Solve column online division. How to divide in a column? How to explain column division to a child? Divide by a single, two-digit, three-digit number, division with a remainder
Dividing by a column, or, more correctly, a written method of dividing by a corner, schoolchildren are already in the third grade of elementary school, but often this topic is given so little attention that not all students can freely use it by grades 9-11. Dividing by a column by a two-digit number takes place in grade 4, as well as dividing by a three-digit number, and then this technique is used only as an auxiliary when solving any equations or finding the value of an expression.
It is obvious that by paying more attention to division by a column than is laid down in the school curriculum, the child will make it easier for himself to complete tasks in mathematics up to grade 11. And for this you need little - to understand the topic and work out, decide, keeping the algorithm in your head, bring the calculation skill to automatism.
Algorithm for dividing by a column by a two-digit number
As with division by a single digit, we will successively move from dividing larger counting units to dividing smaller units.
1. Find the first incomplete dividend. This is a number that is divisible by a divisor to get a number greater than or equal to 1. This means that the first partial divisible is always greater than the divisor. When dividing by a two-digit number, the first incomplete divisible has at least 2 digits.
Examples 76 8:24. First incomplete dividend 76
265:53 26 is less than 53, so it doesn't fit. You need to add the next number (5). The first incomplete dividend is 265.
2. Determine the number of digits in private. To determine the number of digits in the private, it should be remembered that one digit of the private corresponds to the incomplete dividend, and one more digit of the private corresponds to all other digits of the dividend.
Examples 768:24. The first incomplete dividend is 76. It corresponds to 1 private digit. After the first partial divisor, there is one more digit. So there will be only 2 digits in the quotient.
265:53. The first incomplete dividend is 265. It will give 1 digit of the quotient. There are no more numbers in the dividend. So there will be only 1 digit in the quotient.
15344:56. The first incomplete dividend is 153, and after it there are 2 more digits. So there will be only 3 digits in the quotient.
3. Find the numbers in each digit of the private. First, find the first digit of the quotient. We select such an integer that, when multiplied by our divisor, we get a number that is as close as possible to the first incomplete divisible. We write the private number under the corner, and subtract the value of the product in a column from the incomplete divisor. We write down the rest. We check that it is less than the divisor.
Then we find the second digit of the private. We rewrite in a line with a remainder the number following the first incomplete divisor in the dividend. The resulting incomplete dividend is again divided by the divisor and so we find each subsequent private number until the divisor digits run out.
4. Find the remainder(if there is).
If the quotient digits are over and the remainder is 0, then the division is performed without a remainder. Otherwise, the quotient value is written with a remainder.
The division by any multi-digit number (three-digit, four-digit, etc.) is also performed.
Parsing examples for dividing by a column by a two-digit number
First, consider the simple cases of division, when the quotient is a single-digit number.
Let's find the value of the private numbers 265 and 53.
The first incomplete dividend is 265. There are no more numbers in the dividend. So the quotient will be a single-digit number.
To make it easier to pick up the private number, we divide 265 not by 53, but by a close round number 50. To do this, we divide 265 by 10, there will be 26 (remainder 5). And 26 divided by 5 will be 5 (remainder 1). The number 5 cannot be immediately written in private, since this is a trial number. First you need to check if it fits. Multiply 53*5=265. We see that the number 5 came up. And now we can record it in a private corner. 265-265=0. The division is done without a remainder.
The value of the private numbers 265 and 53 is 5.
Sometimes, when dividing, the trial digit of the quotient does not fit, and then it needs to be changed.
Let's find the value of the private numbers 184 and 23.
The quotient will be a single digit.
To make it easier to pick up the private number, we divide 184 not by 23, but by 20. To do this, we divide 184 by 10, it will be 18 (remainder 4). And we divide 18 by 2, it will be 9. 9 is a trial number, we won’t write it in private right away, but we’ll check if it fits. Multiply 23*9=207. 207 is greater than 184. We see that the number 9 does not fit. In private it will be less than 9. Let's try if the number 8 is suitable. Multiply 23 * 8 = 184. We see that the number 8 is suitable. We can record it privately. 184-184=0. The division is done without a remainder.
The value of the private numbers 184 and 23 is 8.
Let's consider more difficult cases of division.
Find the value of the private numbers 768 and 24.
The first incomplete dividend is 76 tens. So, there will be 2 digits in the quotient.
Let's determine the first digit of the quotient. Let's divide 76 by 24. To make it easier to find the private number, we divide 76 not by 24, but by 20. That is, we need to divide 76 by 10, there will be 7 (remainder 6). Divide 7 by 2 to get 3 (remainder 1). 3 is the trial digit of the quotient. Let's check if it fits first. Multiply 24*3=72 . 76-72=4. The remainder is less than the divisor. This means that the number 3 has come up and now we can write it down in place of tens of quotients. 72 we write under the first incomplete divisible, put a minus sign between them, write the remainder under the line.
Let's continue the division. Let's rewrite the number 8 in the line with the remainder, following the first incomplete divisible. We get the following incomplete dividend - 48 units. Let's divide 48 by 24. To make it easier to pick up the private number, we divide 48 not by 24, but by 20. That is, we divide 48 by 10, there will be 4 (remainder 8). And 4 divided by 2 will be 2. This is a trial digit of the private. We must first check if it will fit. Multiply 24*2=48. We see that the number 2 has come up and, therefore, we can write it down in place of the units of the quotient. 48-48=0, the division is done without a remainder.
The value of the private numbers 768 and 24 is 32.
Find the value of the private numbers 15344 and 56.
The first incomplete dividend is 153 hundreds, which means that there will be three digits in the private.
Let's determine the first digit of the quotient. Let's divide 153 by 56. To make it easier to find the private number, we divide 153 not by 56, but by 50. To do this, we divide 153 by 10, there will be 15 (remainder 3). And 15 divided by 5 will be 3. 3 is the trial digit of the quotient. Remember: you cannot immediately write it in private, but you must first check whether it fits. Multiply 56*3=168. 168 is greater than 153. So, in the quotient it will be less than 3. Let's check if the number 2 is suitable. Multiply 56*2=112. 153-112=41. The remainder is less than the divisor, which means that the number 2 is suitable, it can be written in place of hundreds in the quotient.
We form the following incomplete dividend. 153-112=41. We rewrite the number 4 in the same line, following the first incomplete divisible. We get the second incomplete dividend 414 tens. Let's divide 414 by 56. To make it more convenient to choose the number of the quotient, we will divide 414 not by 56, but by 50. 414:10=41(remaining 4). 41:5=8(rest.1). Remember: 8 is a trial number. Let's check it out. 56*8=448. 448 is greater than 414, which means that in the quotient it will be less than 8. Let's check if the number 7 is suitable. Multiply 56 by 7, we get 392. 414-392=22. The remainder is less than the divisor. So, the number came up and in the quotient in place of tens we can write 7.
We write in a line with a new remainder of 4 units. So the next incomplete dividend is 224 units. Let's continue the division. Divide 224 by 56. To make it easier to pick up the quotient, divide 224 by 50. That is, first by 10, it will be 22 (remainder 4). And 22 divided by 5 will be 4 (remainder 2). 4 is a trial number, let's check if it works. 56*4=224. And we see that the figure has come up. We write 4 in place of units in the quotient. 224-224=0, the division is done without a remainder.
The value of the private numbers 15344 and 56 is 274.
Example for division with a remainder
To draw an analogy, let's take an example similar to the example above, and differing only in the last digit
Let's find the value of private numbers 15345:56
We divide first in the same way as in the example 15344:56, until we reach the last incomplete divisible 225. Divide 225 by 56. To make it easier to find the private number, divide 225 by 50. That is, first by 10, there will be 22 (the remainder is 5 ). And 22 divided by 5 will be 4 (remainder 2). 4 is a trial number, let's check if it works. 56*4=224. And we see that the figure has come up. We write 4 in place of units in the quotient. 225-224=1, division is done with a remainder.
The value of the private numbers 15345 and 56 is 274 (remainder 1).
Division with zero in quotient
Sometimes in the quotient one of the numbers turns out to be 0, and children often skip it, hence the wrong solution. Let's figure out where 0 can come from and how not to forget it.
Find the value of private numbers 2870:14
The first partial dividend is 28 hundreds. So the quotient will have 3 digits. We put three points under the corner. This is an important point. If the child loses zero, there will be an extra dot, which will make you think that a number is missing somewhere.
Let's determine the first digit of the quotient. Divide 28 by 14. By selection, we get 2. Let's check if the number 2 fits. Multiply 14*2=28. The number 2 is suitable, it can be written in place of hundreds in private. 28-28=0.
There is a zero remainder. We've marked it in pink for clarity, but you don't need to write it down. We rewrite the number 7 from the dividend into a line with a remainder. But 7 is not divisible by 14 to get an integer, so we write in place of tens in private 0.
Now we rewrite the last digit of the dividend (the number of units) in the same line.
70:14=5 We write the number 5 instead of the last point in the quotient. 70-70=0. There is no rest.
The value of the private numbers 2870 and 14 is 205.
Division must be checked by multiplication.
Examples per division for self-test
Find the first incomplete dividend and determine the number of digits in the quotient.
3432:66 2450:98 15145:65 18354:42 17323:17
You have mastered the topic, and now practice solving a few examples in a column on your own.
1428: 42 30296: 56 254415: 35 16514: 718
With this mathematical program, you can divide polynomials by a column.
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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 \((x^3/x = x^2) \)
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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 a 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 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
\(\large(\frac(x^3-12x^2-42)(x-3)) = x^2-9x-27 + \large(\frac(-123)(x-3)) \)
Division of multi-digit numbers is easiest to do in a column. Column division is also called corner division.
Before we begin performing division by a column, let us consider in detail the very form of recording division by a column. First, we write down the dividend and put a vertical bar to the right of it:
Behind the vertical line, opposite the dividend, we write the divisor and draw a horizontal line under it:
Under the horizontal line, the quotient resulting from the calculations will be written in stages:
Under the dividend, intermediate calculations will be written:
The full form of division by a column is as follows:
How to divide by a column
Let's say we need to divide 780 by 12, write the action in a column and start dividing:
The division by a column is carried out in stages. The first thing we need to do is define the incomplete dividend. Look at the first digit of the dividend:
this number is 7, since it is less than the divisor, then we cannot start dividing from it, so we need to take one more digit from the dividend, the number 78 is greater than the divisor, so we start dividing from it:
In our case, the number 78 will be incomplete divisible, it is called incomplete because it is just a part of the divisible.
Having determined the incomplete dividend, we can find out how many digits there will be in the quotient, for this we need to calculate how many digits are left in the dividend after the incomplete dividend, in our case there is only one digit - 0, which means that the quotient will consist of 2 digits.
Having found out the number of digits that should turn out in a private one, you can put dots in its place. If, at the end of the division, the number of digits turned out to be more or less than the indicated points, then a mistake was made somewhere:
Let's start dividing. We need to determine how many times 12 is contained in the number 78. To do this, we successively multiply the divisor by natural numbers 1, 2, 3, ... until we get a number as close as possible to the incomplete divisible or equal to it, but not exceeding it. Thus, we get the number 6, write it under the divisor, and subtract 72 from 78 (according to the rules of column subtraction) (12 6 \u003d 72). After we subtracted 72 from 78, we got a remainder of 6:
Please note that the remainder of the division shows us whether we have chosen the right number. If the remainder is equal to or greater than the divisor, then we did not choose the correct number and we need to take a larger number.
To the resulting remainder - 6, we demolish the next digit of the dividend - 0. As a result, we got an incomplete dividend - 60. We determine how many times 12 is contained in the number 60. We get the number 5, write it into the quotient after the number 6, and subtract 60 from 60 ( 12 5 = 60). The remainder is zero:
Since there are no more digits left in the dividend, it means that 780 is divided by 12 completely. As a result of performing division by a column, we found the quotient - it is written under the divisor:
Consider an example where zeros are obtained in the quotient. Let's say we need to divide 9027 by 9.
We determine the incomplete dividend - this is the number 9. We write it into the quotient 1 and subtract 9 from 9. The remainder turned out to be zero. Usually, if in intermediate calculations the remainder is zero, it is not written down:
We demolish the next digit of the dividend - 0. We recall that when dividing zero by any number, there will be zero. We write to private zero (0: 9 = 0) and subtract 0 from 0 in intermediate calculations. Usually, in order not to pile up intermediate calculations, the calculation with zero is not written down:
We demolish the next digit of the dividend - 2. In intermediate calculations, it turned out that the incomplete dividend (2) is less than the divisor (9). In this case, zero is written into the quotient and the next digit of the dividend is taken down:
We determine how many times 9 is contained in the number 27. We get the number 3, write it into a quotient, and subtract 27 from 27. The remainder is zero:
Since there are no more digits left in the dividend, it means that the number 9027 is divided by 9 completely:
Consider an example where the dividend ends in zeros. Let's say we need to divide 3000 by 6.
We determine the incomplete dividend - this is the number 30. We write it into the quotient 5 and subtract 30 from 30. The remainder is zero. As already mentioned, it is not necessary to write down zero in the remainder in intermediate calculations:
We demolish the next digit of the dividend - 0. Since when dividing zero by any number there will be zero, we write it to private zero and subtract 0 from 0 in intermediate calculations:
We demolish the next digit of the dividend - 0. We write one more zero into the quotient and subtract 0 from 0 in intermediate calculations. at the very end of the calculation, it is usually written to show that the division is complete:
Since there are no more digits left in the dividend, it means that 3000 is divided by 6 completely:
Division by a column with a remainder
Let's say we need to divide 1340 by 23.
We determine the incomplete dividend - this is the number 134. We write in the quotient 5 and subtract 115 from 134. The remainder turned out to be 19:
We demolish the next digit of the dividend - 0. Determine how many times 23 is contained in the number 190. We get the number 8, write it into a quotient, and subtract 184 from 190. We get the remainder 6:
Since there are no more digits left in the dividend, the division is over. The result is an incomplete quotient of 58 and a remainder of 6:
1340: 23 = 58 (remainder 6)
It remains to consider an example of division with a remainder, when the dividend is less than the divisor. Suppose we need to divide 3 by 10. We see that 10 is never contained in the number 3, so we write it to the quotient 0 and subtract 0 from 3 (10 0 = 0). We draw a horizontal line and write down the remainder - 3:
3: 10 = 0 (remainder 3)
Column Division Calculator
This calculator will help you perform division by a column. Just enter the dividend and divisor and click the Calculate button.
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.
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 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.
Solution.
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 into 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.
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 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 .
Solution.
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.
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 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:
- Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
- Maths. Any textbooks for 5 classes of educational institutions.
Math-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, extracting the root, raising to a power, calculating percentages, and other operations.
Solution:
How to use the math calculator
| Key | Designation | Explanation |
|---|---|---|
| 5 | numbers 0-9 | Arabic numerals. Enter natural integers, zero. To get a negative integer, press the +/- key |
| . | semicolon) | A decimal separator. If there is no digit before the dot (comma), the calculator will automatically substitute a zero before the dot. 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) |
| X | multiplication sign | Multiplication of numbers (integers, decimals) |
| √ | root | Extracting the root from a number. When you press the "root" button again, the root is calculated from the result. For example: square root of 16 = 4; square root of 4 = 2 |
| x2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
| 1/x | fraction | Output to decimals. In the numerator 1, in the denominator the input number |
| % | percent | Get 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 bracket | An open parenthesis to set the evaluation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
| ) | closed bracket | A closed parenthesis to set the evaluation priority. Mandatory open bracket |
| ± | plus minus | Changes sign to opposite |
| = | equals | Displays the result of the solution. Also, intermediate calculations and the result are displayed above the calculator in the "Solution" field. |
| ← | deleting a character | Deletes the last character |
| FROM | reset | Reset button. Completely resets the calculator to "0" |
The algorithm of the online calculator with examples
Addition.
Addition of whole natural numbers ( 5 + 7 = 12 )
Addition of whole natural and negative numbers ( 5 + (-2) = 3 )
Adding decimal fractional numbers ( 0.3 + 5.2 = 5.5 )
Subtraction.
Subtraction of whole natural numbers ( 7 - 5 = 2 )
Subtraction of whole natural and negative numbers ( 5 - (-2) = 7 )
Subtraction of decimal fractional numbers ( 6.5 - 1.2 = 4.3 )
Multiplication.
Product of whole natural numbers ( 3 * 7 = 21 )
Product of whole natural and negative numbers ( 5 * (-3) = -15 )
Product of decimal fractional numbers ( 0.5 * 0.6 = 0.3 )
Division.
Division of whole natural numbers ( 27 / 3 = 9 )
Division of whole natural and negative numbers ( 15 / (-3) = -5 )
Division of decimal fractional numbers ( 6.2 / 2 = 3.1 )
Extracting the root from a number.
Extracting the root of an integer ( root(9) = 3 )
Extracting the root of decimals ( root(2.5) = 1.58 )
Extracting the root from the sum of numbers ( root(56 + 25) = 9 )
Extracting the root of the difference in numbers ( root (32 - 7) = 5 )
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ( (2.2) 2 = 4.84 )
Convert to decimal fractions.
Calculating percentages of a number
Increase 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Decrease the number 510 by 35% ( 510 - 510 * 0.35 = 331.5 )
18% of the number 140 is ( 140 * 0.18 = 25.2 )