dividend is 153 hundreds, which means that the quotient will have three digits.

Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the quotient, let's divide 153 not by 56, but by 50. To do this, divide 153 by 10, the result will be 15 (remainder 3). And divide 15 by 5, it becomes 3. 3 is the test digit of the quotient. Remember: you cannot immediately write it down in private, but you must first check whether it is suitable. Let's multiply. And 168 is greater than 153. This means that the quotient will be less than 3. Let's check if the number 2 is suitable. Let's multiply. A . The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in the place of hundreds in the quotient.

Let us form the following incomplete dividend. That's 414 tens. Let's divide 414 by 56. To make it more convenient to choose the quotient number, let's divide 414 not by 56, but by 50. . . Remember: 8 is a test number. Let's check it out. . And 448 is greater than 414, which means that the quotient will be less than 8. Let’s check if the number 7 is suitable. Multiply 56 by 7, we get 392. . The remainder is less than the divisor. This means that the number fits and in the quotient we can write 7 in place of tens.

Let's continue the division. The next partial dividend is 224 units. Let's divide 224 by 56. To make it easier to find the quotient number, divide 224 by 50. That is, first by 10, there will be 22 (the remainder is 4). And divide 22 by 5, there will be 4 (remainder 2). 4 is a test number, let's check it to see if it fits. . And we see that the number has come up. Let's write 4 in place of units in the quotient.

The value of the quotient of 15,344 and 56 is 274.

Today we learned to divide by two-digit numbers in writing.

Bibliography

1. Mathematics. Textbook for 4th grade. beginning school At 2 o'clock/M.I. Moreau, M.A. Bantova - M.: Education, 2010.
2. Uzorova O.V., Nefedova E.A. Large math problem book. 4th grade. - M.: 2013. - 256 p.
3. Mathematics: textbook. for 4th grade. general education institutions with Russian language training. At 2 p.m. Part 1 / T.M. Chebotarevskaya, V.L. Drozd, A.A. Carpenter; lane with white language L.A. Bondareva. - 3rd ed., revised. - Minsk: Nar. Asveta, 2008. - 134 p.: ill.
4. Mathematics. 4th grade. Textbook. At 2 o'clock/Geidman B.P. and others - 2010. - 120 p., 128 p.

1. Ppt4web.ru ().
2. Myshared.ru ().
3. Viki.rdf.ru ​​().

Homework

Perform division

Unfortunately, children nowadays practically do not know how to do mental calculations. This happened due to the fact that modern technologies offer each child to solve the problem with a couple of clicks. For many children, the Internet has replaced not only textbooks, but also certain skills. You can increasingly hear from the younger generation that it is not at all necessary to know mathematics, since you always have a calculator or phone at hand. But the true significance of this science lies in the development of thinking, and not in overcoming the fear of being deceived by a trader in the market.

Long division helps elementary school students become familiar with number operations. Thanks to it, the multiplication table is fixed in memory, and the skill of performing addition and subtraction operations is honed.

To implement this arithmetic operation, you need to become familiar with its components:

1. Dividend - a number that is divided.

2. Divisor - the number that is divided by.

3. Quotient - the result obtained by division.

4. Remainder is the part of the dividend that cannot be divided.

American and European division models

The rules for long division are the same in all countries. There is only a difference in the graphic part, that is, in its recording. In the European system, the dividing line, or the so-called corner, is placed on the right side of the number being divided. The divisor is written above the corner line, and the quotient is written below the horizontal line of the corner.

Dividing into a column according to the American model involves placing a corner on the left side. The quotient is written above the horizontal line of the angle, directly above the number being divided. The divisor is written under the horizontal line, to the left of the vertical line. The process of performing the action itself does not differ from the European model.

Divide by a two-digit number

To use a two-digit value, you need to write it down according to the diagram, and then carry out the action. Column division begins with the highest digits of the number being divided. The first two digits are taken if the number formed by them is greater in value than the divisor. Otherwise, the first three digits are separated. The number they form is divided by the divisor, the remainder goes down, and the result is written in the dividing corner. After this, the digit from the next digit of the number being divided is transferred, and the procedure is repeated. This continues until the number is completely divided.

If it is necessary to divide a number with a remainder, it is written separately. If you need to completely divide a number, then after the end of the digits of the number a comma is placed in the answer, indicating the beginning of the fractional part, and instead of the digits, a zero is moved down each time.

Division multi-digit or multi-digit numbers are convenient to produce in writing in a column. Let's figure out how to do this. Let's start by dividing a multi-digit number by a single-digit number, and gradually increase the digit of the dividend.

So let's divide 354 on 2 . First, let's place these numbers as shown in the figure:

We place the dividend on the left, the divisor on the right, and the quotient will be written under the divisor.

Now we begin to divide the dividend by the divisor bitwise from left to right. We find first incomplete dividend, for this we take the first digit on the left, in our case 3, and compare it with the divisor.

3 more 2 , Means 3 and there is an incomplete dividend. We put a dot in the quotient and determine how many more digits will be in the quotient - the same number as remained in the dividend after selecting the incomplete dividend. In our case, the quotient has the same number of digits as the dividend, that is, the most significant digit will be hundreds:

In order to 3 divide by 2 remember the multiplication table by 2 and find the number, when multiplied by 2 we get the greatest product, which is less than 3.

2 × 1 = 2 (2< 3)

2 × 2 = 4 (4 > 3)

2 less 3 , A 4 more, which means we take the first example and the multiplier 1 .

Let's write it down 1 to the quotient in place of the first point (in the hundreds place), and write the found product under the dividend:

Now we find the difference between the first incomplete dividend and the product of the found quotient and the divisor:

The resulting value is compared with the divisor. 15 more 2 , which means we have found the second incomplete dividend. To find the result of division 15 on 2 again remember the multiplication table 2 and find the greatest product that is less 15 :

2 × 7 = 14 (14< 15)

2 × 8 = 16 (16 > 15)

The required multiplier 7 , we write it as a quotient in place of the second point (in tens). We find the difference between the second incomplete dividend and the product of the found quotient and divisor:

We continue the division, why we find third incomplete dividend. We lower the next digit of the dividend:

We divide the incomplete dividend by 2, putting the resulting value in the category of units of the quotient. Let's check the correctness of the division:

2 × 7 = 14

We write the result of dividing the third incomplete dividend by the divisor into the quotient and find the difference:

We got the difference equal to zero, which means the division is done Right.

Let's complicate the problem and give another example:

1020 ÷ 5

Let's write our example in a column and define the first incomplete quotient:

The thousands place of the dividend is 1 , compare with the divisor:

1 < 5

We add the hundreds place to the incomplete dividend and compare:

10 > 5 – we have found an incomplete dividend.

We divide 10 on 5 , we get 2 , write the result into the quotient. The difference between the incomplete dividend and the result of multiplying the divisor and the found quotient.

10 – 10 = 0

0 we do not write, we omit the next digit of the dividend – the tens digit:

We compare the second incomplete dividend with the divisor.

2 < 5

We should add one more digit to the incomplete dividend; for this we put in the quotient, on the tens digit 0 :

20 ÷ 5 = 4

We write the answer in the category of units of the quotient and check: we write the product under the second incomplete dividend and calculate the difference. We get 0 , Means example solved correctly.

And 2 more rules for dividing into a column:

1. If the dividend and divisor have zeros in the low-order digits, then before dividing they can be reduced, for example:

As many zeros in the low-order digit of the dividend we remove, we remove the same number of zeros in the low-order digits of the divisor.

2. If there are zeros left in the dividend after division, then they should be transferred to the quotient:

So, let’s formulate the sequence of actions when dividing into a column.

1. Place the dividend on the left and the divisor on the right. We remember that we divide the dividend by isolating incomplete dividends bit by bit and dividing them sequentially by the divisor. The digits in the incomplete dividend are allocated from left to right from high to low.
2. If the dividend and divisor have zeros in the lower digits, then they can be reduced before dividing.
3. We determine the first incomplete divisor:

A) allocate the highest digit of the dividend into the incomplete divisor;

b) compare the incomplete dividend with the divisor; if the divisor is larger, then go to point (V), if less, then we have found an incomplete dividend and can move on to point 4 ;

V) add the next digit to the incomplete dividend and go to point (b).

1. We determine how many digits there will be in the quotient, and put as many dots in place of the quotient (under the divisor) as there will be digits in it. One point (one digit) for the entire first incomplete dividend and the remaining points (digits) are the same as the number of digits left in the dividend after selecting the incomplete dividend.
2. We divide the incomplete dividend by the divisor; to do this, we find a number that, when multiplied by the divisor, would result in a number either equal to or less than the incomplete dividend.
3. We write the found number in place of the next quotient digit (dot), and write the result of multiplying it by the divisor under the incomplete dividend and find their difference.
4. If the difference found is less than or equal to the incomplete dividend, then we have correctly divided the incomplete dividend by the divisor.
5. If there are still digits left in the dividend, then we continue division, otherwise we go to point 10 .
6. We lower the next digit of the dividend to the difference and get the next incomplete dividend:

a) compare the incomplete dividend with the divisor, if the divisor is greater, then go to point (b), if less, then we have found the incomplete dividend and can move on to point 4;

b) add the next digit of the dividend to the incomplete dividend, and write 0 in the place of the next digit (dot) in the quotient;

c) go to point (a).

10. If we performed division without a remainder and the last difference found is equal to 0 , then we did the division correctly.

We talked about dividing a multi-digit number by a single-digit number. In the case where the divider is larger, division is performed in the same way: