Column solution of examples. How to Long Divide? How to explain long division to a child? Division by 1-digit, 2-digit, 3-digit, division with remainder
Division is one of the four basic mathematical operations (addition, subtraction, multiplication). Division, like other operations, is important not only in mathematics, but also in everyday life. For example, you will hand over money to the whole class (25 people) and buy a gift for the teacher, but you will not spend everything, there will be change. So you will need to divide the change among all. The division operation comes in to help you solve this problem.
Division is an interesting operation, as we will see with you in this article!
Division of numbers
So a little theory and then practice! What is division? Division is splitting something into equal parts. That is, it can be a bag of chocolates that needs to be split into equal parts. For example, there are 9 sweets in a bag, and the person who wants to get them - three. Then you need to divide these 9 chocolates among three people.
It is written like this: 9: 3, the answer will be the number 3. That is, dividing the number 9 by the number 3 shows the number of three numbers contained in the number 9. The opposite action, a test, will be multiplication. 3 * 3 = 9. Right? Absolutely.
So consider example 12: 6. First, let's name each component in the example. 12 - dividend, that is. a number that can be divided into parts. 6 is the divisor, this is the number of parts by which the dividend is divided. And the result will be a number called "quotient".
Divide 12 by 6, the answer will be the number 2. You can check the solution by multiplying: 2 * 6 = 12. It turns out that the number 6 is contained 2 times in the number 12.
Division with remainder
What is division with remainder? This is the same division, only the result is not an even number, as shown above.
For example, divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, the answer is 3 and the remainder is 2, and it is written like this: 17: 5 = 3 (2).
For example, 22: 7. In the same way, we determine the maximum number divisible by 7 to 22. This number is 21. The answer then will be: 3 and remainder 1. And it is written: 22: 7 = 3 (1).
Division by 3 and 9
A special case of division will be the division by the number 3 and the number 9. If you want to know whether a number can be divided by 3 or 9 without a remainder, then you need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (whichever you want).
If the answer is obtained without a remainder, then the number will be divided without a remainder.
For example, the number 18. The sum of the digits is 1 + 8 = 9. The sum of the digits is divisible by both 3 and 9. The number 18: 9 = 2, 18: 3 = 6. Divided without remainder.
For example, the number 63. The sum of the digits 6 + 3 = 9. Divisible by both 9 and 3. 63: 9 = 7, and 63: 3 = 21. Such operations are performed with any number to find out whether it is divisible with the remainder 3 or 9 or not.
Multiplication and division
Multiplication and division are opposite operations. Multiplication can be used as a test for division, and division as a test for multiplication. You can learn more about multiplication and master the operation in our article on multiplication. Which describes in detail the multiplication and how to do it correctly. There you will also find the multiplication table and examples for training.
Let's give an example of checking division and multiplication. Let's say the example is 6 * 4. Answer: 24. Then check the answer by division: 24: 4 = 6, 24: 6 = 4. Resolved correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for division 56: 8. Answer: 7. Then the check will be 8 * 7 = 56. Right? Yes. In this case, the check is performed by multiplying the answer by the divisor.
Division 3 class
In the third grade, division is just beginning. Therefore, third-graders solve the simplest problems:
Problem 1... A factory worker was given the task of arranging 56 cakes in 8 packs. How many cakes do you need to put in each package to get the same quantity in each?
Task 2... On New Year's Eve at school, children were given 75 sweets for a class of 15 students. How many sweets should each child get?
Problem 3... Roma, Sasha and Misha collected 27 apples from the apple tree. How many apples will each get if they are to be divided equally?
Problem 4... Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many guys need to buy cookies so that everyone gets 15 pieces?
Division 4 class
The division in the fourth grade is more serious than in the third. All calculations are carried out by the method of division into a column, and the numbers that participate in the division are not small. What is long division? You can find the answer below:
Long division
What is long division? This is a method that allows you to find the answer to the division of large numbers. If prime numbers like 16 and 4 can be divided, and the answer is clear - 4. Then 512: 8 in the mind is not easy for a child. And to tell about the technique for solving such examples is our task.
Consider an example, 512: 8.
Step 1... Let's write the dividend and divisor as follows:
The quotient will be written as a result under the divisor, and the calculations under the dividend.
Step 2... We start division from left to right. First, we take the number 5: 
Step 3... The number 5 is less than the number 8, which means that it cannot be divided. Therefore, we take one more digit of the dividend:
Now 51 is more than 8. This is an incomplete quotient.
Step 4... We put a dot under the divider.
Step 5... After 51 there is another number 2, which means there will be one more number in the answer, that is. the quotient is a two-digit number. We put the second point:
Step 6... We start the division operation. The largest number that can be divided without a remainder by 8 to 51 is 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
7 step... Then we write down the number exactly under the number 51 and put the “-” sign:
Step 8... Then subtract 48 from 51 and get the answer 3.
* 9 step*. We demolish the number 2 and write next to the number 3:
Step 10 Divide the resulting number 32 by 8 and get the second digit of the answer - 4.
So the answer is 64, no remainder. If we were dividing the number 513, then the remainder would be one.
Division of three-digit
Division of three-digit numbers is performed by long division, which was explained in the example above. An example of just the same three-digit number.
Division of fractions
Division of fractions is not as difficult as it seems at first glance. For example, (2/3) :( 1/4). The method for this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but for this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3) * 4, this equals - 8/3 or 2 integers and 2/3 Let's give another example, with an illustration for better understanding. Consider fractions (4/7) :( 2/5):
As in the previous example, flip the divisor 2/5 and get 5/2, replacing division with multiplication. We get then (4/7) * (5/2). We make the reduction and the answer: 10/7, then we take out the whole part: 1 whole and 3/7.
Dividing a number into classes
Let's imagine the number 148951784296 and divide it by three digits: 148 951 784 296. So, from right to left: 296 - class of units, 784 - class of thousands, 951 - class of millions, 148 - class of billions. In turn, in each class, 3 digits have their own category. From right to left: the first digit is ones, the second digit is tens, the third is hundreds. For example, class of units - 296, 6 - units, 9 - tens, 2 - hundreds.
Division of natural numbers
Division of natural numbers is the simplest division described in this article. It can be with or without a remainder. The divisor and divisible can be any non-fractional, whole numbers. 
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Division presentation
Presentation is another way to visually show the topic of division. Below we will find a link to a great presentation that explains well how to divide, what division is, what is the dividend, divisor and quotient. Don't waste your time, but consolidate your knowledge!
Division examples
Easy level
Average level
Difficult level
Games for the development of oral counting
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve the skills of oral counting in an interesting way.
Guess the operation game
The game "Guess the operation" develops thinking and memory. The main point of the game is to choose a mathematical sign for the equality to be true. There are examples on the screen, look carefully and put the desired "+" or "-" sign, so that the equality is correct. The sign "+" and "-" are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you collect points and keep playing.
Simplification game
The Simplification game develops thinking and memory. The main point of the game is to quickly perform a mathematical operation. On the screen, a student is drawn at the blackboard, and a mathematical action is given, the student needs to calculate this example and write an answer. Below there are three answers, count and click the number you need with the mouse. If you answered correctly, you collect points and keep playing.
Fast Add Game
The Fast Addition game develops thinking and memory. The main point of the game is to choose numbers, the sum of which is equal to a given number. This game is given a matrix from one to sixteen. A given number is written above the matrix, you need to select the numbers in the matrix so that the sum of these numbers is equal to the specified number. If you answered correctly, you collect points and keep playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main point of the game is to quickly count the number of painted objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, they must be quickly counted, then they are closed. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you collect points and keep playing.
Piggy bank game
The game "Piggy bank" develops thinking and memory. The main point of the game is to choose which piggy bank has more money. In this game you are given four piggy banks, you need to count which piggy bank has more money and show this piggy bank with the mouse. If you answered correctly, then you collect points and continue to play further.
Fast Add Reload Game
The Fast Addition Reloading game develops thinking, memory and attention. The main point of the game is to choose the correct terms, the sum of which will be equal to a given number. In this game, three numbers are given on the screen and a task is given, add the number, the screen indicates which number needs to be added. You select the desired numbers from three digits and press them. If you answered correctly, then you collect points and continue to play further.
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With this math program, you can divide polynomials with 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 solution process in order to check the knowledge of mathematics and / or algebra.
This program can be useful for senior students of secondary schools in preparation for tests and exams, when checking knowledge before the exam, 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 teaching and / or the teaching of your younger siblings, while the level of education in the field of the problems being solved increases.
If you need or simplify polynomial or multiply polynomials, then for this we have a separate program Simplification (multiplication) of the polynomial
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A bit of theory.
Division of a polynomial by a polynomial (binomial) by a column (angle)
In algebra division of polynomials by a column (corner)- an algorithm for dividing the 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 by hand.
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)) \)
moreover, \ (r (x) \) has a lower degree than \ (g (x) \).
The goal of the algorithm for dividing polynomials into a column (angle) is to find the quotient \ (q (x) \) and remainder \ (r (x) \) for a given dividend \ (f (x) \) and nonzero divisor \ (g (x) \)
Example
We divide one polynomial by another polynomial (binomial) by a column (corner):
\ (\ large \ frac (x ^ 3-12x ^ 2-42) (x-3) \)
The quotient and remainder of the given polynomials can be found by performing the following steps:
1. Divide the first element of the dividend by the leading element of the divisor, place the result under the line \ ((x ^ 3 / x = x ^ 2) \)
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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) \)
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4. We repeat the previous 3 steps, using the polynomial written under the line as the dividend.
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5. Repeat step 4.
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6. End of the algorithm.
Thus, the polynomial \ (q (x) = x ^ 2-9x-27 \) is the quotient of the 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)) \)
Columnar divisions are an integral part of the school curriculum and necessary knowledge for a child. To avoid problems in the classroom and with their implementation, you should give your child basic knowledge from a young age.
It is much easier to explain to a child certain things and processes in a playful way, and not in the format of a standard lesson (although today there are quite a variety of teaching methods in different forms).
From this article you will learn
Division principle for toddlers
Children are constantly faced with different mathematical terms, without even knowing where they are from. After all, many mummies, in the form of a game, explain to the child that dad is more of a plate, to go further to the kindergarten than to the store and other simple examples. All this presents the child with an initial impression of mathematics, even before the child goes to first grade.
To teach a child to divide without a remainder, and later with a remainder, it is necessary to directly invite the child to play games with division. Divide, for example, candy among yourself, and then add the following participants in turn.
First, the child will divide the candies, giving each participant one at a time. And at the end, together you will draw a conclusion. It should be clarified that "to divide" means everyone has the same number of sweets.
If you need to explain this process using numbers, then you can give an example in the form of a game. We can say that the number is candy. It should be explained that the number of chocolates to be divided between the participants is a dividend. And the number of people who share these sweets is the divisor.
Then you should show it all clearly, give "live" examples in order to quickly teach the baby to divide. While playing, he will understand and master everything much faster. It will be difficult to explain the algorithm for now, and now it is not necessary.
How to teach a kid long division
Explaining a tiny bit of math is good preparation for going to class, especially math class. If you decide to move on to teaching your child long division, then such actions as addition, subtraction, and what the multiplication table is, he has already learned.
If this still causes some difficulties for him, then all this knowledge needs to be tightened up. It is worth recalling the algorithm of actions of the previous processes, teach them to freely use their knowledge. Otherwise, the baby will simply get confused in all the processes and stop understanding anything.
To make this easier to understand, there is now a division table for toddlers. Its principle is the same as that of multiplication tables. But is such a table already needed if the kid knows the multiplication table? It depends on the school and the teacher.
When forming the concept of "division", it is imperative to do everything in a playful way, to give all examples on things and objects familiar to the child.
It is very important that all objects are of an even number, so that it is clear to the baby that the result is equal parts. This will be correct, since it will allow the baby to realize that division is the reverse process of multiplication. If the items are of an odd number, then the total will come out with the remainder and the baby will get confused.
Multiply and Divide Using a Table
When explaining to the kid the relationship between multiplication and division, it is necessary to clearly show this all with an example. For example: 5 x 3 = 15. Remember that the result of the multiplication is the product of two numbers.
And only after that, explain that this is the reverse process to multiplication and demonstrate this visually using a table.
Say that you need to divide the result "15" - by some of the factors ("5" / "3"), and the result will be a constantly different factor that did not take part in the division.
It is also necessary to explain to the baby how the categories that perform the division are correctly called: dividend, divisor, quotient. Again, use an example to show which is a specific category.
Long division is not a very difficult thing, it has its own easy algorithm that the baby needs to be taught. After consolidating all these concepts and knowledge, you can proceed to further training.
In principle, parents should learn the multiplication table with their beloved child in reverse order, and memorize it by heart, as this will be necessary when learning long division.
This must be done before going to the first grade, so that the child at school is much easier to get used to and keep up with the school curriculum, and so that the class does not start to tease the child due to minor failures. There is a multiplication table both in school and in notebooks, so there is no need to carry a separate table to school.
Divide with a column
Before starting the lesson, you need to remember the names of the numbers when dividing. What is a divisor, dividend and quotient. The child should divide these numbers into the correct categories without mistakes.
The most important thing when teaching long division is to learn the algorithm, which is, in general, quite simple. But first, explain to your child the meaning of the word "algorithm" if he has forgotten it or has not studied it before.
In the event that the baby is well versed in the multiplication and inverse division tables, he will not have any difficulties.
However, it is impossible to linger on the result obtained for a long time; it is necessary to regularly train the acquired skills and abilities. Move on as soon as it becomes clear that the baby has understood the principle of the method.
It is necessary to teach the baby to divide with a column without a remainder and with a remainder, so that the child is not afraid that he has not succeeded in dividing something correctly.
To make it easier to teach the baby the division process, it is necessary:
- in 2-3 years understanding of the whole-part relationship.
- at 6-7 years old, the baby should be able to freely perform addition, subtraction and be aware of the essence of multiplication and division.
It is necessary to stimulate the child's interest in mathematical processes so that this lesson at school brings him pleasure and a desire to learn, and not to motivate him in some lessons, but in life.
The child should carry different tools for math lessons, learn to use them. However, if it is difficult for a child to carry everything, then you should not overload him.
Math-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimal, root extraction, exponentiation, percent calculation, and other operations.
Solution:
How to work with a math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Input of natural integers, zero. To get a negative integer, press the +/- key |
| . | semicolon) | Separator for decimal fraction. If there is no digit in front of the point (comma), the calculator will automatically substitute zero in front of the point. For example: .5 - 0.5 will be written |
| + | plus sign | Addition of numbers (whole, decimal fractions) |
| - | minus sign | Subtraction of numbers (whole, decimal fractions) |
| ÷ | division sign | Division of numbers (whole, decimal fractions) |
| NS | multiplication sign | Multiplication of numbers (whole, decimal fractions) |
| √ | root | Extracting the root of a number. When you press the "root" button again, the root is calculated from the result. For example: root of 16 = 4; root of 4 = 2 |
| x 2 | squaring | Squaring a number. When you press the "square" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1 / x | fraction | Output in decimal fractions. In the numerator 1, in the denominator the entered number |
| % | percent | Getting a percentage of a number. To work, you must enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
| ( | open parenthesis | An open parenthesis to set the priority of the calculation. A closed parenthesis is required. Example: (2 + 3) * 2 = 10 |
| ) | closed parenthesis | A closed parenthesis to set the priority of the calculation. An open parenthesis is required |
| ± | plus minus | Reverse sign |
| = | equals | Displays the result of the solution. Also, above the calculator, in the "Solution" field, intermediate calculations and the result are displayed. |
| ← | delete character | Removes the last character |
| WITH | discharge | Reset button. Resets the calculator completely to the "0" position |
Algorithm of the online calculator by examples
Addition.
Adding integer natural numbers (5 + 7 = 12)
Adding positive integers and negative integers (5 + (-2) = 3)
Adding decimal fractional numbers (0.3 + 5.2 = 5.5)
Subtraction.
Subtraction of integer natural numbers (7 - 5 = 2)
Subtraction of positive integers and negative integers (5 - (-2) = 7)
Subtraction of decimal fractions (6.5 - 1.2 = 4.3)
Multiplication.
Product of integer natural numbers (3 * 7 = 21)
Product of positive integers and negative integers (5 * (-3) = -15)
Product of decimal fractional numbers (0.5 * 0.6 = 0.3)
Division.
Division of integer natural numbers (27/3 = 9)
Division of integers and negative numbers (15 / (-3) = -5)
Division of decimal fractional numbers (6.2 / 2 = 3.1)
Extracting the root of a number.
Extracting the root of an integer (root (9) = 3)
Extracting the root of decimal fractions (root (2.5) = 1.58)
Extracting the root from the sum of numbers (root (56 + 25) = 9)
Extracting the root from the difference of numbers (root (32 - 7) = 5)
Squaring a number.
Square an integer ((3) 2 = 9)
Squaring decimals ((2.2) 2 = 4.84)
Conversion to decimal fractions.
Calculating percent of a number
Increase the number 230 by 15% (230 + 230 * 0.15 = 264.5)
Decrease the number 510 by 35% (510 - 510 * 0.35 = 331.5)
18% of 140 is (140 * 0.18 = 25.2)
The division of natural numbers, especially multi-valued ones, is conveniently carried out using a special method, which is called division by a column (in a column)... You can also find the name division by corner... Immediately, we note that a column can be used for dividing natural numbers without a remainder, or dividing natural numbers with a remainder.
In this article, we will understand how long division is performed. Here we will talk about both the recording rules and all the intermediate calculations. First, let us focus on dividing a multi-digit natural number by a single-digit number by a column. After that, we will dwell on the cases when both the dividend and the divisor are multivalued natural numbers. The whole theory of this article is supplied with characteristic examples of division by a column of natural numbers with detailed explanations of the solution course and illustrations.
Page navigation.
Long division notation rules
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 perform column division in writing on paper with a checkered lining - this way there is less chance of straying off 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 divisible number is 6 105, and the divisor is 5 5, then their correct record when dividing in a column will be as follows:
Take a look at the following diagram, illustrating the places to write the dividend, divisor, quotient, remainder, and intermediate calculations for long division.
From the above diagram, it can be seen that the desired quotient (or incomplete quotient when dividing with a remainder) will be written below the divisor under the horizontal bar. 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 records of the dividend and the divisor, the more space will be required. For example, when dividing a natural number 614 808 by a column 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 entries 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 the completed records of the division by a column of these natural numbers:
Now you can go 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 in a column. However, it will be helpful to practice your basic long division skills with these simple examples.
Example.
Let's say we need to divide by 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 perform division of these numbers with a column.
First, we write the dividend 8 and the divisor 2 as the method requires:
Now we start to figure 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 division with remainder takes place). If we get a number equal to the dividend, then we immediately write it down under the dividend, and in place of the quotient we write down 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 got a number equal to the dividend, so we write it under the dividend, and instead of the quotient, write the number 4. In this case, the record will take the following form:
There remains the final stage of dividing single-digit natural numbers by a column. Under the number written under the dividend, you need to draw a horizontal line, and subtract numbers above this line, 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 were divided without a remainder.
In our example, we get
Now we have a completed record of dividing the number 8 by 2 with a column. We see that the quotient 8: 2 is 4 (and the remainder is 0).
Answer:
8:2=4 .
Now let's look at how division by a column of single-digit natural numbers with a remainder is carried out.
Example.
Divide by a column 7 by 3.
Solution.
At the initial stage, the record looks like this:
We begin to figure out how many times the divisor 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 of 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 (multiplication was carried out by it at the penultimate step).
It remains to carry out the subtraction, and the column division 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 you can proceed to the division by a column of multi-digit natural numbers by single-digit natural numbers.
Now we will analyze long division algorithm... At each of its stages, we will present the results obtained by dividing the multivalued 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, we will be able to disassemble them in detail.
First, we look at the first digit on the left in the dividend record. If the number determined 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 record of the dividend, and work further with the number determined by the two digits in question. For convenience, let's select in our record the number with which we will work.
The first digit on the left in the record of 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 highlight this number in the dividend record.
The next paragraphs 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 with which we are working (for convenience, we will denote this number as x). To do this, we sequentially multiply the divisor by 0, 1, 2, 3, ... until we get a number x or a number greater than x. When the number x is obtained, then we write it under the selected number according to the notation rules used when subtracting natural numbers with 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 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 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 more 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 paragraph the multiplication was carried out by it.
At this stage, from the selected number, subtract the number below it in 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 (unless the subtraction in this paragraph is the very last action that completely completes the long division process). 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, there was a mistake somewhere.
We need to subtract the number 12 in a column from the number 14 (for correct writing, we must remember to put the minus sign to the left of the numbers to be 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 of 4, you can safely proceed to the next item.
Now, under the horizontal bar to the right of the numbers located there (or to the right of the place where we did not write zero), write the number located in the same column in the dividend record. If there are no numbers in the record of the dividend in this column, then the division by the column ends there. 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, accept 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 subtraction in a column. Since we subtract equal natural numbers, due to the property of subtraction of equal natural numbers, the result is zero. We do not write down the zero (since this is not the final stage of long division), 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, 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 actions 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 down 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 down the number 0 (by 0 we carried out multiplication at the penultimate step).
We perform subtraction in a column, we get the number 2 under the horizontal line. We check ourselves by comparing the resulting number with a divisor of 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 record of 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 attentive until now. Having done all the necessary steps, the following result is obtained.
It remains for the last time to carry out the actions from points 2, 3, 4 (we leave it to you), after which you get a complete picture of dividing the natural numbers 140 288 and 4 into a column:
Please note that the bottom line contains the number 0. If this were not the last step of long division (that is, if there were numbers in the dividend in the columns on the right), then we would not write this zero.
Thus, looking at the complete 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 bottom line).
Of course, when dividing natural numbers with 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 natural number 9.
Solution.
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 column division record will take the form
Repeating the cycle, we will have
Another passage will give us a complete picture of dividing by a column of natural numbers 7 136 and 9
Thus, the incomplete quotient is 792, and the remainder of the division is 8.
Answer:
7 136: 9 = 792 (rest 8).
This example demonstrates how long division should look like.
Example.
Divide the natural number 7,042,035 by the single-digit natural number 7.
Solution.
It is most convenient to perform division by a column. 
Answer:
7 042 035:7=1 006 005 .
Column division of multi-digit natural numbers
We hasten to please you: if you have mastered the column division algorithm well from the previous paragraph of this article, then you almost know how to perform column division of multiple-digit natural numbers... This is indeed the case, since stages 2 to 4 of the algorithm remain unchanged, and only minor changes appear in the first paragraph.
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 record of the dividend, but at as many of them as there are signs in the record of the divisor. If the number determined 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 dividend record. After that, the actions specified 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 column division algorithm for multivalued natural numbers in practice when solving examples.
Example.
Let's carry out division by a column of multivalued natural numbers 5 562 and 206.
Solution.
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 556. Since 556 is greater than the divisor 206, we accept the number 556 as a working number, 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 556 or greater than 556. We have (if the multiplication is difficult, then it is better to multiply natural numbers by 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 556, then under the highlighted number we write down the number 412 (it was obtained at the penultimate step), and in place of the quotient we write down the number 2 (since multiplication was carried out on it at the penultimate step). Long division notation 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 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 1 442, select it, and go through points from the second to the fourth one more time.
Multiply the divisor 206 by 0, 1, 2, 3, ... until you get the number 1 442 or a number that is greater than 1 442. 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 subtraction in a column, we get zero, but we do not write it down right away, but only remember its position, because we do not know whether the division ends there, or we will 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 memorized position, since there are no numbers in the record of the dividend in this column. Therefore, this is where the long division is over, and we complete the recording:
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