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Calculator of examples for actions in a column. Division of natural numbers by a column, examples, solutions

One of the important stages in teaching a child mathematical operations is learning the operation of dividing prime numbers. How to explain division to a child, when can you start mastering this topic?

In order to teach a child division, it is necessary that by the time of learning he has already mastered such mathematical operations as addition, subtraction, and also has a clear understanding of the very essence of the operations of multiplication and division. That is, he must understand that division is the division of something into equal parts. It is also necessary to teach multiplication operations and learn the multiplication table.

I already wrote about how this article can be useful for you.

We master the operation of division (division) into parts in a playful way

At this stage, it is necessary to form in the child the understanding that division is the division of something into equal parts. The easiest way to teach a child to do this is to invite him to share a certain number of items among his friends or family members.

For example, take 8 identical cubes and invite the child to divide into two equal parts - for him and another person. Vary and complicate the task, invite the child to divide 8 cubes not into two, but into four people. Analyze the result with him. Change the components, try with a different number of objects and people into which these objects need to be divided.

Important: Make sure that at first the child operates with an even number of objects, so that the result of division is the same number of parts. This will be useful in the next step, when the child needs to understand that division is the inverse of multiplication.

Multiply and divide using the multiplication table

Explain to your child that, in mathematics, the opposite of multiplication is called division. Using the multiplication table, demonstrate to the student, using any example, the relationship between multiplication and division.

Example: 4x2=8. Remind your child that the result of multiplication is the product of two numbers. Then explain that division is the inverse of multiplication and illustrate this clearly.

Divide the resulting product "8" from the example - by any of the factors - "2" or "4", and the result will always be another factor that was not used in the operation.

You also need to teach the young student how the categories that describe the operation of division are called - “divisible”, “divisor” and “quotient”. Use an example to show which numbers are divisible, divisor and quotient. Consolidate this knowledge, they are necessary for further learning!

In fact, you need to teach your child the multiplication table “in reverse”, and you need to memorize it as well as the multiplication table itself, because this will be necessary when you start teaching long division.

Divide by a column - give an example

Before starting the lesson, remember with your child how the numbers are called during the division operation. What is a "divisor", "divisible", "quotient"? Learn to accurately and quickly identify these categories. This will be very useful while teaching the child to divide prime numbers.

We explain clearly

Let's divide 938 by 7. In this example, 938 is the dividend, 7 is the divisor. The result will be a quotient, and then you need to calculate it.

Step 1. We write down the numbers, dividing them with a "corner".

Step 2 Show the student the number of divisible and ask him to choose from them the smallest number that is greater than the divisor. Of the three numbers 9, 3 and 8, this number will be 9. Invite the child to analyze how many times the number 7 can be contained in the number 9? That's right, just once. Therefore, the first result we write down will be 1.

Step 3 Let's move on to the design of the division by a column:

We multiply the divisor 7x1 and get 7. We write the result obtained under the first number of our dividend 938 and subtract, as usual, in a column. That is, we subtract 7 from 9 and get 2.

We write down the result.

Step 4 The number that we see is less than the divisor, so we need to increase it. To do this, we combine it with the next unused number of our dividend - it will be 3. We attribute 3 to the resulting number 2.

Step 5 Next, we act according to the already known algorithm. Let's analyze how many times our divisor 7 is contained in the resulting number 23? That's right, three times. We fix the number 3 in the quotient. And the result of the product - 21 (7 * 3) is written below under the number 23 in a column.

Step.6 Now it remains to find the last number of our quotient. Using the already familiar algorithm, we continue to do calculations in a column. By subtracting in the column (23-21) we get the difference. It equals 2.

Of the dividend, we have one number left unused - 8. We combine it with the number 2 obtained as a result of subtraction, we get - 28.

Step 7 Let's analyze how many times our divisor 7 is contained in the resulting number? That's right, 4 times. We write the resulting figure in the result. So, we have the quotient obtained as a result of division by a column = 134.

How to teach a child to divide - we consolidate the skill

The main reason why many students have a problem with mathematics is the inability to quickly do simple arithmetic calculations. And on this basis, all mathematics in elementary school is built. Especially often the problem is in multiplication and division.
In order for a child to learn how to quickly and efficiently carry out division calculations in the mind, the correct teaching methodology and consolidation of the skill are necessary. To do this, we advise you to use the currently popular aids in mastering the division skill. Some are designed for children to work with their parents, others for independent work.

"Division. Level 3. Workbook "from the largest international center for additional education Kumon
"Division. Level 4 Workbook by Kumon
“Not mental arithmetic. A system for teaching a child rapid multiplication and division. For 21 days. Notepad simulator.» from Sh. Akhmadulin - the author of best-selling educational books

The most important thing when you teach a child to divide in a column is to master the algorithm, which, in general, is quite simple.

If the child operates well with the multiplication table and "reverse" division, he will not have difficulties. Nevertheless, it is very important to constantly train the acquired skill. Don't stop there as soon as you realize that the child has grasped the essence of the method.

In order to easily teach a child the operation of division, you need:

So that at the age of two or three years he mastered the relationship "whole - part". He should develop an understanding of the whole as an inseparable category and the perception of a separate part of the whole as an independent object. For example, a toy truck is a whole, and its body, wheels, doors are parts of this whole.
So that at primary school age the child freely operates with actions for adding and subtracting numbers, understands the essence of the processes of multiplication and division.

In order for the child to enjoy mathematics, it is necessary to arouse his interest in mathematics and mathematical actions, not only during training, but also in everyday situations.

Therefore, encourage and develop observation in the child, draw analogies with mathematical operations (operations on counting and division, analysis of part-whole relationships, etc.) during construction, games and observations of nature.

Lecturer, child development center specialist
Druzhinina Elena
site specially for the project

Video plot for parents, how to correctly explain the division into a column to the child:

A column calculator for Android devices will be a great helper for modern schoolchildren. The program not only gives the correct answer to a mathematical action, but also clearly demonstrates its step-by-step solution. If you need more complex calculators, you can look at the advanced engineering calculator.

Peculiarities

The main feature of the program is the uniqueness of the calculation of mathematical operations. Displaying the calculation process in a column allows students to get to know it in more detail, understand the solution algorithm, and not just get the finished result and rewrite it in a notebook. This feature has a huge advantage over other calculators. quite often at school, teachers require intermediate calculations to be written down to make sure that the student does them in his mind and really understands the algorithm for solving problems. By the way, we have another program of a similar kind - .

To start using the program, you need to download a calculator in a column on Android. You can do this on our website absolutely free of charge without additional registrations and SMS. After installation, the main page will open in the form of a notebook sheet in a cage, on which, in fact, the calculation results and their detailed solution will be displayed. At the bottom there is a panel with buttons:

Numbers.
Signs of arithmetic operations.
Delete previously entered characters.

Input is carried out according to the same principle as on. All the difference is only in the interface of the application - all mathematical calculations and their results are displayed in a virtual student notebook.

The application allows you to quickly and correctly perform standard mathematical calculations for a student in a column:

multiplication;
division;
addition;
subtraction.

A nice addition to the app is the daily math homework reminder feature. If you want, do your homework. To enable it, go to the settings (press the button in the form of a gear) and check the reminder box.

Advantages and disadvantages

It helps the student not only to quickly get the correct result of mathematical calculations, but also to understand the very principle of calculation.
Very simple, intuitive interface for every user.
You can install the application even on the most budgetary Android device with operating system 2.2 and later.
The calculator saves a history of mathematical calculations, which can be cleared at any time.

The calculator is limited in mathematical operations, so it will not work for complex calculations that an engineering calculator could handle. However, given the purpose of the application itself - to clearly demonstrate the principle of calculating in a column to elementary school students, this should not be considered a disadvantage.

The application will also be an excellent assistant not only for schoolchildren, but also for parents who want to get their child interested in mathematics and teach him how to correctly and consistently perform calculations. If you have already used the Stacked Calculator app, leave your impressions below in the comments.

It is convenient to carry out a special method, which is called column subtraction or column subtraction. This method of subtraction justifies its name, since the minuend, the subtrahend and the difference are written in a column. Intermediate calculations are also carried out in columns corresponding to the digits of the numbers.

The convenience of subtracting natural numbers in a column lies in the simplicity of calculations. Calculations come down to using the addition table and applying the subtraction properties.

Let's see how column subtraction is performed. We will consider the subtraction process together with the solution of examples. So it will be clearer.

Page navigation.

What do you need to know to subtract by a column?

To subtract natural numbers in a column, you need to know, firstly, how subtraction is performed using the addition table.

Finally, it does not hurt to repeat the definition of the discharge of natural numbers.

Subtraction by a column on examples.

Let's start with the recording. The minuend is written first. Below the minuend is the subtrahend. Moreover, this is done in such a way that the numbers are one under the other, starting from the right. A minus sign is placed to the left of the recorded numbers, and a horizontal line is drawn below, under which the result will be recorded after the necessary actions have been taken.

Here are some examples of correct entries when subtracting by a column. Write down the difference in a column 56−9 , difference 3 004−1 670 , as well as 203 604 500−56 777 .

So, with the record sorted out.

We turn to the description of the process of subtraction by a column. Its essence lies in the sequential subtraction of the values of the corresponding digits. First, the values of the units digit are subtracted, then the values of the tens digit, then the values of the hundreds digit, and so on. The results are recorded under the horizontal line at the appropriate places. The number that is formed under the line after the completion of the process is the desired result of subtracting the two original natural numbers.

Imagine a diagram illustrating the process of subtraction by a column of natural numbers.

The above scheme gives a general picture of the subtraction of natural numbers by a column, but it does not reflect all the subtleties. We will deal with these subtleties when solving examples. Let's start with the simplest cases, and then we will gradually move towards more complex cases, until we figure out all the nuances that can occur when subtracting by a column.

Example.

First, subtract a column from the number 74 805 number 24 003 .

Decision.

Let's write these numbers as required by the column subtraction method:

We start by subtracting the values of the digits of units, that is, we subtract from the number 5 number 3 . From the addition table we have 5−3=2 . We write the results obtained under the horizontal line in the same column in which the numbers are located 5 and 3 :

Now subtract the values of the tens digit (in our example, they are equal to zero). We have 0−0=0 (we mentioned this property of subtraction in the previous paragraph). We write the resulting zero under the line in the same column:

Go ahead. Subtract the values of the hundreds place: 8−0=8 (according to the property of subtraction, voiced in the previous paragraph). Now our entry will look like this:

Let's move on to subtracting the thousands place values: 4−4=0 (these are properties of subtraction of equal natural numbers). We have:

It remains to subtract the values of the tens of thousands place: 7−2=5 . We write the resulting number under the line in the right place:

This completes the column subtraction. Number 50 802 , which turned out below, is the result of subtracting the original natural numbers 74 805 and 24 003 .

Consider the following example.

Example.

Subtract a column from the number 5 777 number 5 751 .

Decision.

We do everything in the same way as in the previous example - we subtract the values of the corresponding digits. After completing all the steps, the entry will look like this:

Under the line we got a number in the record of which there are numbers on the left 0 . If these numbers 0 discard, then we get the result of subtracting the original natural numbers. In our case, we discard two digits 0 obtained on the left. We have: difference 5 777−5 751 is equal to 26 .

Up to this point, we have subtracted natural numbers whose records consist of the same number of characters. Now, using an example, let's figure out how natural numbers are subtracted in a column when there are more signs in the record of the reduced than in the record of the subtrahend.

Example.

Subtract from the number 502 864 number 2 330 .

Decision.

We write the minuend and the subtrahend in a column:

Subtract the values of the unit digit one by one: 4−0=4 ; followed by tens: 6−3=3 ; further - hundreds: 8−3=5 ; further - thousand: 2−2=0 . We get:

Now, to complete the column subtraction, we still need to subtract the values of the tens of thousands place, and then the values of the hundreds of thousands place. But from the values of these digits (in our example, from the numbers 0 and 5 ) we have nothing to subtract (since the subtracted number 2 330 does not have digits in these digits). How to be? Very simple - the values of these bits are simply rewritten under the horizontal line:

On this subtraction by a column of natural numbers 502 864 and 2 330 completed. The difference is 500 534 .

It remains to consider the cases when, at some step of column subtraction, the value of the digit of the reduced number is less than the value of the corresponding digit of the subtrahend one. In these cases, you have to "borrow" from the senior ranks. Let's understand this with examples.

Example.

Subtract a column from the number 534 number 71 .

Decision.

At the first step, subtract from 4 number 1 , we get 3 . We have:

In the next step, we need to subtract the values of the tens digit, that is, from the number 3 subtract the number 7 . As 3<7 , then we cannot subtract these natural numbers (the subtraction of natural numbers is defined only when the subtrahend is not greater than the minuend). What to do? In this case, we take 1 unit from the highest order and "exchange" it. In our example, "exchange" 1 a hundred per 10 tens. To visually reflect our actions, we put a thick dot over the number in the hundreds place, and over the number in the tens place we write the number 10 using a different color. The entry will look like this:

We add received after the "exchange" 10 tens to 3 available tens: 3+10=13 , and subtract from this number 7 . We have 13−7=6 . This number 6 write under the horizontal line in its place:

Let's move on to subtracting the values of the hundreds place. Here we see a dot above the number 5, which means that from this number we took one “for exchange”. That is, now we have 5 , a 5−1=4 . From number 4 nothing else needs to be subtracted (since the original subtracted number 71 does not contain digits in the hundreds place). Thus, under the horizontal line we write the number 4 :

So the difference 534−71 is equal to 463 .

Sometimes, when subtracting by a column, you have to “exchange” units from the highest digits several times. In support of these words, we analyze the solution of the following example.

Example.

Subtract from natural number 1 632 number 947 column.

Decision.

In the first step, we need to subtract from the number 2 number 7 . As 2<7 , then you immediately have to "exchange" 1 dozen on 10 units. After that, from the sum 10+2 subtract the number 7 , we get (10+2)−7=12−7=5 :

In the next step, we need to subtract the tens digit values. We see that over the number 3 worth a point, that is, we have not 3 , a 3−1=2 . And from this number 2 we need to subtract the number 4 . As 2<4 , then again you have to resort to "exchange". But now we are exchanging 1 a hundred per 10 tens. In this case, we have (10+2)−4=12−4=8 :

Now we subtract the values of the hundreds place. From the number 6 unit was occupied in the previous step, so we have 6−1=5 . From this number we need to subtract the number 9 . As 5<9 , then we need to "exchange" 1 a thousand per 10 hundreds. We get (10+5)−9=15−9=6 :

The last step remains. From the one in the thousands place we borrowed in the previous step, so we have 1−1=0 . We do not need to subtract anything else from the resulting number. This number is written under the horizontal line:

With this mathematical program, you can divide polynomials by a column.
The program for dividing a polynomial by a polynomial does not just give the answer to the problem, it gives a detailed solution with explanations, i.e. displays the process of solving in order to check the knowledge of mathematics and / or algebra.

This program can be useful for high school students in preparation for tests and exams, when testing knowledge before the Unified State Examination, for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.

In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.

If you need or simplify the polynomial or multiply polynomials, then for this we have a separate program Simplification (multiplication) of a polynomial

Divide polynomials

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A bit of theory.

Division of a polynomial by a polynomial (binomial) with a column (corner)

In algebra division of polynomials by a column (corner)- an algorithm for dividing a polynomial f(x) by a polynomial (binomial) g(x), the degree of which is less than or equal to the degree of the polynomial f(x).

The algorithm for dividing a polynomial by a polynomial is a generalized form of dividing numbers by a column, easily implemented manually.

For any polynomials \(f(x) \) and \(g(x) \), \(g(x) \neq 0 \), there are unique polynomials \(q(x) \) and \(r(x ) \), such that
\(\frac(f(x))(g(x)) = q(x)+\frac(r(x))(g(x)) \)
where \(r(x) \) has a lower degree than \(g(x) \).

The purpose of the algorithm for dividing polynomials into a column (corner) is to find the quotient \(q(x) \) and the remainder \(r(x) \) for given dividend \(f(x) \) and nonzero divisor \(g(x) \)

Example

We divide one polynomial by another polynomial (binomial) with a column (corner):
\(\large \frac(x^3-12x^2-42)(x-3) \)

The quotient and remainder of the division of these polynomials can be found in the course of the following steps:
1. Divide the first element of the dividend by the highest element of the divisor, put the result under the line \((x^3/x = x^2) \)

\(x\)	\(-3 \)
\(x^2 \)

3. Subtract the polynomial obtained after multiplication from the dividend, write the result under the line \((x^3-12x^2+0x-42-(x^3-3x^2)=-9x^2+0x-42) \)

\(x^3 \)	\(-12x^2 \)	\(+0x\)	\(-42 \)
\(x^3 \)	\(-3x^2 \)
	\(-9x^2 \)	\(+0x\)	\(-42 \)

\(x\)	\(-3 \)
\(x^2 \)

4. We repeat the previous 3 steps, using the polynomial written under the line as a dividend.

\(x^3 \)	\(-12x^2 \)	\(+0x\)	\(-42 \)
\(x^3 \)	\(-3x^2 \)
	\(-9x^2 \)	\(+0x\)	\(-42 \)
	\(-9x^2 \)	\(+27x\)
		\(-27x\)	\(-42 \)

\(x\)	\(-3 \)
\(x^2 \)	\(-9x\)

5. Repeat step 4.

\(x^3 \)	\(-12x^2 \)	\(+0x\)	\(-42 \)
\(x^3 \)	\(-3x^2 \)
	\(-9x^2 \)	\(+0x\)	\(-42 \)
	\(-9x^2 \)	\(+27x\)
		\(-27x\)	\(-42 \)
		\(-27x\)	\(+81 \)
			\(-123 \)

\(x\)	\(-3 \)
\(x^2 \)	\(-9x\)	\(-27 \)

6. End of the algorithm.
Thus, the polynomial \(q(x)=x^2-9x-27 \) is a partial division of polynomials, and \(r(x)=-123 \) is the remainder of the division of polynomials.

The result of dividing polynomials can be written as two equalities:
\(x^3-12x^2-42 = (x-3)(x^2-9x-27)-123 \)
or
\(\large(\frac(x^3-12x^2-42)(x-3)) = x^2-9x-27 + \large(\frac(-123)(x-3)) \)

The division of natural numbers, especially multi-valued ones, is conveniently carried out by a special method, which is called division by a column (in a column). You can also see the name corner division. Immediately, we note that the column can be carried out both division of natural numbers without a remainder, and division of natural numbers with a remainder.

In this article, we will understand how division by a column is performed. Here we will talk about the writing rules, and about all intermediate calculations. First, let us dwell on the division of a multi-valued natural number by a single-digit number by a column. After that, we will focus on cases where both the dividend and the divisor are multi-valued natural numbers. The whole theory of this article is provided with characteristic examples of division by a column of natural numbers with detailed explanations of the solution 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 divide in a column in writing on paper with a checkered line - so there is less chance of going astray from the desired row and column.

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

Look at the following diagram, which illustrates the places for writing the dividend, divisor, quotient, remainder, and intermediate calculations when dividing by a column.

It can be seen from the above diagram that the desired 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 of the availability of space on the page in advance. In this case, one 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 is required. For example, when dividing a natural number 614,808 by 51,234 by a column (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 numbers 8 058 and 4 (here the difference in the number of characters is 4−1=3 ). To confirm our words, we present the completed records of division by a column of these natural numbers:

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

Division by a column of a natural number by a single-digit natural number, algorithm for dividing by a column

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 useful to practice the initial skills of division by a column on these simple examples.

Example.

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

Decision.

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 by a column.

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

Now we start to figure out how many times the divisor is in the dividend. To do this, we successively 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 place of the private we write the number by which we multiplied the divisor. If we get a number greater than the divisible, 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 got a number equal to the dividend, so we write it under the dividend, and in place of the private we write the number 4. The record will then look like this:

The final stage of dividing single-digit natural numbers by a column remains. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line in the same way as it is done when subtracting natural numbers with a column. The number obtained after 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 a finished record of division by a column of the number 8 by 2. We see that the quotient 8:2 is 4 (and the remainder is 0 ).

Answer:

8:2=4 .

Now consider how the division by a column of single-digit natural numbers with a remainder is carried out.

Example.

Divide by a column 7 by 3.

Decision.

At the initial stage, the entry looks like this:

We begin to find out how many times the dividend contains a 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 comparison of 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 (it was multiplied 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.

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

Answer:

7:3=2 (rest. 1) .

Now we can move on to dividing multi-valued natural numbers by single-digit natural numbers by a column.

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

First, we look at the first digit from the left in the dividend entry. 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 the next digit to the left in the dividend record, and work further with the number determined by the two digits in question. For convenience, we select in our record the number with which we will work.

The first digit from the left in the dividend 140,288 is the number 1. The number 1 is less than the divisor 4, so we also look at the next digit on the left in the dividend record. At the same time, we see the number 14, with which we have to work further. We select this number in the notation of the dividend.

The following points 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 successively multiply the divisor by 0, 1, 2, 3, ... until we get the number x or a number greater than x. When a number x is obtained, then we write it under the selected number according to the notation rules used when subtracting by a column of natural numbers. The number by which the multiplication was carried out is written in place of the quotient during the first pass of the algorithm (during 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 selected 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).

We multiply the divisor of 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>fourteen . Since at the last step we got the number 16, which is greater than 14, then under the selected number we write the number 12, which turned out at the penultimate step, and in place of the quotient we write the number 3, since in the penultimate paragraph the multiplication was carried out precisely on it.

At this stage, from the selected number, subtract the number below it in a column. Below the horizontal line is the result of the subtraction. However, if the result of the subtraction is zero, then it does not need to be written down (unless the subtraction at this point is the very last action that completely completes the division by a column). Here, for your control, it will not be superfluous to compare the result of subtraction with the divisor and make sure that it is less than the divisor. Otherwise, a mistake has been made somewhere.

We need to subtract the number 12 from the number 14 in a column (for the correct notation, you must not forget to put a minus sign to the left of the subtracted numbers). After the completion of this action, the number 2 appeared under the horizontal line. Now we check our calculations by comparing the resulting number with a divisor. Since the number 2 is less than the divisor 4, you can safely move on to the next item.

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 zero), we write down the number located in the same column in the record of the dividend. If there are no numbers in the record of the dividend in this column, then the division by a column ends here. After that, we select the number formed under the horizontal line, take it as a working number, and repeat with it from 2 to 4 points of the algorithm.

Under the horizontal line to the right of the number 2 already there, we write 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 the actions of the second, third and fourth points of the algorithm with it.

We multiply the divisor of 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 subtraction by a column. Since we subtract equal natural numbers, then, due to the property of subtracting equal natural numbers, we get zero as a result. We do not write down zero (since this is not yet the final stage of dividing by a column), but we remember the place where we could write it down (for convenience, we will mark this place with a black rectangle).

Under the horizontal line to the right of the memorized place, we write down the number 2, since it is she who 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 a working number, mark it, and once again we will have to perform the steps from 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 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 subtraction by 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, we add the number 8 (since it is in this column in the record of the dividend 140 288). Thus, under the horizontal line is the number 28.

We accept this number as a worker, mark it, and repeat steps 2-4 of paragraphs.

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

It remains for the last time to carry out the actions from points 2, 3, 4 (we provide it to you), after which you will get a complete picture of dividing natural numbers 140 288 and 4 in a column:

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

Thus, looking at the completed record of dividing the multi-valued natural number 140 288 by the single-valued natural number 4, we see that the number 35 072 is private (and the remainder of the division is zero, it is on 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 7136 and the divisor is a single natural number 9.

Decision.

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

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

Repeating the cycle, we will have

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

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

Answer:

7 136:9=792 (rest 8) .

And this example demonstrates how long division should look like.

Example.

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

Decision.

It is most convenient to perform division by a column.

Answer:

7 042 035:7=1 006 005 .

Division by a column of multivalued natural numbers

We hasten to please you: if you have well mastered the algorithm for dividing by a column from the previous paragraph of this article, then you already almost know how to perform division by a column of multivalued natural numbers. This is true, since steps 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first step.

At the first stage of dividing into a column of multi-valued natural numbers, you need to look not at the first digit on the left in the dividend entry, but at as many of them as there are digits in the divisor entry. 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 record of the dividend. After that, the actions indicated in paragraphs 2, 3 and 4 of the algorithm are performed until the final result is obtained.

It remains only to see the application of the algorithm for dividing by a column of multi-valued natural numbers in practice when solving examples.

Example.

Let's perform division by a column of multivalued natural numbers 5562 and 206.

Decision.

Since 3 characters are involved in the record of the divisor 206, we look at the first 3 digits on the left in the record of 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 one, select it, and proceed 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 the multiplication is difficult, then it is better to perform the multiplication of natural numbers in a column): 206 0=0<556 , 206·1=206<556 , 206·2=412<556 , 206·3=618>556 . Since we got a number that is greater than the number 556, then under the selected 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 it was multiplied at the penultimate step). The column division entry takes the following form:

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

Under the horizontal line to the right of the number available there, we write the number 2, since it is in the record of the dividend 5 562 in this column:

Now we work with the number 1442, select it, and go through steps two through four again.

We multiply the divisor 206 by 0 , 1 , 2 , 3 , ... until we 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 subtract by a column, we get zero, but we don’t write it down right away, but only remember its position, because we don’t know if the division ends here, or we will have to repeat the steps of the algorithm again:

Now we see that under the horizontal line to the right of the memorized position, we cannot write down any number, since there are no numbers in the record of the dividend in this column. Therefore, this division by a column is over, and we complete the entry: