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 as a whole class (25 people) donate money and buy a gift for the teacher, but you don’t spend it all, there will be change left over. So you will need to divide the change among everyone. The division operation comes into play to help you solve this problem.
Division is an interesting operation, as we will see in this article!
Dividing numbers
So, a little theory, and then practice! What is division? Division is breaking something into equal parts. That is, it could be a bag of sweets that needs to be divided into equal parts. For example, there are 9 candies in a bag, and the person who wants to receive them is three. Then you need to divide these 9 candies 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 reverse action, a check, will be multiplication. 3*3=9. Right? Absolutely.
So let's look at example 12:6. First, let's name each component of the example. 12 – dividend, that is. a number that can be divided into parts. 6 is a divisor, this is the number of parts into which the dividend is divided. And the result will be a number called “quotient”.
Let's 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 a remainder? This is the same division, only the result is not an even number, as shown above.
For example, let's divide 17 by 5. Since the largest number divisible by 5 to 17 is 15, then the answer will be 3 and the remainder is 2, and 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 the remainder 1. And it is written: 22:7 = 3 (1).
Division by 3 and 9
A special case of division would be division by the number 3 and the number 9. If you want to find out whether a number is divisible by 3 or 9 without a remainder, then you will need:
Find the sum of the digits of the dividend.
Divide by 3 or 9 (depending on what you need).
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 is 6+3 = 9. Divisible by both 9 and 3. 63:9 = 7, and 63:3 = 21. Such operations are carried out with any number to find out whether it is divisible by the remainder by 3 or 9, or not.
Multiplication and division
Multiplication and division are opposite friend friend operation. Multiplication can be used as a test for division, and division can be used as a test for multiplication. You can learn more about multiplication and master the operation in our article about multiplication. Which describes multiplication in detail and how to do it correctly. There you will also find the multiplication table and examples for training.
Here is an example of checking division and multiplication. Let's say the example is 6*4. Answer: 24. Then let's check the answer by division: 24:4=6, 24:6=4. It was decided correctly. In this case, the check is performed by dividing the answer by one of the factors.
Or an example is given for the division 56:8. Answer: 7. Then the test will be 8*7=56. Right? Yes. In this case, the test is performed by multiplying the answer by the divisor.
Division 3 class
In third grade they are just starting to go through division. Therefore, third graders solve the simplest problems:
Problem 1. A factory worker was given the task of putting 56 cakes into 8 packages. How many cakes should be put in each package to make the same amount in each?
Problem 2. On New Year's Eve at school, children in a class of 15 students were given 75 candies. How many candies should each child receive?
Problem 3. Roma, Sasha and Misha picked 27 apples from the apple tree. How many apples will each person get if they need to be divided equally?
Problem 4. Four friends bought 58 cookies. But then they realized that they could not divide them equally. How many additional cookies do the kids need to buy so that each gets 15?
Division 4th grade
The division in the fourth grade is more serious than in the third. All calculations are carried out using the column division method, and the numbers involved in the division are not small. What is long division? You can find the answer below:
Column division
What is long division? This is a method that allows you to find the answer to dividing large numbers. If prime numbers like 16 and 4 can be divided, and the answer is clear - 4. Then 512:8 is not easy for a child in his mind. And it’s our task to talk about the technique for solving such examples.
Let's look at an example, 512:8.
1 step. Let's write the dividend and divisor as follows:
The quotient will ultimately be written under the divisor, and the calculations under the dividend.
Step 2. We begin division from left to right. First we take the number 5: 
Step 3. The number 5 is less than the number 8, which means it will not be possible to divide. Therefore, we take another digit of the dividend:
Now 51 is greater than 8. This is an incomplete quotient.
Step 4. We put a dot under the divisor.
Step 5. After 51 there is another number 2, which means there will be one more number in the answer, that is. private – two-digit number. Let's put the second point:
Step 6. We begin the division operation. Largest number, divisible by 8 without a remainder to 51 – 48. Dividing 48 by 8, we get 6. Write the number 6 instead of the first dot under the divisor:
Step 7. Then write the number exactly below the number 51 and put a “-” sign:
Step 8. Then we subtract 48 from 51 and get the answer 3.
* 9 step*. We take down the number 2 and write it next to the number 3:
Step 10 We divide the resulting number 32 by 8 and get the second digit of the answer – 4.
So the answer is 64, without remainder. If we divided the number 513, then the remainder would be one.
Division of three digits
Dividing three-digit numbers is done using the long division method, which was explained in the example above. An example of just a three-digit number.
Division of fractions
Dividing fractions is not as difficult as it seems at first glance. For example, (2/3):(1/4). The method of this division is quite simple. 2/3 is the dividend, 1/4 is the divisor. You can replace the division sign (:) with multiplication ( ), but to do this you need to swap the numerator and denominator of the divisor. That is, we get: (2/3)(4/1), (2/3)*4, this is equal to 8/3 or 2 integers and 2/3. Let's give another example, with an illustration for better understanding. Consider the fractions (4/7):(2/5):
As in the previous example, we reverse the 2/5 divisor and get 5/2, replacing division with multiplication. We then get (4/7)*(5/2). We make a reduction and answer: 10/7, then take out the whole part: 1 whole and 3/7.
Dividing numbers into classes
Let's imagine the number 148951784296, and divide it by three digits: 148,951,784,296. So, from right to left: 296 is the class of units, 784 is the class of thousands, 951 is the class of millions, 148 is the class of billions. In turn, in each class 3 digits have their own digit. From right to left: the first digit is units, the second digit is tens, the third is hundreds. For example, the class of units is 296, 6 is ones, 9 is tens, 2 is hundreds.
Division of natural numbers
Division natural numbers– this is the simplest division described in this article. It can be either with or without a remainder. The divisor and dividend can be any non-fractional, integer numbers. 
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Division presentation
Presentation is another way to visualize the topic of division. Below we will find a link to an excellent presentation that does a good job of explaining how to divide, what division is, what dividend, divisor and quotient are. Don’t waste your time, but consolidate your knowledge!
Examples for division
Easy level
Intermediate level
Difficult level
Games for developing mental arithmetic
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve skills oral counting in an interesting playful way.
Game "Guess the operation"
The game “Guess the Operation” develops thinking and memory. The main point game, you need to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.
Game "Simplification"
The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and given mathematical operation, the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. 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 score points and continue playing.
Game "Piggy Bank"
The Piggy Bank game develops thinking and memory. The main essence of the game is to choose which piggy bank has more money. In this game there are four piggy banks, you need to count which piggy bank has the most money and show this piggy bank with the mouse. If you answered correctly, then you score points and continue playing.
Game "Fast addition reload"
The game “Fast addition reboot” 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 the 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 numbers and press them. If you answered correctly, then you score points and continue playing.
Development of phenomenal mental arithmetic
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In the section on the question of what is done first, multiplication or division in mathematics asked by the author Caucasian the best answer is These actions are equal, so the first thing to do is where the series begins (counting from left to right): A: B*C=(A: B) *C, A*C: B=(A*C): B True, in this case the result is the same (if the calculations are perfectly accurate).
Reply from 22 answers[guru]
Hello! Here is a selection of topics with answers to your question: what is done first, multiplication or division in mathematics
Reply from sleepy[newbie]
what comes first is first
Reply from Disgusting resource[guru]
I think multiplication.. but I don’t remember anymore.. it’s been a long time since I went to school
Reply from Evgenia Nebesnaya[guru]
I'll wash multiplication.
Reply from Drawdown[guru]
multiplication?!)))
Reply from Lyubov Lavrinovich[expert]
doesn't matter. the answer is the same.
Reply from Vitaly Kholodov[newbie]
yyyyy))))) It's the same thing))))
Reply from Gambit 007[master]
From left to right! If multiplication comes first, then multiplication, if division, then division!
Reply from HELEN &&&[expert]
one by one
Reply from Iris-chan[expert]
If there are no parentheses, then it doesn't matter. I usually do it in the order that is easiest, in which smaller numbers need to be multiplied or divided.
Reply from Eldgammel Vind[guru]
It doesn't matter at all if there are no parentheses.
Reply from Zina Evstigneeva[guru]
such examples are solved in order, such that this action comes first and perform
Reply from Andrey Kozlov[newbie]
multiplication
Reply from Yoerezha Talanin[newbie]
multiplication))) =)
Reply from Arthur[active]
6: 2 * 3 = 9 this is in order 6: 2 * 3 = 1 this is from the beginning to multiplication then to division the answers are different, so the order matters. They count from left to right
Reply from Dasha Zaraf[newbie]
The action is performed depending on the order. For example: 200*45/1000=9 (in this case * comes first and division last. And so first we will multiply 200*45 and then divide 9000/1000=9) Another example: 36/9*4=16 (in this case / comes first, and
multiply in any order.
Methodologically, this rule aims to prepare the child to become familiar with the methods of multiplying numbers ending in zeros, so they are introduced to it only in the fourth grade. In reality, this property of multiplication allows you to rationalize mental calculations in both 2nd and 3rd grade.
For example:
Calculate: (7 2) 5 = ...
In this case, it is much easier to calculate the option
7 (2 5) = 7 10 - 70.
Calculate: 12 (5 7) = ...
8 in this case it is much easier to calculate the option (12-5)-7 = 60-7 = 420.
Calculation techniques
1. Multiplication and division of numbers ending in zero: 20 3; 3 20; 60:3; 80:20
The computational technique in this case comes down to multiplying and dividing single-digit numbers expressing the number of tens in given numbers. For example:
20 3 =... 3 20 =... 60:3 = ...
2 dec. 3 = 20 3 = 60 b dec.: 3 = 2 dec.
20 - 3 = 60 3 20 = 60 60: 3 = 20
For the 80:20 case, two calculation methods can be used: the one used in previous cases, and the method of selecting the quotient.
For example: 80: 20 =... 80: 20 =...
8 dec.: 2 dec. = 4 or 20 4 = 80
80: 20 = 4 80: 20 = 4
In the first case, the technique of representing two-digit tens in the form of digit units was used, which reduces the case under consideration to a tabular one (8:2). In the second case, the quotient figure is found by selection and checked by multiplication. In the second case, the child may not immediately select the correct number of the quotient, which means that the check will be performed more than once.
2. Method of multiplying a two-digit number by a single-digit number: 23 4; 4-23
When multiplying a two-digit number by a one-digit number, the following knowledge and skills are updated:
In the case of multiplication of the form 4 23, the permutation of factors is first applied, and then the same multiplication scheme as above is applied.
3. Method of dividing a two-digit number by a single-digit number: 48:3; 48:2
When dividing a two-digit number by a single-digit number, the following knowledge and skills are updated:
4. Method of dividing a two-digit number by a two-digit number: 68: 17
When dividing a two-digit number by a two-digit number, the following knowledge and skills are required:
The difficulty of the last technique is that the child cannot immediately select the desired digit of the quotient and performs several checks of the selected digits, which requires quite complex calculations. Many children spend a lot of time performing calculations of this type, because they begin not so much to select the appropriate quotient number, but rather to sort through all the factors in a row, starting with two.
To facilitate calculations, two techniques can be used:
1) orientation to the last digit of the dividend;
2) rounding method.
First appointment assumes that when selecting a possible digit of a quotient, the child is guided by knowledge of the multiplication table, immediately multiplying the selected digit (number) and the last digit of the divisor.
For example, 3-7 = 21. The last digit of the number 68 is 8, which means there is no point in multiplying 17 by 3, the last digit of the divisor still does not match. Let's try the number 4 in the quotient - multiply 7 4 = 28. The last digit matches, so it makes sense to find the product 17 4.
Second appointment involves rounding the divisor and selecting a quotient digit based on the rounded divisor.
For example, 68:17, the divisor of 17 is rounded to 20. The approximate figure for the quotient 3 gives, when checked, 20 3 = 60< 68, значит имеет смысл сразу проверять в качестве цифры частного 4:17 4 = 68.
These techniques allow you to reduce the cost of effort and time when performing calculations of this type, but require a good knowledge of the multiplication table and the ability to round numbers.
Integers ending in 0,1,2,3,4 are rounded to the nearest whole ten, discarding those digits.
For example, the numbers 12, 13, 14 should be rounded to 10. The numbers 62, 63, 64 should be rounded to 60.
Integers ending in 5, 6, 7,8,9 are rounded up to the nearest whole ten.
For example, the numbers 15,16,17,18,19 are rounded to 20. The numbers 45,47, 49 are rounded to 50.
Order of operations in expressions containing multiplication and division
Rules for the order of actions specify the main characteristics of expressions that should be used when calculating their values.
The first rules defining the order of operations in arithmetic expressions specified the order of actions in expressions containing addition and subtraction operations:
1. In expressions without parentheses containing only addition and subtraction operations, the actions are performed in the order they are written: from left to right.
2. Actions in brackets are performed first.
3. If an expression contains only addition actions, then two adjacent terms can always be replaced by their sum (combinative property of addition).
In grade 3, new rules for the order of performing actions in expressions containing multiplication and division are studied:
4. In expressions without parentheses, containing only multiplication and division, the actions are performed in the order they are written: from left to right.
5. In expressions without parentheses, multiplication and division are performed before addition and subtraction.
In this case, the setting to perform the action in brackets first is preserved. Possible cases of violation of this setting were discussed earlier.
Rules for the order of actions are general rules for calculating the values of mathematical expressions (examples), which are maintained throughout the entire period of studying mathematics at school. In this regard, the formation in a child of a clear understanding of the algorithm of the order of actions is an important successive task of teaching mathematics in elementary school. The problem is that the rules for the order of actions are quite variable and not always clearly defined.
For example, in the expression 48-3 + 7 + 8, as a general rule, rule 1 should be applied for an expression without parentheses containing addition and subtraction operations. At the same time, as an option for rational calculations, you can use the technique of replacing the sum of the part 7 + 8, since after subtracting the number 3 from 48 you get 45, to which it is convenient to add 15.
However, such an analysis of such an expression is not provided in the elementary grades, since there are fears that with an inadequate understanding of this approach, the child will use it in cases of the form 72 - 9 - 3 + 6. In this case, replacing the expression 3 + 6 with a sum is impossible, it will lead to wrong answer.
Great variability in the application of the entire group of rules and variants of rules in determining the order of actions requires significant flexibility of thinking, a good understanding of the meaning of mathematical actions, the sequence of mental actions, mathematical “feeling” and intuition (mathematicians call this “number sense”). In reality, it is much easier to teach a child to strictly adhere to a clearly established procedure for analyzing a numerical expression from the point of view of the features that each rule is focused on.
When determining the course of action, think like this:
1) If there are parentheses, I perform the action written in parentheses first.
2) I perform multiplication and division in order.
3) I perform addition and subtraction in order.
This algorithm sets the order of actions quite unambiguously, although with minor variations.
In these expressions, the order of action is uniquely determined by the algorithm and is the only possible one. Let's give other examples
After performing multiplication and division in this example, you could immediately add 6 to 54, and subtract 9 from 18, and then add the results. Technically, it would be much easier than the path determined by the algorithm; an initially different order of actions in the example is possible:
Thus, the question of developing the ability to determine the order of actions in expressions in elementary school in a certain way contradicts the need to teach the child methods of rational calculations.
For example, in this case, the order of actions is absolutely unambiguously determined by the algorithm, and requires a series of complex mental calculations with transitions through the digits: 42 - 7 and 35 + 8.
If, after performing the division 21:3, you perform the addition 42 + 8 = 50, and then subtract 50 - 7 = 43, which is much easier technically, the answer will be the same. This calculation path contradicts the setting given in the textbook
The operations of multiplication and division were especially complex and difficult in the old days - especially the latter.
“Multiplication is my torment, but division is trouble,” they said in the old days.
IN ancient times and almost until the eighteenth century, Russian people in their calculations did without multiplication and division: they used only two arithmetic operations - addition and subtraction, and also the so-called “doubling” and “bifurcation”. The essence of the ancient Russian method of multiplication is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half (sequential bifurcation) while simultaneously doubling the other number. If in a product, for example 24∙5, the multiplicand is reduced by 2 times (“double”), and the multiplier is increased by 2 times (“double”), then the product will not change: 24∙5=12∙10=120
The division of the multiplicand continues until the quotient turns out to be 1, while doubling the multiplier. The last doubled number gives the desired result. So 32∙17=1∙544=544. In the proposed example, all numbers are divisible by 2 without a remainder.
But what if division by 2 occurs with a remainder?
If the multiplicand is not divisible by 2, then one is first subtracted from it, and then divided by 2. The lines with even multiplicands are crossed out, and the right parts of the lines with odd multiplicands are added.
That is, 21∙17=(20+1)∙17=20∙17+1∙17.
Let us remember the number 17 (the first line is not crossed out), and replace the product 20∙17 with the equal product 10∙34. but the product 10∙34, in turn, can be replaced by an equal product 5∙68, so the second line is crossed out: 5∙68=(4+1) ∙68= 4∙68+68 Remember the number 68 (the third line is not crossed out) , and replace the product 4∙68 with the equal product 2∙136. But the product 2∙136 can be replaced by the equal product 1∙272, so the fourth line is crossed out. This means that in order to calculate the product 21∙17, you need to add 17.68.272 - the right-hand sides with odd multiplicands.
Products with even multiplicands can always be replaced by bifurcating the multiplicand and doubling the factor by equal products. Therefore, such lines are excluded from the calculation of the final product.
Time passed. Almost a dozen were in use at the same time in various ways multiplication and division - techniques are more complicated than the other, which a person of average abilities was not able to firmly remember.
In the book by V. Bellustin “How people gradually reached real arithmetic” (1941), 27 methods of multiplication are outlined, and the author notes; “It is quite possible that there are also (methods) hidden in caches, book depositories, scattered in numerous, mainly handwritten collections.”
And all these methods of multiplication - “chessboard”, “bending”, “back to front”, “diamond” and others, as well as all methods of division, which had no less intricate names, competed with each other in cumbersomeness and complexity.
In the time of M. Lomonosov, the action of multiplication was already written almost the same way as in our time. Only the multiplicand was called “quantity”, and the product was called “product” and, in addition, the multiplication sign was not written.
48 - Majesty. 8 - Multiplier. 384 - Product or work.
It is known that M.V. Lomonosov knew by heart the entire “Arithmetic” of Magnitsky. In accordance with this textbook, little Misha Lomonosov would explain the multiplication of 48 by 8 as follows: “8 times 8 is 64, I write 4 under the line, against 8, and have 6 decimals in my mind. And then 8 times 4 is 32, and I keep 3 in mind, and to 2 I will add 6 decimals, and it will be 8. And I will write this 8 next to 4, in a row to my left hand, and while 3 is in my mind, I will write in a row near 8, to the left hand. And from the multiplication of 48 with 8 the product will be 384.”
Now we explain it in almost the same way, only we speak in modern, not ancient, and, in addition, we name the categories. For example, 3 should be written in third place because it will be hundreds, and not just “in a row next to 8, to the left hand.”
As for division... The textbook by L.F. Magnitsky gives several methods of division. Some of these methods are so difficult that it is very easy to get confused.
Let's look at one of these methods now. Magnitsky considers it elegant and simple.
Suppose we need to divide 598432 by 678. First, write the first digits of the dividend 5984, below it the divisor 678. Divide 59 by 7 (678 is close to 700), get the first digit of the quotient 8 and write it to the right of the dividend, multiply 8 by 678: eight eight 64 , subtract 4 from 4 in your mind and write the remainder 0 over 4; eight is seven 56, and 6 in mind is 62, we subtract 2 from 8, we get 6 as a remainder and write it over 8; 8X6=48, 48 +6=54, 59-54=5, which means that above 59 we write the remainder 5. Now to the remainder 560 we add the next digit of the dividend 3 and continue the action in the same order.
Having completed division with difficulty, our ancestors considered it obligatory to check it once or twice. Magnitsky in this case is limited to one check. He recommends multiplying from the highest digits: 678 x 8 = 5424, again. 678 x 8 = 5424 and 678 x 2 = 1356; Under these numbers he signs the remainder and adds it up. Gets the dividend. “Fairly divided,” they wrote in conclusion in the old days.
This is what the division notation looked like:
598432 correctly divided
As you can see, this method is very similar to the one we use. Probably ours modern way evolved from this. We will not examine other methods; we will only present the form of writing divisions in a “diamond”, which is found in Magnitsky.
Divide 9649378 by 5634:
