Content:
At first glance, dividing a whole number by a decimal is quite difficult. After all, no one knows the multiplication table for decimal fractions, for example, 0.7. The secret is that you need to rewrite the division problem so that only integers remain - in this case, you will only have to divide the two numbers into a column.
Steps
Part 1 Rewrite the problem in a different form
- 1
Write the division problem. If you want to make changes, use a pencil.
- For example, solve the problem: 3 ÷ 1.2.
- 2
Convert a whole number to a decimal. To do this, place a decimal point after the number, and then write so many zeros so that the number of decimal places for both fractions is equal. Keep in mind that adding zeros to a whole number after the decimal point does not change the value of that number.
- In our example, the integer is the number 3. Since the decimal fraction 1.2 has one sign after the decimal point, rewrite 3 as 3.0, that is, add one zero to 3. Now the original problem looks like: 3.0 ÷ 1.2.
- Attention: do not add zeros without a decimal point! Remember that 3 = 3.0 = 3.00, but 3 ≠ 30 ≠ 300.
- 3
Move the decimal point to the right so that the decimals become whole numbers. In division problems, you can move the decimal point of each decimal, but only the same number of places after the decimal point. This will allow you to convert decimals to whole numbers.
- In our example, convert the decimals 3.0 and 1.2 to whole numbers by moving the decimal point one place to the right. Thus, 3.0 will turn into 30, and 1.2 into 12. Now the problem looks like: 30 ÷ 12.
- 4 Rewrite the problem in long division form. To do this, write the dividend (usually the larger number) on the left and the divisor (the number being divided by) on the right. You will receive a column division problem with integers. If you don't remember how to do long division, skip to the next section.
Part 2 Column division
- 1
Find the first digit of the quotient (the result of division). To do this, divide the first digit of the dividend by the divisor. Write the result under the divisor.
- In our example, the first digit of the dividend is 3. Divide 3 by 12. Since 3 is less than 12, the result of division will be 0. Write 0 under the divisor - this is the first digit of the quotient.
- 2
Multiply the result by the divisor. Write the result of the multiplication under the first digit of the dividend, since this is the digit you just divided by the divisor.
- In our example, 0 × 12 = 0, so write 0 under 3.
- 3
Subtract the result of the multiplication from the first digit of the dividend. Write your answer on a new line.
- In our example: 3 - 0 = 3. Write 3 directly below 0.
- 4
Move down the second digit of the dividend. To do this, write down the next digit of the dividend next to the result of the subtraction.
- In our example, the dividend is 30. The second digit of the dividend is 0. Move it down by writing a 0 next to the 3 (the result of the subtraction). You will receive the number 30.
- 5
Divide the result by the divisor. You will find the second digit of the quotient. To do this, divide the number located on the bottom line by the divisor.
- In our example, divide 30 by 12. 30 ÷ 12 = 2 plus some remainder (since 12 x 2 = 24). Write 2 after 0 under the divisor - this is the second digit of the quotient.
- If you can't find a suitable digit, go through the digits until the result of multiplying a digit by a divisor is smaller and closest to the number located last in the column. In our example, consider the number 3. Multiply it by the divisor: 12 x 3 = 36. Since 36 is greater than 30, the number 3 is not suitable. Now consider the number 2. 12 x 2 = 24. 24 is less than 30, so the number 2 is the correct solution.
- 6
Repeat the steps above to find the next number. The described algorithm is used in any long division problem.
- Multiply the second digit of the quotient by the divisor: 2 x 12 = 24.
- Write the result of the multiplication (24) under the last number in the column (30).
- Subtract the smaller number from the larger one. In our example: 30 - 24 = 6. Write the result (6) on a new line.
- 7
If there are still digits in the dividend that can be moved down, continue the calculation process. Otherwise, continue to the next step.
- In our example, you moved down the last digit of the dividend (0). So move on to the next step.
- 8
If necessary, use a decimal point to expand the dividend. If the dividend is divisible by the divisor, then on the last line you will get the number 0. This means that the problem has been solved, and the answer (in the form of an integer) is written under the divisor. But if at the very bottom of the column there is any figure other than 0, it is necessary to expand the dividend by adding a decimal point and adding 0. Let us remember that this does not change the value of the dividend.
- In our example, the last line contains the number 6. Therefore, to the right of 30 (the dividend), write a decimal point, and then write 0. Also, place a decimal point after the found digits of the quotient, which you write under the divisor (don’t write anything after this comma yet!) .
- 9
Repeat the steps described above to find the next number. The main thing is not to forget to put a decimal point both after the dividend and after the found digits of the quotient. The rest of the process is similar to the process described above.
- In our example, move down the 0 (which you wrote after the decimal point). You will get the number 60. Now divide this number by the divisor: 60 ÷ 12 = 5. Write 5 after the 2 (and after the decimal point) under the divisor. This is the third digit of the quotient. So the final answer is 2.5 (the zero before the 2 can be ignored).
- When solving a division problem, you can write the answer with a remainder (in our example: 3 ÷ 1.2 = 2 remainder 6). However, when working with decimals, your teacher will likely expect you to present your answer as a decimal.
- If you do long division correctly, your answer will be either a whole number (when numbers are divided by whole numbers) or a decimal fraction. Do not try to guess the position of the decimal point in the answer - it may differ from its position in the dividend or divisor.
- There are problems where long division can take an infinite amount of time. In this case, stop and round your answer. For example, 17 ÷ 4.20 = 4.047619... In this case, round the result to 4.05.
- Remember the terminology:
- Dividend is the number that is being divided.
- The divisor is the number that is being divided by.
- The quotient is the result of division.
- Dividend ÷ Divisor = Quotient.
Attention
- Remember that the result of dividing 30 ÷ 12 is equal to the result of dividing 3 ÷ 1.2. Don't try to correct your answer by moving the decimal point.
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 with the remainder by 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 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 start dividing 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. quotient is a two-digit number. Let's put the second point:
Step 6. We begin the division operation. The largest number divisible by 8 without a remainder to 51 is 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 into 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 of natural numbers 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
Average level
Difficult level
Games for developing mental arithmetic
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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. 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 a mathematical operation is given; 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.
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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"
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Game "Fast addition reload"
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Children in grades 2-3 are learning a new mathematical operation - division. It is not easy for a student to understand the essence of this mathematical operation, so he needs the help of his parents. Parents need to understand exactly how to present new information to their child. TOP 10 examples will tell parents how to teach children how to divide numbers in a column.
Learning long division in the form of a game
Children get tired at school, they get tired of textbooks. Therefore, parents need to give up textbooks. Present information in the form of a fun game.
You can set tasks this way:
1 Organize a place for your child to learn through play. Place his toys in a circle, and give the child pears or candy. Have the student divide 4 candies between 2 or 3 dolls. To achieve understanding on the part of the child, gradually increase the number of candies to 8 and 10. Even if the baby takes a long time to act, do not put pressure or yell at him. You will need patience. If your child does something wrong, correct him calmly. Then, after he completes the first action of dividing the candies between the participants in the game, he will ask him to calculate how many candies went to each toy. Now the conclusion. If there were 8 candies and 4 toys, then each got 2 candies. Let your child understand that sharing means distributing an equal amount of candy to all toys.
2 You can teach math operations using numbers. Let the student understand that numbers can be classified as pears or candy. Say that the number of pears to be divided is the dividend. And the number of toys that contain candy is the divisor.
3 Give your child 6 pears. Give him a task: to divide the number of pears between grandfather, dog and dad. Then ask him to divide 6 pears between grandpa and dad. Explain to your child the reason why the division result was different.
4 Teach your student about division with a remainder. Give your child 5 candies and ask him to distribute them equally between the cat and dad. The child will have 1 candy left. Tell your child why it happened this way. This mathematical operation should be considered separately, as it can cause difficulties.
Playful learning can help your child quickly understand the whole process of dividing numbers. He will be able to learn that the largest number is divisible by the smallest or vice versa. That is, the largest number is candy, and the smallest number is the participants. In column 1 the number will be the number of candies, and 2 will be the number of participants.
Do not overload your child with new knowledge. You need to learn gradually. You need to move on to new material when the previous material is consolidated.
Learning long division using the multiplication table
Students up to 5th grade will be able to understand division more quickly if they have a good understanding of multiplication.
Parents need to explain that division is similar to the multiplication table. Only the actions are opposite. For clarity, we need to give an example:
- Tell the student to freely multiply the values of 6 and 5. The answer is 30.
- Tell the student that the number 30 is the result of a mathematical operation with two numbers: 6 and 5. Namely, the result of multiplication.
- Divide 30 by 6. The result of the mathematical operation is 5. The student will be able to see that division is the same as multiplication, but in reverse.
You can use the multiplication table to illustrate division if the child has mastered it well.
Learning long division in a notebook
Learning should begin when the student understands the material about division in practice, using games and multiplication tables.
You need to start dividing in this way, using simple examples. So, divide 105 by 5.
The mathematical operation needs to be explained in detail:
- Write an example in your notebook: 105 divided by 5.
- Write this down as you would for long division.
- Explain that 105 is the dividend and 5 is the divisor.
- With a student, identify 1 number that can be divided. The value of the dividend is 1, this figure is not divisible by 5. But the second number is 0. The result is 10, this value can be divided in this example. The number 5 is included in the number 10 twice.
- In the division column, under the number 5, write the number 2.
- Ask your child to multiply the number 5 by 2. The result of the multiplication is 10. This value must be written under the number 10. Next, you need to write the subtraction sign in the column. From 10 you need to subtract 10. You get 0.
- Write down in the column the number resulting from the subtraction - 0. 105 has a number left that was not involved in the division - 5. This number needs to be written down.
- The result is 5. This value must be divided by 5. The result is the number 1. This number must be written under 5. The result of the division is 21.
Parents need to explain that this division has no remainder.
You can start division with numbers 6,8,9, then go to 22, 44, 66 , and then to 232, 342, 345 , and so on.
Learning division with remainder
Once the child has mastered the material about division, you can make the task more difficult. Division with a remainder is the next step in learning. You need to explain using available examples:
- Invite your child to divide 35 by 8. Write the problem in the column.
- To make it as clear as possible for your child, you can show him the multiplication table. The table clearly shows that the number 35 includes the number 8 4 times.
- Write down the number 32 under the number 35.
- The child needs to subtract 32 from 35. The result is 3. The number 3 is the remainder.
Simple examples for a child
We can continue with the same example:
- When dividing 35 by 8, the remainder is 3. You need to add 0 to the remainder. In this case, after the number 4 in the column you need to put a comma. Now the result will be fractional.
- When dividing 30 by 8, the result is 3. This number must be written after the decimal point.
- Now you need to write 24 under the value 30 (the result of multiplying 8 by 3). The result will be 6. You also need to add a zero to the number 6. It will turn out to be 60.
- The number 60 contains the number 8 included 7 times. That is, it turns out to be 56.
- When subtracting 60 from 56, the result is 4. This number also needs to be signed 0. The result is 40. In the multiplication table, a child can see that 40 is the result of multiplying 8 by 5. That is, the number 40 includes the number 8 5 times. There is no remainder. The answer looks like this - 4.375.
This example may seem difficult to a child. Therefore, you need to divide values that will have a remainder many times.
Teaching division through games
Parents can use division games to teach their students. You can give your child coloring books in which you need to determine the color of a pencil by dividing. You need to choose coloring pages with easy examples so that the child can solve the examples in his head.
The picture will be divided into parts containing the results of the division. And the colors to use will be examples. For example, the color red is labeled with an example: 15 divided by 3. You get 5. You need to find the part of the picture under this number and color it. Math coloring pages captivate children. Therefore, parents should try this method of teaching.
Learning to divide by column the smallest number by the largest
Division by this method assumes that the quotient will start at 0 and be followed by a comma.
In order for the student to correctly assimilate the information received, he needs to give an example of such a plan.
In this article we will look at dividing integers without a remainder. Here we will only talk about the division of such integers, the absolute values of which are divisible by a whole (see the meaning of dividing natural numbers without a remainder). We will talk about dividing integers with a remainder in a separate article.
First, we'll introduce the terms and notation we'll use to describe division of integers. Next we will indicate the meaning of dividing integers, which will help us obtain the rules for dividing positive integers, negative integers and integers with different signs. Here we will look at examples of applying the rules for dividing integers. Finally, we'll show how to check the result of division of integers.
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Terms and symbols
The integer that is being divided is called divisible. The integer by which division is carried out is called divider. The result of dividing integers is called private.
Division is indicated by a symbol of the form:, which is located between the dividend and the divisor (sometimes there is a symbol ÷, which also denotes division). The division of an integer a by an integer b can be written using the symbol: as a:b . If dividing an integer a by an integer b results in a number c, then it is convenient to write this fact as the equality a:b=c. the form a:b is also called quotient, as is the meaning of this expression.
The meaning of dividing integers
We know that there is a connection between multiplication and division of natural numbers. From this connection we concluded that division is finding an unknown factor when the second factor and product are known. Let us give the same meaning to the division of integers. That is, dividing integers is finding, using a given product and one of the integer factors, another integer factor.
Based on the meaning of division of integers, we can say that if the product of two integers a and b is equal to c, then the quotient of c divided by a is equal to b, and the quotient of c divided by b is equal to a. Let's give an example. Let's say we know that the product of two integers 5 and −7 is equal to −35, then we can say that the quotient (−35):5 is equal to −7, and the quotient (−35):(−7) is equal to 5.
Note that the quotient of an integer a divided by an integer b is an integer (if a is divisible by b without a remainder).
Rules for dividing integers
The meaning of dividing integers, indicated in the previous paragraph, allows us to state that one of the two factors is quotient from dividing their product by the other factor. But it does not provide a way to find an unknown factor from a known factor and product. For example, the equality 6·(−7)=−42 allows us to say that the quotients (−42):6 and (−42):(−7) are equal to −7 and 6, respectively. However, if we know that the product of two factors is equal to 45 and one of the factors is equal to −5, then the meaning of dividing integers does not give us a direct answer to the question of what the other factor is equal to.
This reasoning leads us to the following conclusion: we need rules that allow us to divide one integer by another. Now we will receive them. These rules will allow us to reduce the division of integers to the division of natural numbers.
Dividing positive integers
Positive integers are natural numbers, therefore the division of positive integers is carried out according to all the rules for dividing natural numbers. There is nothing more to add here; we just need to consider the solution to a couple of examples in which division of positive integers is carried out.
Example.
Divide the positive integer 104 by the positive integer 8.
Solution.
Dividend 104 in this case can be represented as the sum 80+24, and then use the rule of dividing the sum by this number. We get 104:8=(80+24):8=80:8+24:8=10+3=13 .
Answer:
104:8=13 .
Rule for dividing negative integers, examples
The following reasoning will help us formulate the rule for dividing negative integers.
Let us need to divide a negative integer a by a negative integer b. Let us denote by the letter c the required quotient of dividing a by b, that is, a:b=c. Let's first find out what c is equal to.
Due to the meaning of dividing integers, the equality b·c=a must be true. Then . allow us to write the equality , therefore, . From the resulting equality it follows that, that is, the absolute value of the quotient of division is equal to the quotient of the modules of the dividend and divisor.
It remains to determine the sign of the number c. In other words, let's find out whether the result of dividing negative integers is a positive or negative integer.
In the sense of dividing integers, the equality b·c=a is true. Then from the rules for multiplying integers it follows that the number c must be positive. Otherwise, b·c will be a product of negative integers, which, according to the multiplication rule, will be equal to the product of the moduli of factors, therefore, will be a positive number, and our number a is a negative integer. Thus, the quotient c of dividing negative integers is a positive integer.
Now let's combine the conclusions we have drawn into the rule for dividing negative integers. To divide a negative integer by a negative integer, you need to divide the modulus of the dividend by the modulus of the divisor. That is, if a and b are negative integers, then .
Let's consider using the rule for dividing negative integers when solving examples.
Example.
Divide the negative integer −92 by the negative integer −4.
Solution.
According to the rule for dividing negative integers, the desired result is equal to the quotient of the modulus of the dividend divided by the modulus of the divisor. We get.
Answer:
(−92):(−4)=23 .
Example.
Calculate the quotient (−512):(−32) .
Solution.
We need to divide negative integers, let's use the appropriate rule. The module of the dividend is 512, the module of the divisor is 32. All that remains is to divide 512 by 32. Let's do the division by column: 
Answer:
(−512):(−32)=16 .
Rule for dividing integers with different signs, examples
We obtain a rule for dividing integers with different signs.
Let us divide the integer a by the integer b (the signs of the numbers a and b are different, that is, if a is a positive integer, then b is negative, and if a is negative, then b is a positive number) and as a result we get the number c.
In the previous paragraph of this article, we found out that the modulus of the quotient is equal to the quotient of the modulus of the dividend divided by the modulus of the divisor, that is, . Now we can calculate the absolute value of the quotient from dividing integers with different signs. It remains to find out the sign of the number c.
The meaning of dividing integers gives us the equality b·c=a. There are two options: either a is a positive integer, b is a negative integer; or a is a negative integer, b is a positive integer. In any of these cases, due to the rules of multiplication of integers, the number c must be negative. Indeed, according to the rules of multiplication of integers, if both b and c are negative integers, then their product will be a positive number, and if b is positive, c is negative, then their product is a negative number.
Now we can formulate a rule for dividing integers with different signs. To divide integers with different signs, you need to divide the modulus of the dividend by the modulus of the divisor, and put a minus sign in front of the resulting number. That is, if a and b are integers with different signs, then 
.
Let us analyze solutions to examples in which the rule for dividing integers with different signs is applied.
Example.
Divide the positive integer 56 by the negative integer −4.
Solution.
We will act according to the rule for dividing integers with different signs. The module of the dividend is 56, the module of the divisor is 4. Let's calculate the quotient of dividing the modulus of the divisor by the modulus of the divisor: 56:4=14. It remains to put a minus sign in front of the resulting number, we have −14.
Thus, when dividing integers with different signs 56 and −4, we got the number −14.
Answer:
56:(−4)=−14 .
Example.
Divide the integer −1 625 by 25.
Solution.
We need to divide integers with different signs. Let's use the resulting division rule: (1,625 can be divided by 25 in a column, or represent 1,625 as the sum of 1,500+125 and use the rule of dividing the sum by this number).
Answer:
(−1 625):25=−65 .
Dividing zero by an integer
Separately, you need to dwell on dividing zero by an integer other than zero. In these cases, the division rule is: the quotient of zero divided by any integer other than zero is zero. That is, 0:b=0 for any integer and non-zero number b.
Let us give an explanation of the announced rule for dividing zero by an integer. Suppose that dividing zero by the integer b (b is not equal to zero) results in the number c. Then, in the sense of dividing integers, the equality b·c=0 must be true. We know that the product of two integers is equal to zero if and only if at least one of the factors is equal to zero (we mentioned this in the theory section of multiplying an integer by zero). Since b is not equal to zero, this means that the multiplier c must be equal to zero. Therefore, the quotient of zero divided by an integer other than zero is zero.
Let's give a few examples. The quotient of 0 divided by the negative integer −908 is equal to 0, and the quotient of 0:4 is also zero.
You can't divide by zero
Dividing an integer by zero is not defined. In other words, you cannot divide by zero.
Why is this so? Let's assume that dividing an integer a by zero produces an integer c. Then, in the sense of dividing integers, the equality c·0=a is true. From the rule of multiplying an integer by zero it follows that c·0=0, no matter what the number c is. Comparing the two obtained equalities, we conclude that if the dividend a is different from zero, then the equality c·0=a will be incorrect, which indicates that a number other than zero cannot be divided by zero.
Is it possible to divide zero by zero? Let's assume that when dividing zero by zero, the result is an integer c, then, due to the meaning of dividing integers, the equality c·0=0 must be true. This equality is indeed true, but it is true not only for some specific integer c, but also for any number c in general. In other words, the result of dividing zero by zero can be any integer. So, in order to avoid this ambiguity, we decided not to consider division by zero.
So, you can't divide by zero.
Checking the result of dividing integers
Checking the result of dividing integers is done using multiplication. To check whether the division of integers was carried out correctly, you need to multiply the resulting quotient by the divisor; if the result is a number equal to the dividend, then the result of the division is correct.
Let's look at a solution to an example that checks the result of dividing integers.
This article talks about how to divide integers without a remainder, that is, by a whole number. Terms and notations will be introduced to further describe numbers, dividing positive and negative numbers. Finally, we will check the calculations.
Yandex.RTB R-A-339285-1
Terms and symbols
When dividing integers, the same terms are used as when describing natural numbers.
Definition 1
Dividend- this is the number over which division is performed.
Divider– the number by which to divide.
Private- the result of division.
The division sign is indicated by a colon “:” or the sign ÷. Its location is after the dividend and before the divisor. Notation using symbols looks like this: a: b . The result is written after the equal sign “=”. If, when dividing the number a by b, we get c, then the entry looks like the equality a: b = c. Division is otherwise called quotient.
Integer division
There is a connection between multiplication and division of natural numbers. This is due to the fact that when dividing, you can find a quotient, which, when reversed, will be considered a multiplier. Otherwise, we can write that dividing integers serves to find one of the integer factors.
From this we conclude that the product of integers a and b with the quotient equal to c can be represented by the inverse action of dividing c by b with the quotient equal to a. If the product of the numbers 5 and - 7 is equal to - 35, we have that the quotient (− 35) : 5 equals - 7, and (− 35) : (− 7) with the result 5.
The quotient of division is considered an integer when the result is obtained without a remainder, that is, the integer a must be divided by the number b with the integer quotient as a result.
Rules for dividing integers
The meaning of division is necessary to state that one of the two factors is a quotient and the other is simply a factor. Thus, it is impossible to find an unknown factor if you have a known factor and product. The equality 6 · (− 7) = − 42 means that the results of (− 42) : 6 and (− 42) : (− 7) are equal to - 7 and 6, respectively. If the product is known to be 45, and one of the factors is 5, then the meaning of division will not give a direct result of the other factor.
We can conclude that it is necessary to use rules that allow division of integers. They will allow you to divide integers and natural numbers.
Positive integers are natural numbers, so the division of positive integers is carried out based on the rules for dividing natural numbers. Let's look at a few examples for a detailed look at division of positive integers.
Example 1
Divide the positive integer 104 by the positive integer 8.
Solution
To simplify the division process, you can represent the number 104 as the sum of 80 + 24; now you need to apply the rule for dividing the sum by this number. We get 104: 8 = (80 + 24) : 8 = 80: 8 + 24: 8 = 10 + 3 = 13 .
Answer: 104: 8 = 13.
Example 2
Find the quotient of division 308 716: 452.
Solution
When we have a large number, it is best to divide into a column:
Answer: 308,716: 452 = 683.
To formulate a rule, reasoning must be applied. If it is necessary to divide negative integers a by b, then the desired quotient will be equal to c. Notation form: a: b = c. Then you can find out what the absolute value of c is.
Based on the meaning of division, the equality b · c = a is true. So b · c = a. Thanks to the properties of the module, we can write the equality b · c = b · c, which means b · c = a. From here we get that c = a: b. The absolute value of the quotient of division is equal to the quotient of the modules of the dividend and divisor.
To determine the sign of a number c, you need to find out what signs are in front of the dividend and divisor.
Based on the meaning of dividing integers, the equality b · c = a is true. The rule for multiplying integers says that the quotient must be positive. Otherwise, b · c will be produced according to the rules for negative integers. The quotient c of dividing negative integers is a positive number.
Combine into a division rule: To divide a negative integer by a negative number, you must divide the dividend by the modulus of the divisor. This entry will look like this: a: b = a: b, with a and b equal to negative numbers.
Let's look at some examples of dividing negative numbers.
Example 3
Divide - 92 by - 4.
Solution
Using the rules for dividing negative integers, we find that we should divide modulo. We get that - 92: - 4 = - 92: - 4 = 92: 4 = 23
Answer: (− 92) : (− 4) = 23.
Example 4
Calculate - 512: (- 32) .
Solution
To solve, you need to divide the numbers modulo. The division is done in a column.
Answer: (− 512) : (− 32) = 16.
Rule for dividing integers with different signs, examples
Let us highlight the rule for dividing integers containing different signs.
If we divide the integer numbers a and b with different signs, we get the number c. It is necessary to determine the sign of the resulting number. You should write c = a: b.
To determine the meaning of dividing the equality b · c = a, it is necessary to consider two options. Presumably there is an option when a is negative, b is positive, or a is positive and b is negative. Either case ultimately has a negative result. Following from the rules of multiplication, we have that b and c are negative, then the product will be positive. If b is positive and c is negative, then the product is a negative number.
For the formulation, the rule for dividing integers with different signs is applicable. From here we get: in order to divide integers with different signs, you need to divide the dividend by the modulo divisor and put “-” in front of the result. We get that a and b are integers with different signs. Let's write this as a: b = - a: b .
Let us examine in detail examples where it is necessary to apply the rule for dividing integers with different signs.
Example 5
Divide 56 by - 4.
Solution
Based on the rule, we have that 56 must be divided by 4 modulo. So we get that 56: 4 = 14. To determine the sign of the result, you need to look for the presence of “-” before the divisor and dividend. If there is only one minus sign, then we write the result as a negative value. That is, - 14.
Answer: 56: (− 4) = − 14.
Example 5
Divide - 1625 by 25.
Solution
This example shows the correct division of integers with different signs. To do this, you need to apply the rule
1625: 25 = - - 1625: 25 = - 1625: 25 = - 65
The number 1625 can be divided in a column or by representing it as the sum 1500 + 125, applying the rule of dividing the resulting amount by the number.
Answer: (− 1,625): 25 = − 65.
Dividing zero by an integer
Dividing zero by any integer is considered as a separate topic, as it has its own nuances. According to the rule, the quotient of division by any integer other than zero is equal to zero . Otherwise, we can write that 0: b = 0, where the value of the number b is non-zero.
To delve deeper into the rule, let's look at some explanations.
Let us assume that the result of dividing zero by an integer is equal to c, then the equality b · c = 0 is considered true. The product ends up being zero when at least one of them is zero. If by condition b is not equal to zero, then the factor c = 0. It follows that the quotient obtained by dividing zero by an integer other than zero is equal to zero.
For example, when dividing zero by an integer, the quotient is equal to zero: 0: 4 or 0: - 908. Both results will be zero.
Don't divide by zero
Dividing an integer by zero is not defined, and therefore dividing by 0 is prohibited.
For example, if when dividing an integer a by zero we get the number c, then from the meaning of division the equality c · 0 = a should be true. The rule of multiplication by zero says that c · 0 = 0 for any value of c. Comparing both equalities, we find that if the dividend of anne is equal to zero, then the equality c · 0 = a is considered incorrect. Therefore, we can conclude that division by zero cannot be performed.
Is it possible to divide zero by itself? Let us assume that when dividing we obtain an integer c, then the equality c · 0 = 0 must be true. It is considered valid for any value of c. The result of dividing 0 by 0 can be any value. To reduce multitasking, this option is not considered.
Checking the result of dividing integers
The check is carried out by multiplication. To check division, you need to multiply the resulting quotient by the divisor; if the result is a number equal to the dividend, then the result is considered correct.
Let's look at an example of a solution with checking the result.
Example 6
The result of dividing 72 by - 9 is - 7. Check this expression.
Solution
We perform a division check. It is necessary to multiply the resulting quotient and the divisor, that is, (− 7) · (− 9) = 63. The check showed that 63 is different from 72, which means the action was performed incorrectly.
Answer: the division was performed incorrectly.
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