How to count large numbers without a calculator. Quick Mental Counting: A Teaching Method

How to learn to quickly count in your mind? It's not as difficult as many people think. You don't need to be a mathematical genius to do this. It is enough to learn simple rules and mental counting methods to significantly increase the speed of calculations.

1 When adding multivalued terms, add the most significant digit of the smaller number, then the least significant digit. For example, when adding a two-digit number, tens are added first, then ones. When adding two-digit numbers, first add all the tens, then all the ones, then add the ones to the total number of tens.

2 When subtracting multi-digit numbers, first subtract the most significant digits of the subtrahend, then its least significant digits. To learn how to quickly count in your mind, you need to remember that if the value being subtracted is close in value to a round number, then you first need to subtract this round number, and then make a correction.

3 When multiplying by a number that is represented by a unit with zeros, for example, 10 or 100, you need to assign as many zeros to the multiplied number as the multiplier has. When dividing by a number that is represented by a unit with zeros, you need to separate with a comma as many last digits as the divisor has zeros.

4 To learn how to quickly count in your mind, you need to remember that when multiplying a number by 4, you must first multiply it by 2, then again by 2. For example, 214x4=428x2=856. When dividing by 4, first divide the number by 2, then again by 2. For example, 116:4=58:2=29.

5 When dividing by 8 or 16, you need to consecutively divide the number by 2 3 or 4 times. For example, 448:8=224:4=112:2=56.

6 When multiplying by 25, multiply the number by 100 and divide by 4. When dividing by 25, multiply the number by 4 (2 times 2) and divide by 100.

7 When multiplying a number by 50, multiply the number by 100 and divide in half; when dividing a number by 50, double the number first, then divide by 100.

8 When multiplying any number by 9 or 11, increase it by 10 times, then subtract the given number from the resulting number. For example, we multiply 87 by 11: increasing 87 by 10 times, we get 870, we add 87 to this number, we get 957.

More methods:
Cunning tricks of the account in the mind

Multiplication of numbers from 10 to 20

To one of the numbers we add the number of units of the other, multiply the sum by 10 and add the product of units of numbers.

For example:

15 x 17 = (15 + 7) x 10 + 5 x 7 = 220 + 35 = 255

Note. Do not believe? Get a calculator and check it out. I have no cheating. But in the case of, for example, 98 x 12, this rule no longer works, because 98 is more than 20.
Squaring numbers ending in 5

A number ending in 5 is squared like this: 100 x (number of tens of a number) x (number of tens + 1) + 25.

For example:

Let's square 35:

100 x 3 x (3+1) + 25 = 300 x 4 + 25 = 1225
Multiply by 5, 50, 25 and 125

Multiplying the number X by these numbers, it is convenient to use the following expressions:

X x 5 = X x 10: 2

X x 50 = X x 100:2

X x 25 = X x 100:4

X x 125 = X x 1000:8

For example:

22 x 5 = 22 x 10: 2 = 220: 2 = 110

34 x 50 = 34 x 100: 2 = 3400: 2 = 1700

46 x 25 = 46 x 100: 4 = 4600: 4 = 1150

64 x 125 = 64 x 1000: 8 = 64000: 8 = 8000
Division by 5, 50, 25

When dividing the number X by these numbers, it is convenient to keep in mind that:

X:5 = X x 2:10

X: 50= X x 2: 100

X:25=Xx4:100

For example:

75:5 = 75 x 2:10 = 150:10 = 15

4350: 50 = 4350 x 2: 100 = 8700: 100 = 87

8600:25=8600x4:100=34400:100=344
Fast addition and subtraction of natural numbers, trick 1

If one of the terms is increased by several units, then the same number of units must be subtracted from the resulting amount.

For example:

654 + 348 = (654 + 348 + 2) - 2 = 1004 - 2 = 1002
Fast addition and subtraction of natural numbers, trick 2

If one of the terms is increased by several units, and the second is reduced by the same number of units, then the sum will not change.

For example:

334 + 768 = (334 + 6) + (768 - 6) = 340 + 762 = 1102
Fast addition and subtraction of natural numbers, trick 3

If you add (or subtract) the same number of units to the subtracted and reduced, then the difference will not change.

For example:

345 - 229 = (345 + 5) - (229 + 5) = 350 - 234 = 116
Fast multiplication of natural numbers

To get the units of the product, we multiply the units of the factors. To obtain tens of a product, multiply tens of one factor by units of another and vice versa, and add the results. To get hundreds, multiply tens of factors.

For example:

Multiply 43 x 57:

A) 3 x 7 \u003d 21 (we write 1 on the right as a result, but keep 2 in our mind)

B) 4 x 7 + 3 x 5 + 2 (from the mind) (we write 5 to the left of 1 from point "a", keep 4 in mind)

C) 4 x 5 + 4 (from the mind) = 24 (write 24 to the left of 5)

As a result: 43 x 57 = 2451.

For non-two-digit numbers, we proceed similarly.

Note. In general, in elementary school, this method is simply called "column multiplication", but elementary school - it was so long ago, right? ..
Multiply numbers that have the same number of tens and the sum of units is 10

Multiply the number of tens of any of the factors by a number that is greater than 1, then multiply the units of these numbers separately, and then add the second from the right to the first result.

For example:

Multiply 303 by 307:

A) 30 x (30 +1) = 900 + 30 = 930

B) 3 x 7 = 21

We write down the first result, and on the right - the second:

93021
Multiplying a number X by a two-digit number of the form YY

We multiply X by Y (by one digit), and then by 11.

For example:

12 x 44 = (12 x 4) x 11 = 48 x 11 = 480 + 48 = 528

Multiply by 11

To multiply the number X by 11, represent 11 as the sum of 10 + 1.

For example:

15 x 11 = 15 x (10 + 1) = 150 + 15 = 165

123 x 11 = 123 x (10 + 1) = 1230 + 123 = 1353
Multiply by 11 a two-digit number with a sum of digits less than 10

If the sum of the digits of the two-digit number X multiplied by 11 is less than 10, then we “insert” the sum of the digits between the digits of X themselves and, thus, we get the product.

For example:

36 x 11 \u003d 3 (between the numbers we insert the sum 3 + 6 \u003d 9) 6 \u003d 396

17 x 11 \u003d 1 (between the numbers we insert the sum 1 + 7 \u003d 8) 7 \u003d 187

Note. This method is only suitable for two-digit numbers!
Multiply by 111 a two-digit number with a sum of digits less than 10

If the sum of the digits of the two-digit number X multiplied by 111 is less than 10, then we “insert” the sum of the digits between the digits of X twice and, thus, we get the product.

For example:

52 x 111 \u003d 5 (between the numbers we insert the sum 5 + 2 \u003d 7 twice) 2 \u003d 5772
Multiplying by 11 a three-digit number

To multiply a three-digit number X by 11:

1. The product will be four-digit. The thousands digit in the product is the hundreds digit of the number.

2. The hundreds digit of the product is the hundreds digit X plus the tens digit X.

3. The tens digit of the product is the tens digit X plus the units digit X.

4. The digit of the units of the product is the digit of the units of the number X.

For example:

2 is the digit of thousands of the product,

2 + 4 = 6 - the hundreds digit of the product,

4 + 5 \u003d 9 - the tens digit of the product,

5 - the number of units of the product.

245 x 11 = 2695

If the sum of two digits is greater than 9, then 10 is subtracted from the sum and the resulting difference is written instead of the sum, and 1 is added to the highest (left adjacent) digit.

For example:

4 is the number of thousands of the product,

4+8 = 12. 12-10 = 2. 2 is the hundreds digit of the product. We add 1 to the thousands place: 4 + 1 \u003d 5.

8 + 9 \u003d 17. 17-10 \u003d 7. 7 - the number of tens of the product. Add 1 to the hundreds place: 2+1 = 3.

9 - the number of units of the product.

489 x 11 = 5379
Multiply by a number consisting of only digits 9

Let's say you need to multiply 154 by 999 (99, 9999, or any other number of nines). We calculate like this:

154 x 999 = 154 x (1000 -1) = 154000 - 154 = 153999 - 153 = 153846

Note. Pay attention to 154000-154 = 153999 - 153. This is not a required step, but another way to make the calculation easier.
Addition of numbers that are close in magnitude

Suppose you need to add a sequence of numbers that are close to each other in value:

23 + 21 + 19 + 22 + 17 + 24 = ?

We write the numbers in the following form:

Then the sum of these numbers is:

20 x 6 + (3+1-1+2-3+4) = 120 + 6 = 126
Subtraction from 100, 1000, 10000 and other powers of 10

We all remember, I hope that column subtraction is performed starting from the lowest (leftmost) digit. But when subtracting from 100, 1000, 10000 and other powers of ten, this rule can be violated.

Starting with the highest (rightmost), subtract each digit from 9. Subtract the last, leftmost digit from 10.

For example:

1) 100 - 57 = ?

10 - 7 \u003d 3 (the last digit is subtracted from 10, not from 9)

2) 1000000 - 546721 = ?

Answer: 453279

3) 100000 - 548 = ?

100000 - 548 = 100000 - 00548

Answer: 99542

Note. Do you want to surprise your friends? Ask them to write down a number with any number of zeros and any other number that needs to be subtracted from it. As soon as the task is written down, without wasting a second to think, start dictating the answer by number. :-)

Lesson 1

To learn how to count really quickly in your head, you need to be able to concentrate on a specific example. This skill is useful not only for performing mathematical operations, but also for solving any life problems. The ability to be attentive at the right moment is a skill that distinguishes great scientists, athletes, politicians, and will undoubtedly come in handy for you.

Sequence of arithmetic operations in the mind

To get started, try the following problem in your mind and write your answer in the box on the right:

Take 3000. Add 30. Add 2000 more. Add 10 more. Plus 2000. Add 20 more. Plus 1000. And plus 30. Plus 1000. And plus 10. Your answer:

Check your solution →

Answer: 9 100. If you solved the problem correctly and quickly, then you were able to concentrate on the numbers and avoided the temptation to get a beautiful answer. It is this approach that is needed for oral counting.

Try to solve other similar tasks for training subtraction, division and multiplication in your head.

tasks for attention

3000 - 700 - 60 - 500 - 40 - 300 -20 - 100 Your answer: 1*2*3*4*3*2*1 Your answer: 100:2:2*3*2 + 50 - 100 + 200 - 30 Your answer: 26+88+13+19 Your answer:

Check your solution →

Answers: 1280, 144, 270, 146

Mind counting training

If solving these examples is difficult for you, you can use special exercises and techniques to help you concentrate. You can find many of these techniques in other trainings. It also describes exactly those techniques that are useful for concentrating attention in the process of oral counting.

Visualization. When counting in the mind, it is important to have a clear idea of ​​the problem to be solved. You need to memorize intermediate results not by ear, but as they look if you wrote them down. There are many ways to train visual perception. Part of the visualization of the solution comes with experience. In addition, the techniques described below will also help increase your ability to visualize the necessary arithmetic operations when solving any given problem.

Games. Try to always find something interesting in the routine, turning any action into a game. This is what good parents do who want their child to do some boring work. Games are characteristic of many living beings, it is invested in us at the genetic level. In the game, excitement is important!

Excitement(French hasard) - passion, enthusiasm, fuse, excessive ardor. To create a game of chance, you must decide on the rules of this game and set clear conditions for winning this game. Then your excitement will force you to be more attentive and concentrated.

Competitiveness. The vast majority of people are gambling in an attempt to "be better" than the opponent. Therefore, individual lessons are not as effective as group ones. And in the mental account, you can find an opponent for yourself and try to surpass him.

Personal records. Another factor that creates excitement when counting can be a struggle with oneself to achieve a certain result. Personal records can be set in counting speed, in the number of solved examples, and in many other ways.

Boring job. Some experts advise looking out the window or watching the clock when doing boring work. So, if you try to do a very boring job every day for some time, your body itself will begin to look for ways to adapt to this routine.

external stimuli. Some people have one very important ability: they can do something when there is noise and turmoil around them. Often this is a matter of habit, for example, when a person lives in a small apartment or hostel, and he has to adapt to difficult conditions and be able to practice without paying attention to anything. Difficult conditions make a person more attentive, teach him to disconnect from external stimuli and do what he needs. Try to artificially create difficult conditions for yourself and try to concentrate on counting in your mind when you listen to music, when people are walking around, when the TV is on.

A state of trance, according to the observations of hypnosis specialist M. Erickson, is characterized by increased attention, the ability not to respond to external stimuli, and the ability to ignore the signals of some sense organs. So, in a state of trance, a person can take a position that is uncomfortable in the normal state, and spend quite a long time in this position. For example, while reading an interesting book and cross-legged, after half an hour during a break, we may find that one leg is very numb. But while reading, you did not think about the leg, you were in a state of increased attention to the book, your visual perception worked so strongly that the signals of the other senses were simply not perceived by the brain.

The square of the sum, the square of the difference

In order to square a two-digit number, you can use the formulas of the square of the sum or the square of the difference. For example:

23 2 = (20+3) 2 = 20 2 + 2*3*20 + 3 2 = 400+120+9 = 529

69 2 = (70-1) 2 = 70 2 – 70*2*1 + 1 2 = 4 900-140+1 = 4 761

Squaring numbers ending in 5

To square numbers ending in 5. The algorithm is simple. The number up to the last five, multiply by the same number plus one. Add 25 to the remaining number.

15 2 = (1*(1+1)) 25 = 225

25 2 = (2*(2+1)) 25 = 625

85 2 = (8*(8+1)) 25 = 7 225

This is true for more complex examples as well:

155 2 = (15*(15+1)) 25 = (15*16)25 = 24 025

Multiply numbers up to 20

1 step. For example, let's take two numbers - 16 and 18. To one of the numbers we add the number of units of the second - 16 + 8 = 24

2 step. The resulting number is multiplied by 10 - 24 * 10 \u003d 240

The technique for multiplying numbers up to 20 is very simple:

In short, then:

16*18 = (16+8)*10+6*8 = 288

It is easy to prove the correctness of this method: 16*18 = (10+6)*(10+8) = 10*10+10*6+10*8+6*8 = 10*(10+6+8) +6* eight. The last expression is a demonstration of the method described above.

In fact, this method is a private way of using pivot numbers (which will be discussed in the next lesson link). In this case, the reference number is 10. In the last expression of the proof, it can be seen that it is by 10 that we multiply the bracket. But any other numbers can be used as a reference number, of which the most convenient are 20, 25, 50, 100 ... Read more about the method of using the reference number in the next lesson.

reference number

Look at the essence of this method using the example of multiplying 15 and 18. Here it is convenient to use the reference number 10. 15 is more than ten by 5, and 18 is more than ten by 8. In order to find out their product, you need to perform the following operations:

1. To any of the factors, add the number by which the second factor is greater than the reference. That is, add 8 to 15, or 5 to 18. In the first and second cases, the same thing is obtained: 23.
2. Then we multiply 23 by the reference number, that is, by 10. Answer: 230
3. To 230 we add the product 5 * 8. Answer: 270.

0

Lesson 5

The most popular technique for multiplying large numbers in your head is the technique of using the so-called reference number. In the last lesson, when we showed how to multiply a number up to 20, in fact, we used the pivot number 10. It is also worth noting that you can read more about the methodology for using the pivot number in the book "" by Bill Handley.

General rules for using the reference number

The reference number is useful when multiplying close numbers and when squaring. You already understood how you can use the pivot number method from the last lesson, now let's summarize everything that has been said.

The reference number in multiplication is the number to which both factors are close and by which it is convenient to multiply. When multiplying numbers up to 100 with reference numbers, it is convenient to use all multiples of 10, and especially 10, 20, 50 and 100.

The technique for using the reference number depends on whether the factors are greater or less than the reference number. There are three possible cases here. We will show all 3 methods with examples.

Both numbers are less than the reference (under the reference)

Let's say we want to multiply 48 by 47. These numbers are close enough to 50 that it's convenient to use 50 as a reference number.

To multiply 48 by 47 using the reference number 50, you need:

1. From 47, subtract as much as 48 is missing to 50, that is, 2. It turns out 45 (or subtract 3 from 48 - it's always the same)
2. Then 45 times 50 = 2250
3. Then we add 2*3 to this result and voila - 2256!

Schematically in the mind it is convenient to imagine the table below.

(reference number)

48

*

47

(48-3)*50 = 45*50 = 2 250

(or (47-2)*50 = 45*50 remember that multiplying by 5 is the same as dividing by 2)

2

*

3

+6

Answer:

2 250 + 6 = 2 256

The reference number is written to the left of the product. If the numbers are less than the reference, then the difference between them and the reference is written below these numbers. To the right of 48 * 47 we write the calculation with the reference number, to the right of the remainders 2 and 3 we write their product.

If we use a simplified scheme, then the solution looks like this: 47*48=45*50 + 6= 2 256

Let's see other examples:

Multiply 18*19

(reference number)

18

*

19

(18-1)*20 = 340

2

*

1

+2

Answer:

342

Short entry: 18*19 = 20*17+2 = 342

Multiply 8*7

(reference number)

8

*

7

(8-3)*10 = 50

2

*

3

+6

Answer:

56

Short entry: 8*7 = 10*5+6 = 56

Multiply 98*95

(reference number)

98

*

95

(95-2)*100 = 9300

2

*

5

+10

Answer:

9310

Short entry: 98*95 = 100*93 + 10 = 9 310

Multiply 98*71

(reference number)

98

*

71

(71-2)*100 = 6900

2

*

29

+58

Answer:

6958

Short entry: 98*71 = 100*69 + 58 = 6 958

Both numbers are greater than the reference (above the reference)

Let's say we want to multiply 54 by 53. These numbers are close enough to 50 that it's convenient to use 50 as a reference number. But unlike the previous examples, these numbers are larger than the reference. In fact, the model of their multiplication does not change, but now you need not to subtract the remainders, but to add them.

1. To 54 add as much as 53 exceeds 50, that is, 3. It turns out 57 (or add 4 to 53 - it's always the same)
2. Then multiply 57 by 50 = 2850 (multiplying by 50 is similar to dividing by 2)
3. Then we add 4*3 to this result. Answer: 2862

+12

(reference number)

54

*

53

(54+3)*50 = 2 850

or (53+4)*50 = 57*50 (remember that multiplying by 5 is the same as dividing by 2)

Answer:

2 862

The short solution looks like this: 50*57+12 = 2862

For clarity, examples are given below:

Multiply 23*27

+21

(reference number)

23

*

27

(23+7)*20 = 600

Answer:

621

Short entry: Short entry: 23*27 = 20*30 + 21 = 621

Multiply 51*63

+13

(reference number)

51

*

63

(63+1)*50 = 3 200

Answer:

3 213

Short entry: Short entry: 51*63 = 64*50 + 13 = 3213

One number is below the pivot and the other is above

The third use case for the reference number is when one number is greater than the reference number and the other is smaller. Such examples are not more difficult to solve than the previous ones.

Multiply 45*52

The product of 45 * 52 is considered as follows:

1. Subtract 5 from 52 or add 2 to 45. In either case, we get: 47
2. Then multiply 47 by 50 = 2350 (multiplying by 50 is similar to dividing by 2)
3. Then we subtract (and not add, as before!) 2 * 5. Answer: 2 340

2

(reference number)

45

*

52

(45+2)*50 = 2 350

5

-10

Answer:

2 340

Short entry: 45*52 = 47*50-10 = 2340

We also do with similar examples:

Multiply 91*103

3

(reference number)

91

*

103

(91+3)*100 = 9400

9

-27

Answer:

9 373

Only one number is close to the reference, and the other is not

As you have already seen from the examples, it is convenient to use the pivot number even if only one number is close to the pivot. It is desirable that the difference between this number and the reference number should be no more than 2 or 3, or be equal to the number by which it is convenient to multiply (for example, 5, 10, 25 - see the second lesson)

Multiply 48*73

23

(reference number)

48

*

73

(73-2)*50 = 3 550

2

-46

Answer:

3 504

Short solution: 48*73 = 71*50 – 23*2 = 3 504

Multiply 23*69

3

49

147

(reference number)

23

*

69

(3+69)*20 = 1440

Answer:

1 587

Short entry: Short solution: 23*69 = 72*20 + 147 = 1587 - a bit more difficult

Multiply 98*41

(reference number)

98

*

41

(41-2)*100 = 3900

2

*

59

+118

Answer:

4018

Short entry: Short entry: 98*41 = 100*39 + 118 = 4018

Thus, by using one reference number, a large combination of two-digit numbers can be multiplied. If you are good at multiplying by 30, 40, 60, 70, or 80, then you can use this technique to multiply any number (up to 100 and even more).

Using Multiple Reference Numbers

The multiplication technique using reference numbers allows the use of 2 reference numbers. This is convenient when the reference number of one factor can be expressed in terms of the reference number of another. For example, in the product "23 * 88" it is convenient to use the reference number 20 for 23 and 80 for 88. Multiplying these numbers using two reference numbers is convenient because 20=80:4.

The technique of 2 reference numbers is that we first divide 88 by 4 and get 22, multiply 23 by 22 and multiply the product again by 4. That is, we first divide the product by 4, and then multiply by 4. It turns out: 23*22 = 250*2+6= 506 and 506*4 = 2024 is the answer!

For visualization, you can use the already familiar scheme. The product of 23 * 88 is calculated as follows:

1. We write down a convenient reference number "20" and next we attribute a factor of 4, with which you can express 80 through 20.
2. Then we do, as before, we write how much 23 exceeds 20 (3), and 88 exceeds 80 (8).
3. Above the triple we write the product 3 by 4 (that is, 3 by the reference multiplier).
4. To 88 we add the product of 3 by 4 and multiply by the reference (20), it turns out 100 * 20 \u003d 2000
5. We add to 2000 by the product of 3 and 8. Result: 2024

3*4=12

3

*

8

+24

(reference number)

23

*

88

(88+12)*20 = 2 000

Answer:

2 024

Short entry: 23*88 = (88+3*4)*20 + 24 = 2024

Now let's try to multiply 23*88 using the reference number 100 for 88 and 25 for 23. In this case, the main reference number is 100. And 25 can be written as 100:4=25

(reference number)

23

*

88

(23-3)*100 = 2 000

2

12

+24

12:4=3

Answer:

2 024

Short entry: 23*88 = (23+12:4)*100 + 24 = 2024

As you can see, the answer is the same.

The method using two reference numbers is somewhat more complicated and requires additional steps. First, you need to figure out which 2 base numbers are comfortable for you to use. Secondly, you need to perform an additional action to find the number that needs to be multiplied by the reference.

Use this technique better when you have already mastered multiplication with one reference number quite well.

The process of mental counting can be considered as a counting technology that combines human ideas and skills about numbers, mathematical algorithms of arithmetic.

There are three types mental arithmetic technologies, which use various physical capabilities of a person:

audio motor counting technology;

visual counting technology.

characteristic feature audiomotor mental counting is to accompany each action and each number with a verbal phrase like "twice two - four." The traditional counting system is precisely the audio-motor technology. The disadvantages of the audio-motor method of conducting calculations are:

the absence in the memorized phrase of relationships with neighboring results,

the impossibility to separate tens and units of the product in phrases about the multiplication table without repeating the entire phrase;

the inability to reverse the phrase from the answer to factors, which is important for performing division with a remainder;

slow playback speed of a verbal phrase.

Supercomputers, demonstrating high speeds of thinking, use their visual abilities and excellent visual memory. People who are proficient in speed calculations do not use words in the process of solving an arithmetic problem in their minds. They show reality visual technology of mental counting, devoid of the main drawback - the slow speed of performing elementary operations with numbers.

Perhaps our methods of multiplication are not perfect; maybe even faster and more reliable will be invented.

Of course, it is impossible to know all the methods of quick counting, but the most accessible ones can be studied and applied.

Practice counting.

There are people who can perform simple arithmetic operations in their minds. Multiply a two-digit number by a one-digit number, multiply within 20, multiply two small two-digit numbers, and so on. - they can perform all these actions in the mind and quickly enough, faster than the average person. Often this skill is justified by the need for constant practical use. As a rule, people who are good at mental arithmetic have a mathematical education or at least experience in solving numerous arithmetic problems.

Undoubtedly, experience and training plays a crucial role in the development of any ability. But the skill of mental counting is not based on experience alone. This is proved by people who, unlike those described above, are able to calculate in their minds much more complex examples. For example, such people can multiply and divide three-digit numbers, perform complex arithmetic operations that not every person can count in a column.

What does an ordinary person need to know and be able to master in order to master such a phenomenal ability? Today, there are various techniques that help you learn how to quickly count in your mind. Having studied many approaches to teaching the skill of counting orally, we can distinguish3 main components of this skill:

1. Ability. The ability to concentrate attention and the ability to keep several things in short-term memory at the same time. Predisposition to mathematics and logical thinking.

2. Algorithms. Knowledge of special algorithms and the ability to quickly select the desired, most effective algorithm in each specific situation.

3. Training and experience, the value of which for any skill has not been canceled. Constant training and the gradual complication of tasks and exercises will allow you to improve the speed and quality of mental arithmetic.

It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others with a fast score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, because having the ability and a set of necessary algorithms in your arsenal, you can outdo even the most experienced "bookkeeper", provided that you have been training for the same time.

Several ways of oral counting:

1. Multiply by 5 it's more convenient like this: first multiply by 10, and then divide by 2

2. Multiply by 9. In order to multiply a number by 9, you need to add 0 to the multiplicand and subtract the multiplier from the resulting number, for example 45 9=450-45=405.

3. Multiply by 10. Assign zero on the right: 48 10 = 480

4. Multiply by 11. two-digit number. Move the numbers N and A apart, enter the sum (N + A) in the middle.

e.g. 43 11 === 473.

5. Multiply by 12. is done in approximately the same way as for 11. We double each digit of the number and add the neighbor of the original digit to the right to the result.

Examples.Let's multiplyon the.

Let's start with the rightmost number - this is. Let's doubleand add a neighbor (it does not exist in this case). We get. Let's write downand remember.

Move left to the next digit. Let's double, we get, add a neighbor,, we get, add. Let's write downand remember.

Let's move to the left to the next digit,. Let's double, we get. Add a neighborand get. Let's add, which was memorized, we get. Let's write downand remember.

Let's move to the left to a non-existent figure - zero. Double it, get and add a neighbor, , which will give us . Finally, add , which was remembered, we get . Let's write . Answer: .

6. Multiplication and division by 5, 50, 500, etc.

Multiplying by 5, 50, 500, etc. is replaced by multiplying by 10, 100, 1000, etc., and then dividing by 2 of the resulting product (or dividing by 2 and multiplying by 10, 100, 1000, etc. ). (50 = 100: 2 etc.)

54 5=(54 10):2=540:2=270 (54 5 = (54:2) 10= 270).

To divide a number by 5.50, 500, etc., you need to divide this number by 10,100, 1000, etc. and multiply by 2.

10800: 50 = 10800:100 2 =216

10800: 50 = 10800 2:100 =216

7. Multiplication and division by 25, 250, 2500, etc.

Multiplying by 25, 250, 2500, etc. is replaced by multiplying by 100, 1000, 10000, etc. and the result is divided by 4. (25 = 100: 4)

542 25=(542 100):4=13550 (248 25=248: 4 100 = 6200)

(if the number is divisible by 4, then the multiplication does not take time, any student can do it).

To divide a number by 25, 25,250,2500, etc., this number must be divided by 100,1000,10000, etc. and multiply by 4: 31200: 25 = 31200:100 4 = 1248.

8. Multiplication and division by 125, 1250, 12500, etc.

Multiplication by 125, 1250, etc. is replaced by multiplication by 1000, 10000, etc., and the resulting product must be divided by 8. (125 = 1000 : 8)

72 125=72 1000: 8=9000

If the number is divisible by 8, then first we perform the division by 8, and then the multiplication by 1000, 10000, etc.

48 125 = 48: 8 1000 = 6000

To divide a number by 125, 1250, etc., you need to divide this number by 1000, 10000, etc. and multiply by 8.

7000: 125 = 7000: 10008 = 56.

9. Multiplication and division by 75, 750, etc.

To multiply a number by 75, 750, etc., you need to divide this number by 4 and multiply by 300, 3000, etc. (75=300:4)

4875 = 48:4300 = 3600

To divide a number by 75,750, etc., you need to divide this number by 300, 3000, etc. and multiply by 4

7200: 75 = 7200: 3004 = 96.

10. Multiply by 15, 150.

When multiplying by 15, if the number is odd, multiply it by 10 and add half of the resulting product:

23 15=23 (10+5)=230+115=345;

if the number is even, then we act even simpler - add half of it to the number and multiply the result by 10:

18 15=(18+9) 10=27 10=270.

When multiplying a number by 150, we use the same trick and multiply the result by 10, because 150=15 10:

24 150=((24+12) 10) 10=(36 10) 10=3600.

Similarly, quickly multiply a two-digit number (especially an even one) by a two-digit number ending in 5:

24 35 = 24 (30 +5) = 24 30+24:2 10 = 720+120=840.

11. Multiply two digit numbers less than 20.

To one of the numbers you need to add the number of units of the other, multiply this amount by 10 and add to it the product of the units of these numbers:

18 16=(18+6) 10+8 6= 240+48=288.

In the described way, you can multiply two-digit numbers less than 20, as well as numbers in which the same number of tens: 23 24 \u003d (23 + 4) 20 + 4 6 \u003d 27 20 + 12 \u003d 540 + 12 \u003d 562.

Explanation:

(10+a) (10+b) = 100 + 10a + 10b + a b = 10 (10+a+b) + a b = 10 ((10+a)+b) + a b .

12. Multiplying a two-digit number by 101 .

Perhaps the simplest rule is: add your number to itself. Multiplication completed.
Example: 57 101 = 5757 57 --> 5757

Explanation: (10a+b) 101 = 1010a + 101b = 1000a + 100b + 10a + b
Similarly, three-digit numbers are multiplied by 1001, four-digit numbers by 10001, etc.

13. Multiply by 22, 33, ..., 99.

In order to multiply a two-digit number 22.33, ..., 99, this multiplier must be represented as a product of a single-digit number by 11. Perform multiplication first by a single-digit number, and then by 11:

15 33= 15 3 11=45 11=495.

14. Multiply two-digit numbers by 111 .

First, let's take a multiplicand such a two-digit number, the sum of the digits of which is less than 10. Let's explain with numerical examples:

Since 111=100+10+1, then 45 111=45 (100+10+1). When multiplying a two-digit number, the sum of the digits of which is less than 10, by 111, it is necessary to insert twice the sum of the digits (i.e., the numbers they represent) of its tens and units 4 + 5 = 9 in the middle between the digits. 4500+450+45=4995. Therefore, 45 111=4995. When the sum of the digits of a two-digit multiplier is greater than or equal to 10, for example 68 11, add the digits of the multiplicand (6 + 8) and insert 2 units of the resulting sum in the middle between the numbers 6 and 8. Finally, add 1100 to the compiled number 6448. Therefore, 68 111 = 7548.

15. Squaring numbers consisting of only 1.

11 x 11 =121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

Some non-standard methods of multiplication.

Multiplying a number by a single digit factor.

To multiply a number by a single-digit factor (for example, 34 9) orally, you must perform actions starting from the most significant digit, sequentially adding the results (30 9=270, 4 9=36, 270+36=306).

For effective mental counting, it is useful to know the multiplication table up to 19 * 9. In this case, the multiplication 147 8 is performed in the mind like this: 147 8=140 8+7 8= 1120 + 56= 1176 . However, without knowing the multiplication table up to 19 9, in practice it is more convenient to calculate all such examples by reducing the multiplier to the base number: 147 8=(150-3) 8=150 8-3 8=1200-24=1176, with 150 8=(150 2) 4=300 4=1200.

If one of the multiplied is decomposed into single-valued factors, it is convenient to perform the action by successively multiplying by these factors, for example, 225 6=225 2 3=450 3=1350. Also, it might be simpler 225 6=(200+25) 6=200 6+25 6=1200+150=1350.

Multiplication of two-digit numbers.

1. Multiply by 37.

When multiplying a number by 37, if the given number is a multiple of 3, it is divided by 3 and multiplied by 111.

27 37=(27:3) (37 3)=9 111=999

If this number is not a multiple of 3, then 37 is subtracted from the product or 37 is added to the product.

23 37=(24-1) 37=(24:3) (37 3)-37=888-37=851.

It is easy to remember the product of some of them:

3 x 37 = 111 33 x 3367 = 111111

6 x 37 = 222 66 x 3367 = 222222

9 x 37 = 333 99 x 3367 = 333333

12 x 37 = 444 132 x 3367 = 444444

15 x 37 = 555 165 x 3367 = 555555

18 x 37 = 666 198 x 3367 = 666666

21 x 37 = 777 231 x 3367 = 777777

24 x 37 = 888 264 x 3367 = 888888

27 x 37 = 999 297 x 3367 = 99999

2. If tens of two-digit numbers start with the same digit, and the sum of units is 10 , then when they are multiplied, we find the product in this order:

1) multiply the ten of the first number by the ten of the second larger number by one;

2) multiply units:

8 3x 8 7= 7221 ( 8x9=72 , 3x7=21)

5 6x 5 4=3024 ( 5x6=30 , 6x4=24)

1. Algorithm for multiplying two-digit numbers close to 100

For example:97 x 96 = 9312

Here I use the following algorithm: if you want to multiply two

two-digit numbers close to 100, then do this:

1) find the shortcomings of factors up to a hundred;

2) subtract from one factor the disadvantage of the second up to a hundred;

3) add the product of the shortcomings to the result with two digits

factors up to hundreds.

The relevant literature mentions such methods of multiplication as "bending", "lattice", "back to front", "rhombus", "triangle" and many others. I wanted to know what other non-standard multiplication techniques exist in mathematics? It turns out there are a lot of them. Here are some of these tricks.

Peasant method:

One of the factors doubles while the other decreases in parallel by the same amount. When the quotient becomes equal to one, the product obtained in parallel is the desired answer.

If the quotient turns out to be an odd number, then one is discarded from it and the remainder is divided. Then the works that stood opposite the odd quotients are added to the answer received

"Method of the Cross".

In this method, the factors are written under each other and their numbers are multiplied in a straight line and crosswise.

3 1 = 3 is the last digit.

2 1 + 3 3 = 11. The penultimate digit is 1, 1 more in the mind.

2 3 = 6; 6 + 1 = 7 is the first digit of the product

The desired product is 713.

Sino-Japanese multiplication method.

It is no secret that different countries have different teaching methods. It turns out that in Japan, first-grade students can multiply three-digit numbers without knowing the multiplication table. For this is used. The logic of the method is clear from the figure. After drawing, you just need to count the number of intersections in each area.

Even three-digit numbers can be multiplied using this method. Probably, when children later learn the multiplication table, they will be able to multiply in a simpler and faster way, in a column. Moreover, the above method is too time consuming when multiplying numbers like 89 and 98, because you have to draw 34 stripes and count all the intersections. On the other hand, in such cases, you can use a calculator. It will seem to many that this way of Japanese or Chinese multiplication is too complicated and confusing, but this is only at first glance. It is visualization, that is, the image of all points of intersection of lines (multipliers) on the same plane, that gives us visual support, while the traditional method of multiplication involves a large number of arithmetic operations only in the mind. Chinese or Japanese multiplication helps not only to quickly and efficiently multiply two-digit and three-digit numbers without a calculator, but also develops erudition. Agree, not everyone can boast that in practice he owns the ancient Chinese multiplication method ( ), which is relevant and works great in the modern world.

Multiplication can be done using a matrix table c :

43219876=?

First, we write the products of numbers.
2. Find the sums along the diagonal:
36, 59, 70, 70, 40, 19, 6
3. We get the answer from the end, adding the "extra" digits to the front digit:2674196

Lattice method.

A rectangle divided into squares is drawn. Following are square cells, divided diagonally. In each line we write the product of the numbers above this cell and to the right of it, while the number of tens of the product is written above the slash, and the number of units is below it. Now add up the numbers in each slash by doing this operation, from right to left. If it turns out to be more than 10, then we write only the number of units of the sum, and add the number of tens to the next amount.

6

5

2

4

1 7

3

7

7

We write the answer numbers from left to right: 4, 5, 17, 20, 7, 5. Starting from the right, we write, adding “extra” numbers to the “neighbor”: 469075.

Got: 725 x 647 = 469075.

No matter how ashamed I was, but by the age of 30 I realized that I was very bad at counting elementary numbers in my mind and wasting a lot of time on it. I decided to correct this shortcoming and found tools on the Internet that helped me learn how to count in my head.

In arithmetic, there are key patterns that need to be brought to automatism.

Subtraction 7,8,9 To subtract 9 from any number, you need to subtract 10 from it and add 1. To subtract 8 from any number, you need to subtract 10 from it and add 2. To subtract 7 from any number, you need to subtract 10 from it and add 3. If usually If you think differently, then for the best result you need to get used to this new way.

Multiply by 9. You can quickly multiply any number by 9 as follows: first multiply this number by 10 (just add 0 at the end), and then subtract the number itself from the result. For example 89*9=890-89=801. This operation must be brought to automaticity.

Multiplication by 2. For mental counting, it is very important to be able to quickly multiply any number by 2. To multiply non-round numbers by 2, try rounding them to the nearest more convenient ones. So 139*2 is easier to calculate if you first multiply 140*2 (140*2=280). and then subtract 1*2=2 (exactly 1 needs to be added to 139 to get 140) Total: 140*2-1*2=278

Division by 2. For mental counting, it is also important to be able to quickly divide any number by 2. Despite the fact that many people find multiplication and division by 2 quite simple, in difficult cases also try to round numbers. For example, to divide 198 by 2, you must first divide 200 (this is 198 + 2) by 2 and subtract 1 (we got 1 by dividing the added 2 by 2) Total: 198/2=200/2-2/2=100- 1=99.

Division and multiplication by 4 and 8. Division (or multiplication) by 4 and 8 are two or three division (or multiplication) by 2. It is convenient to perform these operations sequentially. For example, 46*4=46*2*2=922*2=184

Multiply by 5. Multiplying by 5 is very easy. Multiplying by 5 and dividing by 2 are basically the same thing. So 88*5=440 and 88/2=44, so always multiply a number by 5 by dividing the number by 2 and multiplying it by 10.

Multiplication by single digits. To quickly count in your head, it is useful to be able to multiply two-digit and three-digit numbers by one-digit numbers. To do this, you need to multiply a two- or three-digit number bit by bit. For example, let's multiply 83*7. To do this, first multiply 8 by 7 (and add 0, since 8 is the tens place) and add the product of 3 and 7 to this number. Thus, 83*7=80*7+3*7=560+21=581. Let's take a more complex example of 236*3. So, we multiply a complex number by 3 bit by digit: 200*3+30*3+6*3=600+90+18=708.

Definition of ranges. In order not to get confused in the algorithms and give a completely wrong answer by mistake, it is important to be able to build an approximate range of answers. So the multiplication of single-digit numbers by each other can give a result of no more than 90 (9 * 9 = 81), two-digit numbers - no more than 10,000 (99 * 99 = 9801), three-digit numbers - no more than 1,000,000 (999 * 999 \u003d 998001)

Dividing 1000 by 2,4,8,16. And finally, it is useful to know the division of numbers that are multiples of 10 by numbers that are multiples of two: 100=2*500=4*250=8*125=16*62.5

It's no secret that there are some people who can perform medium-complex arithmetic operations in their minds with enviable speed. It is not difficult for them, for example, to multiply two two-digit numbers or divide several three-digit values ​​​​by each other. They do it quickly and without the help of additional devices and do not even use notes, that is, they do calculations in their minds! Of course, for many it is not difficult to learn how to quickly count in the mind - this is a daily practice, forced work or type of activity. But this does not mean that any of us who want to learn how to learn to count in our mind are obliged to graduate from a mathematical university. So, today we will talk about how to learn to count. Count fast!

Learning to count quickly, the necessary preparation

Without a doubt, your experience and ability training will play an important role in the development of such abilities. But this in no way means that the skill of quick counting is available only to people with experience. Calculating in the mind is a rationalization path based on basic arithmetic. Following our tips on how to quickly learn to count, you will be able to surprise others with quick solutions to examples that not everyone can solve even with a calculator.

What do you need to quickly master the technique of instant mental counting? The main components of success can be divided into three groups:

• dispositions and abilities. Your analytical mindset will be a good help. The ability to keep several values ​​in memory at a time is a must.
• Directly algorithms of your thinking. You can learn to count quickly only by strict algorithmization of your actions, their rationalization and the ability to choose the necessary method in a particular situation. We will talk about situations and other things a little later.
• Training and practice of skills. No one canceled the importance of these actions in any direction of activity, and especially in mental activity. The more you train and perform various calculations, the better you will get it.

Attention should be paid to the third factor in the development of the skill of quick counting. Even if you are well versed in all existing algorithms, you are unlikely to be able to learn how to count quickly if there is not enough practice.

Tricks and basic algorithms, how to quickly count

Consider a few common counting simplifications, with their help you will be able to learn how to count quickly. I will also draw your attention to the fact that no one forbids you to improvise - mathematics is remarkable in that, with all its accuracy and rigor, it does not forbid you to act beautifully, like art. And the ability to count quickly is precisely an art! So, some tricks on how to learn to count quickly.

Let's say you need to add multi-valued terms. Easily! Add by digits: add the highest digit of the smaller number to the larger number, then add with the lower digits. Let's say you need to add 361 and 523. It will not be easy to keep in memory right away, agree? Therefore, our course of action will be as follows:

1. A smaller number was determined - 361.
2. What is 361? This is 300+60+1. It's hard to argue if you're trying to be rational.
3. First add 300 to 523. We get 823.
4. Then we add 60 - we get 883.
5. And in the end - our one, added to the amount received earlier, will give us the result 884.

You see, it was much easier to keep 3 numbers in your head than to add two three-digit numbers at the same time! We are starting to get fast counting in our minds!

Do the same with subtraction, but only by successive subtraction of digits we will not achieve the necessary speed! You can cheat a little by adding one more skill to our arsenal - increase / subtract to a round (convenient number).

For example, you need to subtract 93 from 250. Well, it's inconvenient!

What is 93? That's right, it's 100-7!

250 – 100 = 150.

We make an adjustment for our "correction" of the number. If we added - you need to add to the private, and vice versa. In our case, we “increased” the number 93 to 100 by adding 7. So, we add 7 to the quotient.

Check with a calculator. Noticeably more time was spent on typing numbers than on calculation? This is a sign that you are already pretty good at counting fast in your head!

Now with multiplication. There are many ways to speed up the count. For example, when multiplying numbers, break the factors into second-level factors.

For example:

Lots of ways to solve! And here your algorithm may differ from the ways of other people - do not be afraid, that's why we, geniuses, people and unique =)

You can do this: 12 \u003d 3x4. We multiply 150 x 4 = 600, then 600 x 3 = 1800.

Without hesitation, I began to count as follows: 12 = 10 + 2. And now it’s elementary: (150 x 10) + (150 x2). All these are elementary school rules, which, unfortunately, we forget. It is easy to see that in this case you practically don’t have to count - add zero to 150, getting one and a half thousand, and multiply 150 by 2, getting 300. The result is the same, 1800.

Based on the experience of rapid multiplication, it is not difficult to guess how to quickly divide numbers in your mind. You can again go in different ways, from parallel division by a simplified divisor of the dividend to rounding the dividend up to the elementaryization of division with correction.

For example:

To begin with, discard the same number of zeros. In this example, it's just 39:4. Our brain is much more willing to operate with small numbers than with multi-digit values.

You probably noticed that the number 39 just wants to be rounded up to 40. So what's stopping us? (39+1):4 = 10.

But having changed the dividend, we need to correct the answer. So, it is obvious that it will be less than 10, since we added a certain number 1 to the dividend. Now we need to subtract from 10 the result of dividing the corrector number by the divisor (4). If we were taking away, the procedure would be reversed, it goes without saying.

So 1:4 = 0.25

Answer: 9.75 (9 3 / 4)

It is much easier for our brain to perceive natural fractions, that is, we represent 0.25 as 1/4 (one fourth, a quarter), and then it will be very easy to quickly calculate the result in your mind!

Remember, it is not so difficult to understand how to quickly learn to count. It is much more difficult to quickly choose a method for a specific situation, but this is solved with the help of colossal practice.