To multiply a column, it is enough to know the multiplication table from 1 to 10 and non-relieved rule: Multivissal numbers can be multimed by discharge. Tribe in more detail about the rules of multiplication in the column.

How to multiply in Column: Basic Rules

Take a simple example for oral account.

First, 16 is multiplied by 1, we get 16. Then 16 multiply by 20, we get 320. We fold these two results:

This is a multiplication of discharges: the first factor is multiplied in turn on all the numbers of the second multiplier, starting with the youngest discharge, and then the results are folded.

If you write an example 1 to the column, we get the following:

Here the most important thing is a neat record. Discharges of units should be written under units, dozens - under dozens, etc. Then there is an addition on discharges:

6 + 0 \u003d 6; 1 + 2 \u003d 3. Figure 3 of the older discharge is not with anything, it remains a triple.

0 When multiplying to 20, it is not necessary to write, you can simply multiply by 2, but the results move to the left for 1 discharge.

More complex example: 24 x 328. More is better to make multiplier, and the smaller - multiplier: so it will be necessary to add only 2 numbers, and not 3. Although it is possible and vice versa, because From changes to places of terms or multipliers, the results do not change. So:

Here multiplication turned out more difficult. 8 x 4 \u003d 32. We recorded only 2, and 3 Keep in the mind: this triple will need to add to the result of multiplication of dozens.

Then multiplied 4 x 2 \u003d 8, yes 3 in our mind. We fold dozens, we get: 8 + 3 \u003d 11. And again in the category of dozens, we write only 1, and the second unit that will go to the category of hundreds, keep in mind, do not forget.

4 x 3 \u003d 12 and 1 in the mind - only 13. Because There are no more numbers for multiplication, this is the number and write.

Now you need to multiply the same 328 by 20 or 2 with a recording shift to 1 category left. And fold the results.

Multiplication in the column allows you to quickly give the solution of examples even with multivalued numbers. For the score you only need to know by heart the multiplication table.

How to multiply a column

As in the case of adding and subtraction in the column, when the number multiplying is written in each other. Each category in its place: units under units, dozens are under dozens, etc. At the bottom there is a horizontal trait, the answer is written under it.

Take the number 78 and 12. For a better understanding: We write 78 at the top, 12 - down. We start from the unit of the lower number, that is, with numbers 2.

First we consider 8 × 2 \u003d 16. The number turned out to be more than 10, it means, as in addition, we write the last digit (6), and the unit is kept in the mind. Now we turn to the top ten, that is, we consider 7 × 2 \u003d 14. We kept the unit in the mind, it means that we now add it to the result, it turns out 14 + 1 \u003d 15. The figure 5 is written under dozens, and 1 goes into a new category - hundreds. In other words, under the horizontal line should be written "156".

Go to the next discharge. Now our answer will be recorded differently: the last digit of the answer should be exactly under the upper dozens, that is, under the number 5. It turns out that each subsequent intermediate number is shifted to 1 discharge left.

We consider 8 × 1 \u003d 8. The figure is less than 10, we write 8 under the top five, among the "156". We consider 7 × 1 \u003d 7. Seed switches to the category of hundreds, that is, it must be written under the unit in response "156". Under the six, nothing is written, it is possible to put zero to convenience.

The resulting expression fold in the column: 156 + 78. To 6 nothing is added (0), it means that they rewrite it in former form. Then we consider 5 + 8 \u003d 13, we write 3, one in the mind. Finally, 1 + 7 \u003d 8, we add a unit - it turns out 9.

Thus, the answer is: 936.

It is better to train on a sheet into a cell to get used to the location of multipliers

Similarly, other multivalued numbers are multimed.

If there are zeros in multipliers, they do not change, but simply transferred to right part Final answer.

Options for cards

For clarity, you can print cards with examples of different levels of complexity. So children will be easier to remember the principle of the account.Examples for practice can be used in the first study of multiplication, and to repeat after vacation.

At first, the solution of examples will occupy a lot of time, but gradually the speed will increase. Even if there is a calculator, it is better to be manually assumed: it develops mental activity.

Photo Gallery: Examples of lesson cards

Video: multiplication of numbers in the column

Permanent practice is a key to success, and over time you can learn to multiply even large numbers. But to start, of course, better with simple examples, gradually increasing the level of complexity.

At school, these actions are studied from simple to complex. Therefore, it is certainly necessary to well assimilate the algorithm for the fulfillment of these operations on simple examples. So that there are no difficulties with the division of decimal fractions in the column. After all, it is very complex option Similar tasks.

This subject requires a consistent study. Spaces in knowledge are unacceptable here. Such a principle must learn every student in the first grade. Therefore, with a pass of several lessons in a row, the material will have to master on its own. Otherwise, the problems will arise not only with mathematics, but also other objects associated with it.

The second prerequisite for the successful study of mathematics is to move to examples to divide into a column only after the addition, subtraction and multiplication are mastered.

It will be difficult for a child if he did not learn the multiplication table. By the way, it is better to learn it on the Tipagora table. There is nothing superfluous, and it is absorbed by multiplication in this case.

How are natural numbers multiply in the column?

If there is a difficulty in solving examples in a division and multiplication column, then start changing the problem relying from multiplication. Since division is a reverse operation of multiplication:

1. Before multiplying two numbers, they need to carefully look at. Choose the one in which more discharges (longer), write it first. Under it to place the second. Moreover, the figures of the corresponding discharge should be under the same discharge. That is, the right figure of the first number should be above the right second.
2. Multiply the extreme right digit of the lower number for each digit of the top, starting on the right. Write down the answer below the line so that its last digit is under that which is multiplied.
3. The same repeat on another digital lower number. But the result from multiplication should be shifted to one digit to the left. At the same time, its last digit will be under the one that is multiplied.

Continue this multiplication in the column until the figures are run out in the second multiplier. Now they need to be folded. This will be the desired answer.

Algorithm multiplication in the columns of decimal fractions

First, it is supposed to imagine that there are not decimal fractions, but natural. That is, to remove commas from them and then act as described in the previous case.

The difference begins when the answer is recorded. At this point, you must count all the numbers that are standing after commas in both fractions. It is so much that they need to be counted from the end of the answer and put a comma there.

It is convenient to illustrate this algorithm for example: 0.25 x 0.33:

How to start learning a division?

Before deciding for dividing in a column, it is supposed to remember the names of the numbers that are in the example for division. The first of them (then that is divided) is divisible. The second (divided into it) is a divider. The answer is private.

After that, on a simple everyday example, explain the essence of this mathematical operation. For example, if you take 10 candies, then divide them equally between mom and dad easily. And what if you need to distribute them to parents and brother?

After that, you can get acquainted with the rules of division and master them on specific examples. First, simple, and then go to everything more complex.

Algorithm for dividing numbers in the column

First, imagine the procedure for natural numbersdivided by an unambiguous number. They will be the basis for multivalued dividers or decimal fractions. Only then it is supposed to make minor changes, but this is later:

• Before making division into a column, you need to find out where the divider and divider.
• Write a divide. To the right of it - the divider.
• Dig up to the left and below near the last corner.
• Determine incomplete divisible, that is, the number that will be minimal for division. It usually consists of one digit, a maximum of two.
• Choose a number that will be the first to be recorded in response. It should be how many times the divider is placed in division.
• Record the result from multiplying this number per divider.
• Write it under incomplete division. Perform subtraction.
• To demolish the first digit to the residue after that part that is already divided.
• To recall the number to answer again.
• Repeat multiplication and subtraction. If the residue is zero and the divisible ended, the example is made. IN otherwise Repeat actions: demolish the number, pick up the number, multiply, deduct.

How to solve division in a column if in the divider more than one number?

The algorithm itself fully coincides with what was described above. The difference will be the number of numbers in incomplete division. Their at least two must be two, but if they turn out to be less divider, then work relies with the first three numbers.

There is another nuance in this division. The fact is that the residue and the number demolished to it are sometimes not divided into a divider. Then it is supposed to attribute another digit in order. But at the same time, it is necessary to put zero in response. If the division of three-digit numbers in the column is carried out, then it may be necessary to carry more than two digits. Then the rule is introduced: noise in response should be one less than the number of demolished digits.

Consider such a division by example - 12082: 863.

• An incomplete divisible in it is the number 1208. The number 863 is placed only once. Therefore, in response, it is necessary to put 1, and under 1208 record 863.
• After subtraction, the residue is obtained 345.
• It is necessary to demolish the number 2.
• Among 3452, 863 fits four times.
• Four need to write in response. Moreover, when multiplying on 4 it turns out exactly this number.
• The residue after subtraction is zero. That is, the division is completed.

The answer in the example will be the number 14.

How to be if divisible ends on zero?

Or a few nobles? In this case, the zero residue is obtained, and in Delim, there are still zeros. It is not necessary to despair, everything is easier than it may seem. It is enough just to attribute to the answer all zeros, which remained not divided.

For example, you need to divide 400 to 5. Incomplete divisible 40. The top 8 placed in it. So, in response, it is necessary to write 8. When subtracting the residue does not remain. That is, the division is completed, but a zero remained in Delim. He will have to attribute to the answer. Thus, when dividing 400 per 5 is obtained 80.

What if you need to share a decimal fraction?

Again, this number is similar to the natural, if it were not for a comma separating the whole part of the fractional. This suggests that the division of decimal fractions in the column is similar to that described above.

The only difference will be a semicolon. It is supposed to put in response immediately as soon as the first digit of the fractional part is demolished. In a different way, this can be said like this: the division of the whole part is over - put the comma and continue the decision on.

During the solution of examples of dividing in a column with decimal fractions, it is necessary to remember that in part after the comma it is possible to attribute any number of nonols. Sometimes it is necessary in order to let the numbers to the end.

Division of two decimal fractions

It may seem complex. But only at the beginning. After all, how to make division in the column fractions on a natural number, it is already clear. So you need to reduce this example to the already familiar mind.

Make it easy. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe a million, if this requires a task. The multiplier should be chosen based on how many zoli is in the decimal part of the divider. That is, as a result, it turns out that it will have to divide on a natural number.

And it will be in the worst case. After all, it may turn out that dividable from this operation will become an integer. Then the solution of an example with division in the column fractions will be reduced to himself simple version: Operations with natural numbers.

As an example: 28.4 divide by 3.2:

• First, they must be multiplied by 10, since in the second number after the comma, there is only one digit. Multiplication will give 284 and 32.
• They should be divided. And immediately all the number 284 to 32.
• The first selected number for the answer is 8. From its multiplication it turns out 256. The residue will be 28.
• The division of the whole part is over, and in response it is necessary to put a comma.
• Demolish to the residue 0.
• Take 8 again.
• Rest: 24. To him to attribute one more 0.
• Now you need to take 7.
• The result of multiplication is 224, the residue is 16.
• To demolish another 0. Take 5 and it turns out just 160. The residue is 0.

The division is completed. The result of an example 28.4: 3.2 is 8,875.

What if the divider is 10, 100, 0.1, or 0.01?

As well as with multiplication, the division in the column is not needed here. Just just transfer the comma in needide on a certain number of numbers. Moreover, according to this principle, examples can be solved with both integers and decimal fractions.

So, if you need to divide by 10, 100 or 1,000, the comma is transferred to the left of the number of numbers as zeros in the divider. That is, when the number is divided into 100, the comma should be shifted to the left into two digits. If divisible is a natural number, then it is understood that the comma stands at its end.

This action gives the same result as if the number was needed to multiply by 0.1, 0.01 or 0.001. In these examples, the comma is also transferred to the left of the number of numbers equal to the length of the fractional part.

When divided by 0.1 (, etc.) or multiplication by 10 (, etc.), the comma should move to one digit (or two, three, depending on the number of zeros or the length of the fractional part).

It is worth noting that the number of numbers, data in the division may be insufficient. Then on the left (in the whole part) or on the right (after the comma) you can attribute the missing zeros.

Division of periodic fractions

In this case, it will not be possible to obtain an accurate answer when dividing in the column. How to solve an example if you met a fraction with a period? Here it is necessary to move to ordinary fractions. And then perform their division according to the previously studied rules.

For example, it is necessary to divide 0, (3) by 0.6. The first fraction is periodic. It is converted into a fraction 3/9, which after the reduction will give 1/3. The second fraction is the ultimate decimal. It is even easier to burn it: 6/10, which is 3/5. The rule of division of ordinary fractions prescribes to replace the division by multiplication and the divider - inverse. That is, an example is reduced to a multiplication of 1/3 to 5/3. The answer will be 5/9.

If in the example, different fractions ...

Then there are several solution options. Firstly, ordinary fraction You can try to translate into decimal. Then we already divide two decimal on the algorithm specified above.

Secondly, each finite decimal Can be recorded in the form of an ordinary. Only it is not always convenient. Most often, such fractions are huge. Yes, and the answers are cumbersome. Therefore, the first approach is considered more preferable.

Multiplication of natural numbers is convenient to carry out a special way, which was called " multiplication of column" or " multiplication in Column" The whole charm of this method is that multiplication of multivalued natural numbers is reduced to the sequential multiplication of two unambiguous numbers.

In this article we are the most in detail We will analyze the algorithm of multiplication by the column of two natural numbers. The sequence of actions will be described by step by step, at the same time showing solutions of examples.

Navigating page.

What you need to know for multiplication of natural numbers by the column?

Intermediate calculations with multiplication by the column are carried out using the multiplication table, so it is desirable to know by heart, so as not to spend time on the search for the desired result.

Sooner or later, when multiplying the column, we will face multiplying the unambiguous natural number to zero. In this case, we will use the property of multiplying a natural number to zero: a · 0 \u003d 0where a. - arbitrary natural number ..

We recommend to deal with the material of the article by adding a column. This is due to the fact that at one of the stages of multiplication in the column, it is necessary to add intermediate results (which are called incomplete works), using the principle of addition by the column.

Finally, it is desirable to recall the concept of a natural number discharge.

Recording multiplies when multiplying in a column.

Let's start with the multipliers recording rules when multiplying a column.

The second factor is written under the first multiplier so that the first to the right numbers other than the figure 0 Move each other. A horizontal line is carried out below the recorded multipliers, and the multiplication sign of the type "×" is put on the left. We give examples of correctly recording multiplied by a column. Below are the records in the number of numbers 352 and 71 , 550 and 45 002 , as well as 534 000 and 4 300 .

With the record dealt with the record.

Now you can move directly to the multiplication process of two natural numbers by the column. First, consider multiplication of a multi-valued number on an unambiguous number. After that, we will analyze the crowd of two multi-valued natural numbers.

Multiplying a multi-valued number of natural numbers per unambiguous number.

Now we give algorithm Multiplication Stage The multi-valued natural number on an unambiguous natural number. We will do this, while simultaneously describing the solution of the example.

Let us need to multiply this multi-valued natural number 45 027 on this unambiguous number 3 .

We write down the multipliers as it suggests a multiplication of a column (at the same time, an unambiguous number turns out to be extremely different from a multi-valued number).

For our example, the record will have the following form:

Now we are multiplying the value of the discharge of the units of this multi-valued number to this unambiguous number. If you get a number of less 10 , I write it under the horizontal feature in the same column in which this multiply unambiguous number is located. If we get a number 10 or number greater than 10 , then a horizontal feature is recorded by the value of the discharge of the numbers of the resulting number, and the discharge value of the dozens remember (the stored number will add to the result of multiplication in the next step, after which the stored number is removed from the memory).

That is, multiply 7 (This is the value of the discharge of the units of the first factor 45 027 ) on the 3 . Receive 21 . As 21 more 10 , then under the line write the number 1 (This is the value of the discharge of the numbers received 21 ) and remember the number 2 (this is the value of the discharge of tens of numbers 21 ). At this step, the record will take the following form:

Go to the next stage of the multiplication algorithm to the column. We perform multiplying the value of the discharge value of dozens of this multi-valued number on this unique number and to the work even add a number that is remembered in the previous step (if we remembered it). If, as a result, we get a number of less than ten, then write it under the horizontal line to the left of the number recorded there. If, as a result, we obtain the number of ten or the number of more than ten, then a horizontal feature is recorded in the discharge of the units of the obtained number, and the value of the discharge of dozens remember (it is also used in the next step).

So, multiply 2 (This is the meaning of the discharge of the top of the first factor 45 027 ) on the 3 , have 6 . To this number, add the number that is remembled at the previous step. 2 Receive 6+2=8 . As 8 less than 10 , then under the horizontal feature write the number 8 At the desired position (we do not need to memorize any number, that is, now there are no numbers in my memory). We have:

In the next step, we act likewise, but already perform multiplying the value of the discharge of hundreds of this multi-valued number on this unambiguous natural number. Add a memorized number to this work (if it is remembered); Compare the result with a number 10 ; If necessary, remember a new number, and write down the desired number under the horizontal line to the left of the numbers already located there.

Multiply 0 on the 3 Receive 0 . Since we do not have any number in memory, then to the resulting number 0 No need to add anything. Number 0 less 10 , so recording 0 Under the horizontal line in the desired position:

After that, we turn to the multiplication of the value of the next discharge of this multi-valued natural number and this unambiguous natural number. Similarly, we act in the same time until you multiply the values \u200b\u200bof all the discharges of this multi-valued number on this unambiguous natural number.

So, multiply 5 on the 3 Receive 15 . As 15>10 , then write under the line 5 And remember the number 1 :

Finally, multiply 4 on the 3 Receive 12 . TO 12 add a number stored at the previous stage 1 , have 12+1=13 . As 13 more than 10 then write the number 3 on the right place And remember the number 1 :

Note that if at the last stage we had to remember the number, then it should be recorded under the horizontal line to the left of the numbers already located there.

We have a number in memory 1 So it is necessary to write it in the right place below the line:

On this process, the multiplication process of a multi-valued number by a single-valued natural number is ends, and the multiplication result is the number recorded under the horizontal line.

Thus, multiplication of a column of natural numbers 45 027 and 3 led us to the result 135 081 .

For clarity, it is schematically shown by the multiplication algorithm for a multi-valued number by a single-valued natural number (this figure reflects only the overall picture, but does not show all the nuances).

It remains to deal with the multiplication of a crowd of a multi-valued natural number, in which the number on the right is on the right 0 or several digits 0 In a row, on an unambiguous number. We will also consider all the steps of multiplication by the column in such cases on the example. And take the numbers from the previous example, but in the recording of a multi-valued number, I add several digits 0 on right.

So, multiply natural numbers 4 502 700 (We completed two digits 0 ) by number 3 .

In this case, first write down the multiply numbers as it implies a multiplication by a column:

After that, we spend the multiplication of the column as if the numbers 0 There is no right.

We use the result from the example already solved above:

At the final stage of multiplication by a column under a horizontal line, to the right of the already existing numbers there, write as many numbers 0 how many of them are on the right in the initial multipatal number.

In our example, you need to add two numbers 0 . The record will take the form:

On this multiplication of the column completed.

The result of multiplying multi-valued natural numbers 4 502 700 , the record of which ends with zeros, to the unambiguous natural number 3 is an 13 508 100 .

Multiplying in the column of two multi-valued natural numbers.

We describe all the stages of the multiplication algorithm of two multi-valued natural numbers by the column.

Description We will conduct together with the solution of the example. Now we assume that in the records of multiplying natural numbers, the numbers are not numbered on the right 0 . Multiplying multi-valued natural numbers whose records end with zeros, consider at the end of this item.

Multiply a column of numbers 207 on the 8 063 .

We start with the record of multipliers in each other. Note that it is more convenient to place a multiplier, the record of which consists of more signs (in our example we will install the number 8 603 because in his record 4 sign and number 207 Three digit). If multiplier records contain the same number of characters, it does not matter which factors write from above. So, we have multipliers to each other so that the numbers of the first factor are under the numbers of the second factor on the right to left:

Now at each next step we will get the so-called incomplete works.

The first stage of the algorithm is the multiplication of the first factor by the column (in our example this number 8 063 ) to the value of the discharge of the units of the second factory (in our example, the value of the discharge of the number of numbers 207 is the number 7 ). All actions are similar to the multiplication of a multi-valued number on an unambiguous number (if necessary, return to the previous paragraph of this article), as a result, under the horizontal line we have the first incomplete work. At this stage, the record will take the following form:

Go to the second stage. At this stage, we multiply the first factor (in our example this number 8 063 ) The value of the discharge of tens of a second multiplier, if it is not zero. If the value of the discharge of tens of second factors is zero, then we turn to the next step (in our example, the discharge value of the tens of numbers 207 Equally zero, therefore, we turn to the third stage). The results are recorded under the line below already recorded there, starting from a position that corresponds to the discharge of tens.

In the third, fourth, and so on, we act in the same way, multiplying the first factor (number 8 063 ) To the value of the discharge of hundreds of the second multiplier (if it is not zero), further to the value of the discharge of thousands (if it is not zero) and so on. The results are recorded under the line below the numbers already recorded there, starting from the position that meets the discharge of the unique number to which multiplication is carried out at this stage.

So, multiply the number 8 063 on the value of the discharge of hundreds of numbers 207 , that is, the number 2 . We receive the second incomplete work, and the solution of the example will take the following form:

So, all incomplete works are calculated. Remains final stage The algorithm on which all incomplete works are folded, and it is done in the same way as when adding to the column. The addition is made using the already available record (incomplete works remain in the places where they are recorded, that is, they are not shifted anywhere), it takes another horizontal line to the bottom, the "+" sign is installed, and the results of the addition are recorded under the bottom line . If only one number is located in the column, and at the same time there is no number that is remerected in the previous step, it is written under the horizontal line.

In our example we get:

The number formed below is the result of multiplying the initial multivalued natural numbers. So, the product of numbers 8 063 and 207 equally 1 669 041 .

For clarity, schematically depict the process of multiplication by a column of two natural numbers.

We show the solution of another example to secure the material.

Guys, let's repeat what is an unambiguous, double-digit and three-digit number.

Unambiguous - This is a number for which one sign is needed.
For example: 1, 3, 5, 4, ...
Probably you already guessed that the numbers are unambiguous, when they are recorded as a number. They consist of units.

Two-digit number - This is the number for which two signs need two signs. For example, all numbers from 10 to 99 are double-digit numbers. They consist of dozens and units.

When do children begin to break the numbers?

The separation is carried out at key stage 1 so that the children know that the two-digit number consists of dozens and units. The idea is that the child connects the arrows together to match the numbers. These are two frequently used methods for adding large numbers.

A teacher can start teaching children to add double-digit and three-digit numbers per year 3 by dividing the sections. The reason for this is that it helps children mentally add a multiple ten and multiple 100. Children in the 3rd year should also add to add three-digit numbers with, so your child will probably face both of these methods.

Three digit number - This is the number for which three characters are needed. You have already guessed that all numbers from 100 to 999 are three-digit. They contain units, tens and hundreds.
Guys, answer the question: how many three digits are there?

Let's look at the example, how to perform the operation of multiplying a multi-valued number per unambiguous number.

First of all, remember the rule of multiplication to zero and one.
This rule says:
Number * 0 \u003d 0
Number * 1 \u003d number

Separation in multiplication

Children of the 3rd year also need to multiply double-digit numbers per one-digit number. They are usually trained to this partition, for example. As soon as the teachers are very confident that the child knows how to multiply a multiple ten and a hundred, they will often allow the child to switch to a faster column method.

In the 6th year, children should start computing. To make it easier, the teacher can show them how to divide decimal numbers. It is read as four times six are twenty four or just four times six - twenty-four. Multiplication knowledge is very important. So, if you are weak in multiplication, you must try to achieve the level of ownership of the following "Time Table".

Examples.
5 * 0 = 0;
18 * 0 = 0;
4506 * 0 = 0
1 * 34 = 34;
2384 * 1 = 2384;
1 * 47586 = 47586

For multiplying multivalued numbers, multiplication method is often used by a column, which we will apply in our examples.

Multiply multivalued number for a number other than 0 or 1.
Consider examples.
Take the numbers 348 and 4. For our convenience, we write them into the column. Let's start multiplying from the extreme right column and change the number 4 and 8. We obtain the number 32. The number 2 is recorded strictly under numbers 8 and 4. And the number 30 is translated into the next discharge (a discharge of a tenth). When transferring the number to an older discharge, for example, from units in tens, this number loses 0. Now we multiply 4 and 4 and we get 16. I will add 3 from the previous multiplication. As a result, we obtain 19. The number 9 is writing under Number 4 (left number 2), and 1 translate into the next discharge (the discharge of hundreds). Then, then change the number 3 and 4 and add 1 from the previous action to the result. As a result, we get 13. We write it completely, because This is our last action. As a result, we get the product of numbers 348 to 4, which is equal to 1392.

Multiplying large numbers

Your confidence and ability to learn mathematics will largely depend on your knowledge of reproduction. So, you should strive to cope with the above "Time Table".

• The product is the result of multiplying two numbers.
• To calculate 8 hours 9, we recall the "Table eight times".

To multiply big number To another number, we can use short multiplication or long-lasting multiplication.

To multiply a large number on a single-digit number, enter the numbers vertically, and the larger number will be multiplied by a smaller number. To calculate 89 hours 7, install it vertically with a smaller number placed under a large number, as shown below. Now, calculate 7 hours 8 and add 6 to get it written, as shown below.

Examples of multiplication of a multi-valued number on a two-digit number

In this example, consider multiplication of a three-digit number on a double-digit. Take numbers 925 and 38.
The whole process multiplication is divided into several parts.
The first part is the multiplication of the number 925 to the number 8. For convenience, we write them into the column.
As usual, when you multiply in the column, we will begin your actions from the extreme right column. There are recorded numbers 5 and 8, moving, which we obtain the number 40. Record the number 0 under Number 5 and 8. Do not forget 40 transfer to the next discharge (discharge of a dozen). Now I turn the numbers 2 and 8. We get 16. Do not forget to add a number 4, which remained after the previous action (with multiplication 8 and 5). We obtain the number 20. The number 0 is recorded at a number of 3 next to the previous number 0, and 20 are transferred to the next discharge (the discharge of hundreds). And the last action of the first part is to multiply the numbers 9 and 8. The product of these numbers is 72. We will add the number 2 to the product and get the number 74. We will write it completely.
The second part is the multiplication of the number 925 to the number 3. We will not consider this part as in detail as the previous one, but simply write the result of the product of these numbers. When writing a piece of numbers of the second part, you need to remember that the recording should be started not from the extreme right column, but with a displacement by one. In our example, the first number must be recorded strictly under numbers 2, 3.0. See the picture.
The third part is to obtain the amount of numbers. it the final stagewhere we need to get the amount from the first work - 7400 and from the second work - 2775. We summarize, complying with the rules that are used when adding to the column. The last figure shows the result of multiplication of a two-digit number 38 per three-digit number 925.

The most important rule from which we are starting to learn multiplication in the column are:

We often set out the solution as follows. Multiplication 38 to 60 is faster than multiplication by 60 to 38, since 60 contains zero. Multiplication by 385 by 500 faster than multiplication by 500 to 385, since 500 contains two zero. To multiply two large numbers, write numbers by vertical, and a larger number will be multiplied by a smaller number, which is called the multiplier. We use the table times to find a product of a larger number with each digit in the multiplier by adding results. For example, if the multiply digit is in the hundreds column, add two zeros for a column of tens and columns of units.

• So, place 3 in the units column and carry 6.
• Then calculate 7 hours 8 and add 6 to get 62.
• In the column of units is placed zero.
• Then we calculate 6 hours 38, as shown above.
• In the column of units is placed zero, as well as a column of dozens.
• Then we calculate 5 hours 385, as shown above.
• Do not forget to add zero for each location value after the multiplying digit.
• To multiply 269 to 78, place 78 below.
• Then we calculate 8 hours 269 and 70 hours 269, as shown above.

This is known as a commutative law for multiplication.

Multiplication in the column on a two-digit number

Example: 46 Multiply by 73

Under Number 46 Record number 73 by rule:

Units write under units, and dozens are under dozens

1 Multiply begin with units.

3 Multiply by 6. It will turn out 18.

• 18 units are 1 tens and 8 units.
• 8 units we write under units, and 1 dozen remember and add to dozens.

Now 3 multiply by 4 dozen. It turns out 12.

Label number 1: Room quadrataging in the 50s

Anyone can clearly understand in mathematics with the labels of the T-shirt of the Bister. Now, if the number from step 2 is less than 10, you must put zero in front of it.

Label 2: multiplication of two numbers in the 90s together

When you multiply two numbers in the 90s together, in parentheses next to each number indicated how far this number is far from.

Multiply a three-digit number for two-digit

This is one of my favorite tricks, because he is simple and striking anyone who sees him. Ask someone to choose two numbers below 10 and write one on top of another. Ask a person adding them and put the answer directly under two numbers. Ask a person to continue adding the lower two numbers to the column and continue the summation of the total amount until you have only ten numbers. Then add it the entire column. Example: someone chooses numbers 4 and 7 and writes 4 tops. The next number in the series will be because 4 7 \u003d then adding the lower two numbers to the column, the following number will be 18, because 7 11 \u003d it must continue to do this until it has only ten numbers, and then he will add all Column.

12 dozen, yes more than 1, only 13 dozen.

There are no hundreds in this example, so it's right at the place of hundreds we write 1.

138 - that's the first incomplete work.

2 Multiply dozens.

7 dozen multiply by 6 units will be 42 dozen.

• 42 dozen are 4 hundred and 2 dozen.
• 2 dozen drink under dozens. 4 We remember and add to the hundreds.

7 dozen multiply by 4 dozen will be 28 hundred. 28 hundred, and 4 more will get 32 \u200b\u200bhundred.

The column may look like this. You quickly take a look at the numbers and tell him that all ten numbers add. All you need to do is look at 76 and add dozens of digit to it, 76 7 \u003d Then impose on one digit 76 to the end. If a person has chosen two large numbers, such as 8 and 9, the seventh number can be a three-digit number. The column will look like that.

What errors with multiplication can be done and how to avoid them

The seventh number in this case. Here we will look at how to multiply double-digit numbers. At first, the method was used to be called the direct method of Yakov-Tractenberg, and the second one - the "two fingers" method. Both of these methods will work for any combinations of two-digit numbers.

• 32 hundreds are 3 thousand and 2 hundred.
• 2 hundred we write under hundreds, and 3 thousand remember and add to thousands.

There are no thousand in this example, so I write 3 at once in place.

3220 - that's the second incomplete work.

3 We fold the first and second incomplete works according to the rule of addition in the column.

138 Plus 3220 will be 3358.

If you are interested in multiplying the numbers to twelve, take a look at them. The direct method is rarely taught in schools, but is known for centuries. In school, you usually teach to record the result of multiplying each multiplier number to a separate line, and then summation of the total amount.

Multiplication of a multi-valued number into multivalued

Instead, you write only the answer. To do this, you make a couple of computing at every step. Couples that equal to anything are ignored. These pairs are called external and inner pairs. The outer pair always connects the unit number of the multiplier with the number in which we are now looking. The inner pair always connects dozens of digits with the digit to the right of the figure, which we work in the multiplier.

We read the answer: 46 Multiply to 73 will succeed in 3358

(Click on the picture)

Components of the action of multiplication

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Sample reasoning
During recording
multiplication in Column

Division of periodic fractions

This method is essentially the same as in Vedic mathematics when they use the "vertical and transverse" sutra with multiplication of two-digit numbers. The style of the equation is the only real difference. In the Vedic Mathematics, the equation is written on two lines, as shown below. For direct method The equation is on the same line with the response under the animation.

You can watch a video about direct multiplication using two-digit multipliers or continue reading the following examples. The number of initial zeros always coincides with the number of numbers in the multiplier, so when multiplying into 2-digit numbers, we always add 2 senior zero. Next: We multiply two single numbers together.

Carefully review and apply in your actions!

What errors with multiplication
You can make I.
How to avoid them

Look carefully

in order not to make mistakes!

Rules for other cases of multiplication

Multiplication in the column for a single number

This step includes multiplication of tens of numbers of one number to the number of other units. When writing an equation on one line, if we draw curved connecting lines between multiplied numbers, we get an outer pair and an inner pair. When writing an equation on two lines, we get a cross when we draw straight connecting lines between multiplied numbers.

Multiplying in the column of two multi-valued natural numbers

By adding the results of these two equations, we obtain 14, so we write 4 and transferred. At this step, we multiply dozens of numbers of each number. When writing an equation on one line, the outer steam on this step is connected to zero, so the result of this pair is zero and can be ignored. In this example, mental calculations that we need to do is relatively simple, and since we do less steps than traditional method Multiplication, it happens faster. However, there is a lack of this approach, especially when the participating numbers are greater.

This example can be recorded in the column.

Under Number 34 Record number 2 by rule:

Under the number 68, you write the number 2 by rule:

We multiply two single numbers together. So, we write 2 and carry. This is where it becomes tough, especially if you are trying to mentally calculate the calculation. So, we write 4 and carry. We have 63, to which we add transfer 14 to give us. We write 7 and carry.

How to multiply in Column: Basic Rules

Follow original method And the cause of leading zeros, we have an extra step due to the transfer. So, we have zero plus transfer 7, which we write 7, which gives us our answer. This step may seem unnecessary, and we could simply write the transfer in the last step, but as the method studies it is better to follow the entire equation until you are familiar with the method to take small labels.

Units write under units, and dozens, if they are under dozens

1 Multiply begin with units.

2 Multiply on 8. It will be 16.

• 16 units are 1 tens and 6 units.
• 6 units We write under units. And 1 dozen we remember and add to dozens.

Now 2 multiply by 6 dozen. It turns out 12.

12 dozen yes another 1 is only 13 dozen.

As you can see, when the numbers contain numbers 7, 8 and 9, mathematics becomes more complex, especially if you are trying to make it mentally. Yakov also understood this, and he put himself a task to find a simpler way to achieve this. Enter the "two fingers" method, as he called it, which simplifies the calculations that you need to perform. Before moving to the method with two fingers, we need to get an additional reference information For one-time multiplication.

Examples of multiplying multi-valued numbers per unambiguous number

When multiplying two digits per digit, the result can be only one or two digits. If we put zero before the result of any digit, we can process all the results of multiplying two numbers with one digit in the form of two-digit results, numbers of units and dozens of digits.

• 13 dozen is 1 hundred and another 3 dozen.
• 3 dozen writing under dozens. And 1 hundred we remember and add to the hundreds.

Hundreds in this example is not, so at once in the place of hundreds write 1.

We read the answer: 68 Multiply to 2 will be 136.