Let's take a look at how you can multiply two-digit numbers using the traditional methods we are taught in school. Some of these methods can allow you to quickly multiply double-digit numbers in your head with enough practice. It is helpful to know these methods. However, it is important to understand that this is just the tip of the iceberg. This lesson discusses the most popular techniques for multiplying two-digit numbers.
The first method is a layout into tens and units.
The easiest way to understand how to multiply two-digit numbers is the one that we were taught in school. It consists in dividing both factors into tens and ones, followed by multiplying the resulting four numbers. This method is quite simple, but it requires the ability to keep in memory up to three numbers at the same time and at the same time perform arithmetic operations in parallel.
For example: 63 * 85 = (60 + 3) * (80 + 5) = 60 * 80 + 60 * 5 + 3 * 80 + 3 * 5 = 4800 + 300 + 240 + 15 = 5355
Such examples are easier to solve in 3 steps. First, tens are multiplied by each other. Then 2 products of units by tens are added. Then the product of units is added. It can be schematically described as follows:
- First action: 60 * 80 = 4800 - remember
- Second action: 60 * 5 + 3 * 80 = 540 - remember
- Third action: (4800 + 540) + 3 * 5 = 5355 - answer
For the fastest effect, you will need a good knowledge of the multiplication table for numbers up to 10, the ability to add numbers (up to three-digit), as well as the ability to quickly switch attention from one action to another, keeping the previous result in mind. It is convenient to train the last skill by visualizing the arithmetic operations performed, when you have to imagine a picture of your solution, as well as intermediate results.
Output. It is not difficult to make sure that this method is not the most effective, that is, it allows you to get the right result with the least actions. Other methods should be taken into account.
Second way - arithmetic fits
Reducing an example to a convenient form is a fairly common way of counting in your head. Customizing an example is useful when you need to quickly find an approximate or precise answer. The desire to fit examples to certain mathematical patterns is often brought up in mathematics departments at universities or in schools in classes with a mathematical bias. People are taught to find simple and convenient algorithms for solving various problems. Here are some examples of fit:
Example 49 * 49 can be solved like this: (49 * 100) / 2-49. First, it is considered 49 per hundred - 4900. Then 4900 is divided by 2, which equals 2450, then 49 is subtracted. Total is 2401.
The product 56 * 92 is solved as follows: 56 * 100-56 * 2 * 2 * 2. It turns out: 56 * 2 = 112 * 2 = 224 * 2 = 448. Subtract 448 from 5600 to get 5152.
This method can be more effective than the previous one only if you own an oral account based on the multiplication of two-digit numbers by single-digit numbers and can keep several results in mind at the same time. In addition, you have to spend time looking for a solution algorithm, and it also takes a lot of attention for the correct observance of this algorithm.
Output. The way when you try to multiply 2 numbers, decomposing them into simpler arithmetic procedures, trains your brains perfectly, but it is associated with large mental costs, and the risk of getting the wrong result is higher than with the first method.
The third way - mental visualization of multiplication in a column
56 * 67 - count in a column.
Probably, column counting contains the maximum number of actions and requires constantly keeping auxiliary numbers in mind. But it can be simplified. In the second lesson, you learned that it is important to be able to quickly multiply single-digit numbers by two-digit numbers. If you already know how to do it on the machine, then the column in the mind for you will not be so difficult. The algorithm is as follows
First action: 56 * 7 = 350 + 42 = 392 - remember and do not forget until the third action.
Second action: 56 * 6 = 300 + 36 = 336 (well, or 392-56)
Third action: 336 * 10 + 392 = 3360 + 392 = 3,752 - this is more complicated, but you can start to call the first number in which you are sure - "three thousand ...", but while you say, add 360 and 392.
Output: counting in a column is directly difficult, but you can simplify it if you have the skill of quickly multiplying two-digit numbers by single-digit numbers. Add this method to your arsenal as well. In a simplified form, column counting is a modification of the first method. Which is better is an amateur question.
As you can see, none of the methods described above allows you to count in your head quickly and accurately all examples of multiplication of two-digit numbers. You need to understand that using traditional methods of multiplication for mental counting is not always rational, that is, allowing you to achieve the maximum result with the least effort.
Don't like math? You just don't know how to use it! In fact, it's a fascinating science. And our selection of unusual multiplication methods confirms this.
Multiply on your fingers like a merchant
This method allows you to multiply numbers from 6 to 9... First, bend both hands into fists. Then on the left hand, bend as many fingers as the first factor is greater than 5. On the right hand, do the same for the second factor. Count the number of fingers extended and multiply by ten. Now multiply the sum of the folded fingers of the left and right hand. Adding both sums together, you get the result.
Example. Multiply 6 by 7. Six is more than five by one, which means we bend one finger on the left hand. And seven - for two, which means on the right - two fingers. In total - this is three, and after multiplying by 10 - 30. Now we multiply the four bent fingers of the left hand and three - the right. We get 12. The sum of 30 and 12 gives 42.
Actually, here we are talking about a simple multiplication table, which it would be good to know by heart. But this method is good for self-test, and it is useful to stretch your fingers.
Multiply like Ferrol
This method was named after the name of the German engineer who used it. Method allows you to quickly multiply numbers from 10 to 20... If you practice, you can even do it in your mind.
The essence is simple. As a result, you will always get a three-digit number. So first we count units, then tens, then hundreds.
Example. We multiply 17 by 16. To get units, we multiply 7 by 6, tens - we add the product of 1 and 6 with the product of 7 and 1, hundreds - we multiply 1 by 1. As a result, we get 42, 13 and 1. For convenience, we write them in a column and add up. That's the bottom line!
Multiply like a Japanese
This graphic way, which Japanese schoolchildren use, makes it easy to multiply two- and even three-digit numbers. To try it out, have some paper and a pen ready.
Example. Let's multiply 32 by 143. To do this, let's draw a grid: the first number will be reflected with three and two lines with an indent horizontally, and the second - with one, four and three lines vertically. Put points at the intersection of the lines. As a result, we should get a four-digit number, so let's conditionally divide the table into 4 sectors. And we will count the points that fall into each of them. We get 3, 14, 17 and 6. To get the answer, add the extra ones at 14 and 17 to the previous number. We get 4, 5 and 76 - 4576.
Multiply like an Italian
Another interesting graphic method is used in Italy. Perhaps it is simpler than the Japanese one: you definitely won't get confused when transferring tens. To multiply large numbers with it, you need to draw a grid... We write down the first factor horizontally at the top, and the second one vertically on the right. In this case, there must be one cell for each digit.
Now we multiply the numbers in each row by the numbers in each column. We write the result into a cell (divided in two) at their intersection. If you get a single-digit number, then write 0 to the top of the cell, and the result to the bottom.
It remains to add up all the numbers in the diagonal stripes. We start from the bottom right cell. At the same time, we add tens to the units in the adjacent column.
This is how we multiplied 639 by 12.
Fun, isn't it? No boring mathematics! And remember that humanities in IT are also needed!
Multiplying large numbers by writing them to a string sooner or later becomes a rather complicated and tedious process. It is much easier to use a special algorithm for long multiplication: you do not have to keep numbers in your head and memorize anything. You can mark over the column so you can always see how the numbers need to be transferred. If you are trying to teach a child this way, then it is very important that the multiplication table bounces off his teeth, otherwise, the process will drag on for a long time, and the baby himself will make many mistakes, which will stretch along the whole example in a string. Read the article carefully and take such an algorithm into your arsenal.
Write an example on a line and see which factor is smaller? The smaller one will be lower in the column multiplication notation, and the large factor will be at the top.Write down an example in the same way as shown in the picture below.
- Write a larger number at the top.
- On the left, put the multiplication sign in the form of a cross.
- Write down the lower number below.
- Draw a straight line under the example.
- Zeros should be taken as an example.
- Write the numbers under the numbers.
In this case, you simply carry that number of zeros immediately into the response. If both the first factor and the second have zeros, then add up their number and write in the answer.
- You multiply the entire top number by the last digit of the bottom one. Remember that there is no multiplication by the last zeros.
- To make it easier for you, write down the numbers to be carried over above the entire example. Later, you can simply erase them, but in the process you do not have to remember the transfer numbers.
- Once you finish calculating, write down the resulting number below the line.
Once you multiply the top number by the last digit of the bottom one and write down your answer, start multiplying the next one.
As you might have guessed, you need to multiply the top number by all the numbers on the bottom, starting from the end. Each time the record of the answer is moved one cell to the left.
Multiply all the numbers with each other in this way. Now draw a line under the column again. Place an addition sign between all solutions.
- Add all the numbers on the same vertical line.
- If the number is two-digit, then you transfer the number of tens to the next vertical strip.
Under some numbers there will not be others at all - in that case, you simply write this number in response. Do not forget to carry all zeros at the end of the multipliers in your answer.
Long multiplication is very convenient and fast, especially if you need to multiply large numbers. You can easily check that the multiplication is correct by simply dividing the answer by one of the factors. To do this, use a calculator, or the method of dividing with a corner. At first, such a multiplication takes a significant fraction of the time, but with experience, the whole action takes just a couple of seconds.
In school, these actions are studied from simple to complex. Therefore, it is imperative that you learn well the algorithm for performing these operations using simple examples. So that later there are no difficulties with dividing decimal fractions in a column. After all, this is the most difficult version of such tasks.
This subject requires consistent study. Knowledge gaps are unacceptable here. This principle should be learned by every student already in the first grade. Therefore, if you skip several lessons in a row, you will have to master the material yourself. Otherwise, later there will be problems not only with mathematics, but also with other subjects related to it.
The second prerequisite for successful study of mathematics is to switch to long division examples only after you have mastered addition, subtraction and multiplication.
It will be difficult for a child to divide if he has not learned the multiplication table. By the way, it is better to learn it according to the Pythagorean table. There is nothing superfluous, and multiplication is assimilated in this case easier.
How are natural numbers multiplied in a column?
If there is a difficulty in solving examples in a column for division and multiplication, then you should start to fix the problem with multiplication. Since division is the inverse of multiplication:
- Before you multiply two numbers, you need to look at them carefully. Choose the one with more digits (longer), write it down first. Place the second under it. Moreover, the numbers of the corresponding category should be under the same category. That is, the rightmost digit of the first number should be above the rightmost digit of the second.
- Multiply the rightmost digit of the bottom number by each digit of the top, starting from the right. Write the answer under the line so that its last digit is under the one multiplied by.
- Repeat the same with the other digit of the lower number. But the result from the multiplication must be shifted one digit to the left. In this case, its last digit will be under the one by which it was multiplied.
Continue this multiplication in a column until the numbers in the second multiplier run out. Now they need to be folded. This will be the desired answer.
Algorithm for multiplication in a column of decimal fractions
First, it is supposed to imagine that not decimal fractions are given, but natural ones. That is, remove commas from them and then proceed as described in the previous case.
The difference begins when the answer is recorded. At this moment, it is necessary to count all the numbers that come after the commas in both fractions. That is how many of them you need to count from the end of the answer and put a comma there.
It is convenient to illustrate this algorithm with an example: 0.25 x 0.33:
Where to start learning division?
Before solving the long division examples, it is necessary to remember the names of the numbers that stand in the division example. The first of these (the one that is divided) is the dividend. The second (divided by) is the divisor. The answer is private.
After that, using a simple everyday example, we will explain the essence of this mathematical operation. For example, if you take 10 candies, then it is easy to divide them equally between mom and dad. But what if you need to distribute them to parents and brother?
After that, you can get acquainted with the division rules and master them with specific examples. First, simple, and then move on to more and more complex.
Algorithm for dividing numbers into a column
First, we present the procedure for natural numbers divisible by a single digit. They will also be the basis for multi-digit divisors or decimal fractions. Only then is it supposed to make small changes, but more on that later:
- Before doing long division, you need to figure out where the dividend and the divisor are.
- Write down the dividend. To the right of it is the divider.
- Draw a corner to the left and below near the last.
- Determine the incomplete dividend, that is, the number that will be the minimum for division. It usually consists of one digit, maximum two.
- Choose the number that will be the first to be written in the answer. It should be the number of times the divisor fits in the dividend.
- Write the result from multiplying this number by the divisor.
- Write it under an incomplete dividend. Subtract.
- Remove to the remainder the first digit after the part that has already been divided.
- Pick up the number for the answer again.
- Repeat multiplication and subtraction. If the remainder is zero and the dividend is over, then the example is done. Otherwise, repeat the steps: demolish a digit, pick up a number, multiply, subtract.
How to solve long division if there is more than one digit in the divisor?
The algorithm itself completely coincides with what was described above. The difference will be the number of digits in the incomplete dividend. Now there should be at least two of them, but if they turn out to be less than the divisor, then it is supposed to work with the first three digits.
There is one more nuance in this division. The fact is that the remainder and the digit taken down to it are sometimes not divisible by the divisor. Then it is supposed to assign one more figure in order. But at the same time, you must put zero in the answer. If you are dividing three-digit numbers into a column, then it may be necessary to demolish more than two digits. Then a rule is introduced: there should be one less zeros in the answer than the number of digits removed.
You can consider such a division using the example - 12082: 863.
- The incomplete divisible in it turns out to be the number 1208. The number 863 is placed in it only once. Therefore, in response, it is supposed to put 1, and under 1208, write 863.
- Subtraction gives a remainder of 345.
- To him you need to demolish the number 2.
- Of the 3452, 863 fits four times.
- A four must be written in response. Moreover, when multiplied by 4, this is the number obtained.
- The remainder after subtraction is zero. That is, the division is over.
The answer in the example will be the number 14.
What if the dividend ends in zero?
Or a few zeros? In this case, a zero remainder is obtained, and there are still zeros in the dividend. You should not despair, everything is easier than it might seem. It is enough to simply assign all the zeros that were not separated to the answer.
For example, you need to divide 400 by 5. Incomplete dividend 40. Five is placed in it 8 times. This means that the answer is supposed to write 8. When subtracting the remainder, there is no remainder. That is, the division is complete, but zero remains in the dividend. It will have to be attributed to the answer. So when you divide 400 by 5, you get 80.
What if you need a decimal to divide?
Again, this number looks like a natural number, if not for the comma separating the integer part from the fractional part. This suggests that long divisions are similar to the one described above.
The only difference is the semicolon. It is supposed to be answered as soon as the first digit from the fractional part is taken down. In another way, it can be said this way: the division of the whole part has ended - put a comma and continue the solution further.
When solving examples for long division with decimal fractions, you need to remember that in the part after the decimal point, you can assign any number of zeros. Sometimes this is needed in order to complete the numbers to the end.
Division of two decimal fractions
It may sound complicated. But only at the beginning. After all, it is already clear how to perform column division of fractions by a natural number. Hence, it is necessary to reduce this example to the already familiar form.
This is easy to do. You need to multiply both fractions by 10, 100, 1,000 or 10,000, and maybe by a million, if the task requires it. The factor is supposed to be chosen based on how many zeros are in the decimal part of the divisor. That is, as a result, it turns out that the fraction will have to be divided by a natural number.
And this will be the worst case. After all, it may happen that the dividend from this operation becomes an integer. Then the solution of the example with column division of fractions will be reduced to the simplest option: operations with natural numbers.
As an example, divide 28.4 by 3.2:
- First, they must be multiplied by 10, since there is only one digit in the second number after the decimal point. Multiplication will give 284 and 32.
- They are supposed to be separated. Moreover, the whole number is 284 by 32 at once.
- The first matched number for the answer is 8. It multiplies 256. The remainder is 28.
- The division of the whole part is over, and in response it is supposed to put a comma.
- Carry out to remainder 0.
- Take 8 again.
- Remainder: 24. Add one more 0 to it.
- Now you need to take 7.
- The result of the multiplication is 224, the remainder is 16.
- Take down another 0. Take 5 each and you get exactly 160. The remainder is 0.
The division is over. The result of example 28.4: 3.2 is 8.875.
What if the divisor is 10, 100, 0.1, or 0.01?
As with multiplication, long division is not needed here. It is enough just to move the comma in the desired direction by a certain number of digits. Moreover, according to this principle, you can solve examples with both whole numbers and decimal fractions.
So, if you need to divide by 10, 100 or 1,000, then the comma is shifted to the left by as many digits as there are zeros in the divisor. That is, when a number is divisible by 100, the comma must move two digits to the left. If the dividend is a natural number, then it is assumed that the comma is at its end.
This action gives the same result as if the number were to be multiplied by 0.1, 0.01, or 0.001. In these examples, the comma is also wrapped to the left by the number of digits equal to the length of the fractional part.
When dividing by 0.1 (etc.) or multiplying by 10 (etc.), the comma must move to the right 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 digits given in the dividend may be insufficient. Then, to the left (in the integer part) or to the right (after the decimal point), you can assign the missing zeros.
Division of periodic fractions
In this case, you will not be able to get an exact answer with long division. How to solve an example if a fraction with a period is encountered? Here we are supposed to go over to ordinary fractions. And then perform their division according to the previously learned rules.
For example, you need to divide 0, (3) by 0.6. The first fraction is periodic. It is converted to 3/9, which, when canceled, will give 1/3. The second fraction is the final decimal. It is even easier to write it down as an ordinary one: 6/10, which is equal to 3/5. The division rule for ordinary fractions prescribes to replace division by multiplication and divisor - by its reciprocal. That is, the example boils down to multiplying 1/3 by 5/3. The answer is 5/9.
If the example has different fractions ...
Then several solutions are possible. First, you can try to convert an ordinary fraction to decimal. Then divide two decimal places according to the above algorithm.
Secondly, each final decimal fraction can be written in the form of an ordinary one. Only it is not always convenient. Most often, these fractions are huge. And the answers are cumbersome. Therefore, the first approach is considered more preferable.
>> Lesson 13. Multiplication by a three-digit number