Why do we need mental arithmetic if this is the 21st century, and all sorts of gadgets are capable of performing any arithmetic operations almost at lightning speed? You don’t even have to point your finger at your smartphone, but give a voice command and immediately receive the correct answer. Now this is successfully done even by elementary school students who are too lazy to divide, multiply, add and subtract on their own.

But this medal also has back side: scientists warn that if you don’t train, don’t load him with work and make his tasks easier, he begins to be lazy and his performance declines. In the same way, without physical training, our muscles weaken.

Mikhail Vasilyevich Lomonosov also spoke about the benefits of mathematics, calling it the most beautiful of sciences: “You have to love mathematics because it puts your mind in order.”

Oral arithmetic develops attention and reaction speed. It is not for nothing that more and more new methods of rapid mental calculation are appearing, intended for both children and adults. One of them is the Japanese mental counting system, which uses the ancient Japanese soroban abacus. The methodology itself was developed in Japan 25 years ago, and now it is successfully used in some of our mental counting schools. It uses visual images, each of which corresponds to a specific number. Such training develops right hemisphere brain, responsible for spatial thinking, constructing analogies, etc.

It is curious that in just two years, students of such schools (they accept children aged 4–11 years) learn to perform arithmetic operations with 2-digit and even 3-digit numbers. Kids who don't know multiplication tables can multiply here. They add and subtract large numbers without writing them down. But, of course, the goal of training is the balanced development of the right and left.

You can also master mental arithmetic with the help of the problem book “1001 problems for mental arithmetic at school,” compiled back in the 19th century by a rural teacher and famous educator Sergei Aleksandrovich Rachinsky. This problem book is supported by the fact that it went through several editions. This book can be found and downloaded on the Internet.

People who practice quick counting recommend Yakov Trachtenberg’s book “The Quick Counting System.” The history of the creation of this system is very unusual. To survive the concentration camp where he was sent by the Nazis in 1941, and not lose his mental clarity, a Zurich mathematics professor began developing algorithms for mathematical operations that allow him to quickly count in his head. And after the war, he wrote a book in which the quick counting system is presented so clearly and accessiblely that it is still in demand.

There are also good reviews about Yakov Perelman’s book “Quick Counting. Thirty simple examples oral counting." The chapters of this book are devoted to multiplying by single-digit and two-digit numbers, in particular multiplying by 4 and 8, 5 and 25, by 11/2, 11/4, *, dividing by 15, squaring, and formula calculations.

The simplest methods of mental counting

People who have certain abilities will master this skill faster, namely: the ability to think logically, the ability to concentrate and store several images in short-term memory at the same time.

No less important is knowledge of special action algorithms and some mathematical laws that allow, as well as the ability to choose the most effective one for a given situation.

And, of course, you can’t do without regular training!

Some of the most common quick counting techniques are:

1. Multiplying a two-digit number by a one-digit number

The easiest way to multiply a two-digit number by a single-digit number is to split it into two components. For example, 45 - by 40 and 5. Next, we multiply each component by the required number, for example, by 7, separately. We get: 40 × 7 = 280; 5 × 7 = 35. Then we add the resulting results: 280 + 35 = 315.

2. Multiplying a three-digit number

Multiplying a three-digit number in your head is also much easier if you break it down into its components, but present the multiplicand in such a way that it is easier to perform mathematical operations with it. For example, we need to multiply 137 by 5.

We represent 137 as 140 − 3. That is, it turns out that we now have to multiply by 5 not 137, but 140 − 3. Or (140 − 3) x 5.

Knowing the multiplication table within 19 x 9, you can count even faster. We decompose the number 137 into 130 and 7. Next, we multiply by 5, first 130, and then 7, and add the results. That is, 137 × 5 = 130 × 5 + 7 × 5 = 650 + 35 = 685.

You can expand not only the multiplicand, but also the multiplier. For example, we need to multiply 235 by 6. We get six by multiplying 2 by 3. Thus, we first multiply 235 by 2 and get 470, and then multiply 470 by 3. Total 1410.

The same action can be done differently by representing 235 as 200 and 35. It turns out 235 × 6 = (200 + 35) × 6 = 200 × 6 + 35 × 6 = 1200 + 210 = 1410.

In the same way, by breaking down numbers into their components, you can perform addition, subtraction and division.

3. Multiplying by 10

Everyone knows how to multiply by 10: simply add zero to the multiplicand. For example, 15 × 10 = 150. Based on this, it is no less simple to multiply by 9. First, we add 0 to the multiplicand, that is, multiply it by 10, and then subtract the multiplicand from the resulting number: 150 × 9 = 150 × 10 = 1500 − 150 = 1,350.

4. Multiplication by 5

It is easy to multiply by 5. You just need to multiply the number by 10, and divide the resulting result by 2.

5. Multiplying by 11

It’s interesting to multiply two-digit numbers by 11. Let’s take 18, for example. Let’s mentally expand 1 and 8, and between them write the sum of these numbers: 1 + 8. We get 1 (1 + 8) 8. Or 198.

6. Multiply by 1.5

If you need to multiply a number by 1.5, divide it by two and add the resulting half to the whole: 24 × 1.5 = 24 / 2 + 24 = 36.

These are just the simplest ways of mental counting with which we can train our brains in everyday life. For example, counting the cost of purchases while standing in line at the checkout. Or perform mathematical operations with numbers on the license plates of passing cars. Those who like to “play” with numbers and want to develop their thinking abilities can turn to the books of the above-mentioned authors.

IN modern world With many ultra-progressive devices, mental arithmetic has not lost its relevance.

Sometimes we come across people who can add, multiply and divide complex numbers with lightning speed. Such people do not have supernatural abilities, they simply know the simplified counting formulas and regularly train their skills.

Three components of successful learning

1. Capabilities. In order to learn to count in your head, you should be able to concentrate on the task at hand and retain complex numbers in memory.
2. Formulas. To easily and simply perform calculations in your head, you should remember the basic mathematical formulas.
3. Practice. Frequent training will allow you to develop and improve the skill.

There are several simple ways multiplying a number by 11.

Method 1

When multiplying a 2-digit number by 11, we expand the digits of the multiplier.

For example (54 * 11):
5 _ 4 * 11=…

Now we sum up the units and tens, and write the resulting result in the answer:
5 (5+4) 4 * 11 = 5 (9) 4 = 594

If, when summing tens and ones, you get a 2-digit number, leave only the ones, and add “1” to the tens.

For example (89 * 11):
8 _ (8+9) _9 = 8 _ (17) _ 9 = _ (8+1) _ 79 = 979

Method 2

When multiplying by 11, we decompose the number 11 into the sum: 10+1, and multiply the parts.

For example:
12 * 11 = 12 * (10+1) = 120 + 12 = 132

The same goes for 3-digit numbers:
114 * 11 = 114 * (10+1) = 1140 + 114 = 1254

Multiply by 9 and 11

When multiplying by "9", we simply multiply the number by 10 and then subtract the same original number. If we multiply by “11”, the number should be multiplied by “10” and add the original number.

Examples:
15 * 9 = 15 * 10 – 15 = 150 - 15 = 135
57 * 11 = 57 * 10 + 57 = 570 + 57 = 627
Squaring a number ending in 5

Enough simple technique. Multiply ten by itself +1, and add “25” at the end.

For example (35 * 35):
35 * 35 = 3 * (3+1)_25 = 1225
Verbal multiplication by 5, 25, 50, 125

Multiplying numbers up to 10 by 5 is no problem

Let's learn how to multiply two-digit and three-digit numbers just as easily.

Method 1

Let's divide our multiplier by "2". Did you get a whole number? This means we add “0” to it at the end; if the number is not equally divisible, we discard the remainder and add “5” at the end.

For example (1482 * 5):
1482 * 5 = (1482/2) _ (+0 or +5) = 741 _(+0) = 7410 – the number is divisible by 2 without a remainder
2269 * 5 = (2269/2) _ (+0 or +5) = 1134.5 _ (+5) = 11345 – the number is divisible by 2 with a remainder

Method 2

When multiplying a number by 5, 25, 50, 125, you can use the following formulas:
A * 5 = A * 10 / 2
A * 50 = A * 100 / 2
A * 25 = A * 100 / 4
A * 125 = A * 1000 / 8

Examples:
44 * 5 = 44 * 10 / 2 = 440 / 2 = 220
24 * 50 = 24 * 100 / 2 = 2400 / 2 = 1200
26 * 25 = 26 * 100 / 4 = 2600 / 4 = 650
54 * 125 = 54 * 1000 / 8 = 54000 / 8 = 6750

Learning to multiply by 4 orally

A fairly simple method that does not require much effort.

We multiply the number by “2”, and then multiply the resulting result again by “2”.

For example:
27 * 4 = 27 * 2 * 2 = 54 * 2 = 108

Calculate 15% of the number in your head

Find 10% of the number and add ½ of 10%.

For example:
15% of 664 = (10%) + (10% / 2) = 66.4 + 33.2 = 99.6

Multiply large numbers in your head, one of which is even

When multiplying large numbers, one of which is even, we will use the method of simplifying factors. An even number is halved, and an odd number is increased by the same amount.

For example:
48 * 125 = 24 * 250 = 12 * 500 = 6 * 1000 = 6000

Learning to divide by 5, 50, 25

One simple trick will help you quickly divide in your head: multiply our number by “2” and move the decimal point back one digit.

145 / 5 = 145 * 2 = 290 (shift the comma) = 29
1200 / 5 = 1200 * 2 = 2,400 (offset the decimal point) = 240

When dividing by 50, 25, it is convenient to use the formulas:

A / 50 = A * 2 / 100
A / 25 – A * 4 / 100

Examples:
2350 / 50 = 2350 * 2 / 100 = 4700 / 100 = 47
2600 / 25 = 2600 * 4 / 100 = 10400 / 100 = 104

Subtract from 1000

To subtract a number from 1000, subtract each digit of the number from “9” and subtract the last digit from 10.

For example:
1000 – 248 = (9-2) _ (9-4) _ (10-8) = 752

Multiplying prime numbers

This method is often called diagonal. Above the numbers we add how much they lack to “10”, subtract diagonally and get the 1st digit of the number, then multiply the upper numbers and write down the 2nd digit.

Example, multiply 7 by 8: 3 __ 2
7 8
8 – 3 = 5 _
3 * 2 = 6
Total: 56

Multiply numbers from 10 to 20

In order to quickly multiply numbers from 10 to 20 in your head, you should know one trick: add the units of another to one number, multiply the sum by 10, and add the product of the units to the resulting result.

Example:
13 * 15 = (13 + 5) * 10 + 3 * 5 = 180 + 15 = 195

Adding and subtracting natural numbers

1. If a term is increased by a certain number, then the same number should be subtracted from the resulting amount.

For example:
650 + 346 = (650 + 346 + 4) – 4 = (650 + 350) – 2 = 1000 – 2 = 998

2. If one term is reduced by a certain number and the same number is added to the second term, the sum will not change.

For example:
335 + 765 = (335 + 5) + (765 - 5) = 340 + 760 = 1100

3. If you add the same number to the minuend and subtrahend, the result will not change.

For example:
225 - 339 = (225 + 5) - (339 + 5) = 230 - 344 = 114

We multiply numbers with the same number of tens, the sum of whose units = 10

The arithmetic is quite simple: we multiply the tens of one of the factors by the number greater than “1”, multiply the units, and write down the result one by one.

For example:
302 * 308 = ..
1). 30 * (30 + 1) = 900 + 30 = 930
2). 2 * 8 = 16
Multiply by a number consisting of digits 9

How to multiply by the number 9, 99, 999?

To do this, simply add the missing units and perform the calculation.

Example:
154 * 99 = 154 * (100 - 1) = 15400 - 154 = 15246
Add numbers that are close in size

We calculate a series of numbers that are close in value

They can be expanded and folded in parts.

For example:
19 + 22 + 23 + 21+ 24 + 17=…

Let's expand the terms:
19 = 20 - 1
22 = 20 + 2
23 = 20 + 3
21 = 20 + 1
24 = 20 + 4
17 = 20 -3

Total: 20 * 6 + (2-1+3+1+4-3) = 120 + 6 = 126

We hope that our tips will help you master the techniques of quick mental counting. It should be remembered that theory is only 20% of success. The remaining 80% is your desire and practice.

Effective mental arithmetic or brain exercise

• Mathematics

This article is inspired by the topic and is intended to spread the techniques of S.A. Rachinsky for mental counting.
Rachinsky was a wonderful teacher who taught in rural schools in the 19th century and showed from his own experience that it is possible to develop the skill of rapid mental calculation. For his students it was not a particular problem to count similar example in my mind:

Using round numbers

Because on 10 , 100 , 1000 etc. it’s faster to multiply round numbers, in your mind you need to reduce everything to these simple operations, How 18 x 100 or 36 x 10. Accordingly, it is easier to add by “splitting off” a round number and then adding a “tail”: 1800 + 200 + 190 .
Another example:
31 x 29 = (30 + 1) x (30 - 1) = 30 x 30 - 1 x 1 = 900 - 1 = 899.

Let's simplify multiplication by division

Squaring double digit number

Rachinsky's technique is as follows. In order to find the square of any two-digit number, you need the difference between this number and 25 multiply by 100 and add the square of the complement to the resulting product given number before 50 or the square of its excess over 50 -Yu. For example,
37^2 = 12 x 100 + 13^2 = 1200 + 169 = 1369; 84^2 = 59 x 100 + 34^2 = 5900 + 9 x 100 + 16^2 = 6800 + 256 = 7056;
In general ( M- two-digit number):

Let's try to apply this trick when squaring a three-digit number, first breaking it into smaller terms:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 70 x 100 + 45^2 = 10000 + (90+5) x 2 x 100 + + 7000 + 20 x 100 + 5^2 = 17000 + 19000 + 2000 + 25 = 38025.
Hmm, I wouldn’t say that it’s much easier than erecting it in a column, but perhaps over time you can get used to it.
And, of course, you should start training by squaring two-digit numbers, and from there you can even get to disassembling in your head.

Multiplying two-digit numbers

For example, let's calculate 77 x 13. The sum of the units of these numbers is equal to 10 , because 7 + 3 = 10 . First we put the smaller number before the larger one: 77 x 13 = 13 x 77.
To get round numbers, we take three units from 13 and add them to 77 . Now let's multiply the new numbers 80 x 10, and to the result we add the product of the selected 3 units by the difference of the old number 77 and a new number 10 :
13 x 77 = 10 x 80 + 3 x (77 - 10) = 800 + 3 x 67 = 800 + 3 x (60 + 7) = 800 + 3 x 60 + 3 x 7 = 800 + 180 + 21 = 800 + 201 = 1001.
This technique has special case: everything becomes much simpler when two factors have the same number of tens. In this case, the number of tens is multiplied by the number following it and the product of the units of these numbers is added to the resulting result. Let's see how elegant this technique is with an example.
48 x 42. Tens number 4 , next number: 5 ; 4 x 5 = 20 . Product of units: 8 x 2 = 16 . So 48 x 42 = 2016.
99 x 91. Tens number: 9 , next number: 10 ; 9 x 10 = 90 . Product of units: 9 x 1 = 09 . So 99 x 91 = 9009.
Yeah, that is, to multiply 95 x 95, just count 9 x 10 = 90 And 5 x 5 = 25 and the answer is ready:
95 x 95 = 9025.
Then the previous example can be calculated a little simpler:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 9025 = 10000 + (90+5) x 2 x 100 + 9000 + 25 = 10000 + 19000 + 1000 + 8000 + 25 = 38025.

Instead of a conclusion

References:
“1001 problems for mental arithmetic at the school of S.A. Rachinsky".

IN Lately In Russia, a new method for developing intelligence is beginning to gain popularity in our country. Instead of the usual chess sections, parents send their children to mental arithmetic schools. How kids are taught to count in their heads, how much such classes cost and what experts say about them - in the material "AiF-Volgograd".

What is mental arithmetic?

Mental arithmetic is Japanese technique development of a child’s intellectual abilities through calculations on special abacus “soroban”, which is sometimes called “abacus”.

“When performing actions with numbers in their minds, children imagine these abacus and in a split second they mentally add, subtract, multiply and divide any numbers - even three-digit, even six-digit,” says Natalya Chaplieva, teacher of the Volga club, where children are taught using this method.

According to her, when children are just learning all these actions, they count the numbers directly on the soroban, fingering the bones. Then they gradually move from counting to a “mental map” - a picture depicting them. At this stage of learning, they stop touching the abacus and begin to imagine in their minds how they move the bones on it. Then, the children stop using the mental map and begin to completely visualize the soroban for themselves.

Abacus soroban. Photo: AiF/ Evgeniy Strokan

“We recruit children from 4 to 12 years old into groups. At this age, the brain is most plastic; the child absorbs information like a sponge, and therefore easily masters learning methods. It’s much more difficult for an adult to learn mental arithmetic,” says Ekaterina Grigorieva, teacher of the mental arithmetic club.

How much does it cost?

The abacus has a rectangular frame that contains 23-31 spokes, each of which has 5 bones strung on them, separated by a transverse crossbar. Above it there is one domino, which denotes “five”, and below it there are 4 dominoes, denoting ones.

You need to move the bones with only two fingers - the thumb and forefinger. The counting on the soroban starts from the very first knitting needle on the right. It stands for units. The knitting needle to the left of it is tens, the next one is hundreds, etc.

Soroban is not sold in regular stores. You can buy such accounts on the Internet. Depending on the number of knitting needles and material, the price of soroban can range from 170 to 1,000 rubles.

At the first stage, children work with abacus. Photo: AiF/ Evgeniy Strokan

If you don’t want to spend money on bills at all, you can download a free application for your phone - an online simulator that simulates an abacus.

Classes mental arithmetic for children in Volgograd they cost about 500-600 rubles per hour. You can buy a subscription for 8 classes for 4,000 rubles and 16 classes for 7,200 rubles. Classes are held 2 times a week. The Volga school gives out abacus, mental maps and notebooks to children free of charge, and students can take them home. At the end of the course, the child can keep the soroban as a souvenir.

Children have to learn mental arithmetic for about 1-2 years, depending on their abilities.

Assignments for students. Photo: AiF/ Evgeniy Strokan

If you don't have money for classes at special school, then you can try to search for video tutorials on YouTube. True, some of them are posted on the website by organizations providing lessons for money for the purpose of self-promotion. Their videos are very short - 3 minutes long. With their help you can learn the basics of mental arithmetic, but nothing more.

What do experts say about this?

Teachers who conduct mental arithmetic classes are confident that the training is worth the money spent on it.

“Mental arithmetic develops well the child’s imagination, creativity, thinking, memory, fine motor skills, attentiveness, perseverance. Classes are aimed at ensuring that the child develops both hemispheres at the same time, which is very important, because the traditional preparation of a child for school develops only the right hemisphere of the brain,” believes teacher Natalya Chaplieva.

Psychologist Natalya Oreshkina believes that in the case of children 4-5 years old, mental arithmetic classes will be effective only if they take place in a playful way.

“Children of this age generally have difficulty concentrating for such a time, unless we're talking about not about watching a cartoon, says the expert. - But if the lesson is structured in a playful way, if children practice abacus and color something, then they will learn knowledge while being in their natural environment - in a game. In addition, it should not be difficult for children; they should not exceed the permissible load level. For example, for 4-year-olds, classes should last no more than 30 minutes. I can say that mental arithmetic for children it is very interesting. But if a child lags behind his peers in some way, then such activities will be too difficult for him. If a child does not have an internal resource for activities, then it will be a waste of time, effort and money.”

Quick counting techniques: magic accessible to everyone

In order to understand what role numbers play in our lives, perform a simple experiment. Try to do without them for a while. Without numbers, without calculations, without measurements... You will find yourself in strange world, where you will feel absolutely helpless, tied hand and foot. How to make it to a meeting on time? Can you tell one bus from another? Make a phone call? Buy bread, sausage, tea? Cook soup or potatoes? Without numbers, and therefore without counting, life is impossible. But how difficult this science is sometimes! Try quickly multiplying 65 by 23? Does not work? The hand itself reaches for a mobile phone with a calculator. Meanwhile, semi-literate Russian peasants 200 years ago calmly did this, using only the first column of the multiplication table - multiplication by two. Don't believe me? But in vain. This is reality.

Stone Age "computer"

Even without knowing the numbers, people were already trying to count. If our ancestors, who lived in caves and wore skins, needed to exchange something with a neighboring tribe, they did it simply: they cleared the area and laid out, for example, an arrowhead. A fish or a handful of nuts lay nearby. And so on until one of the exchanged goods ran out, or the head of the “trade mission” decided that enough was enough. It’s primitive, but very convenient in its own way: you won’t get confused and won’t be deceived.

With the development of cattle breeding, the tasks became more complicated. A large herd had to be counted somehow in order to know whether all the goats or cows were there. The “calculating machine” of the illiterate but smart shepherds was a hollowed-out pumpkin with pebbles. As soon as the animal left the pen, the shepherd placed a pebble in the pumpkin. In the evening the herd returned, and the shepherd took out a pebble with each animal that entered the pen. If the pumpkin was empty, he knew that the herd was all right. If there were stones left, he went to look for the loss.

When the numbers came in, things got better. Although for a long time our ancestors had only three numerals in use: “one”, “pair” and “many”.

Is it possible to count faster than a computer?

Overtake a device performing hundreds of millions of operations per second? Impossible... But the one who says this is cruelly disingenuous, or simply deliberately overlooks something. A computer is just a set of chips in plastic; it does not count on its own.

Let's pose the question differently: can a person, counting in his head, outperform someone who does calculations on a computer? And here the answer is yes. After all, in order to receive a response from the “black suitcase”, the data must first be entered into it. This will be done by a person using his fingers or voice. And all these actions have time limits. Insurmountable restrictions. Nature itself supplied them to the human body. Everything - except one organ. Brain!

The calculator can perform only two operations: addition and subtraction. For him, multiplication is multiple addition, and division is multiple subtraction.

Our brains act differently.

The class where the future king of mathematics, Carl Gauss, studied, once received a task: add all the numbers from 1 to 100. Carl wrote the absolutely correct answer on his board as soon as the teacher finished explaining the task. He did not diligently add the numbers in order, as any self-respecting computer would do. He applied the formula he himself discovered: 101 x 50 = 5050. And this is far from the only technique that speeds up mental calculations.

The simplest techniques for quick counting

They are studied at school. The simplest thing: if you need to add 9 to any number, add 10 and subtract 1 if 8 (+ 10 - 2), 7 (+ 10 - 3), etc.

54 + 9 = 54 + 10 - 1 = 63. Fast and convenient.

Two-digit numbers add just as easily. If the last digit in the second term is greater than five, the number is rounded to the next ten, and then the “extra” is subtracted. 22 + 47 = 22 + 50 - 3 = 69. If the key number is less than five, then you need to add the tens first, then the ones: 27 + 51 = 20 + 50 + 7 + 1 = 78.

WITH three digit numbers in the same way, no difficulties arise. We add them up as we read, from left to right: 321 + 543 = 300 + 500 + 20 + 40 + 1 + 3 = 864. Much easier than in a column. And much faster.

What about subtraction? The principle is the same: we round what is subtracted to a whole number and add what is missing: 57 - 8 = 57 - 10 + 2 = 49; 43 - 27 = 43 - 30 + 3 = 16. Faster than using a calculator - and no complaints from the teacher, even during the test!

Do I need to learn the multiplication table?

Children, as a rule, cannot stand this. And they do it right. There's no point in teaching her! But don’t rush to be indignant. No one is saying that you don't need to know the table.

Its invention is attributed to Pythagoras, but, most likely, the great mathematician only gave a complete, laconic form to what was already known. At the excavations of ancient Mesopotamia, archaeologists found clay tablets with the sacramental: “2 x 2”. People have been using this extremely convenient system of calculations for a long time and have discovered many ways that help to comprehend the internal logic and beauty of the table, to understand it - and not to stupidly, mechanically memorize it.

IN ancient China We started learning the table by multiplying by 9. It’s easier this way, not least because you can multiply by 9 “on your fingers.”

Place both hands on the table, palms down. The first finger on the left is 1, the second is 2, etc. Let's say you need to solve the example 6 x 9. Raise your sixth finger. The fingers on the left will show tens, on the right - ones. Answer 54.

Example: 8 x 7. Left hand- the first multiplier, the right one - the second. There are five fingers on the hand, but we need 8 and 7. We bend three fingers on the left hand (5 + 3 = 8), on the right hand 2 (5 + 2 = 7). We have five bent fingers, which means five dozen. Now let's multiply the remaining ones: 2 x 3 = 6. These are units. Total 56.

This is just one of the simplest “finger” multiplication techniques. There are many of them. You can operate with numbers up to 10,000 on your fingers!

The “finger” system has a bonus: the child perceives it as fun game. Engages willingly, experiences a lot positive emotions and as a result, very soon he begins to perform all operations in his mind, without the help of his fingers.

You can also divide using your fingers, but it is a little more difficult. Programmers still use their hands to convert numbers from decimal to binary - it is more convenient and much faster than on a computer. But within school curriculum You can learn to quickly divide even without fingers, in your mind.

Let's say we need to solve example 91: 13. Column? There is no need to dirty the paper. The dividend ends in one. And the divisor is by three. What is the very first thing in the multiplication table that involves a three and ends with a one? 3 x 7 = 21. Seven! That's it, we caught her. You need 84: 14. Remember the table: 6 x 4 = 24. The answer is 6. Simple? Still would!

The magic of numbers

Most fast counting techniques are similar to magic tricks. Take at least famous example multiplying by 11. To, for example, 32 x 11, you need to write 3 and 2 at the edges, and put their sum in the middle: 352.

To multiply a two-digit number by 101, you simply write the number twice. 34 x 101 = 3434.

To multiply a number by 4, you need to multiply it by 2 twice. To divide, divide it by 2 twice.

Many witty and, most importantly, quick tricks help raise a number to a power, extract Square root. The famous "30 techniques of Perelman" for mathematical thinking people will be cooler than Copperfield's shows, because they also UNDERSTAND what is happening and how it is happening. Well, the rest can just enjoy the beautiful focus. For example, you need to multiply 45 by 37. Write the numbers on a sheet of paper and divide them with a vertical line. Divide the left number by 2, discarding the remainder until we get one. Right - multiply until the number of lines in the column is equal. Then we cross out from the RIGHT column all those numbers opposite which in the LEFT column we got an even result. We add up the remaining numbers from the right column. The result is 1665. Multiply the numbers in the usual way. The answer will fit.

"Charge" for the mind

Quick counting techniques can greatly make life easier for a child at school, for a mother in a store or in the kitchen, and for a father at work or in the office. But we prefer a calculator. Why? We don't like to strain ourselves. It's hard for us to keep numbers, even two-digit ones, in our heads. For some reason they don't hold up.

Try going to the middle of the room and doing the splits. For some reason it doesn’t “plant”, right? And the gymnast does it completely calmly, without straining. Need to train!

The easiest way to train and, at the same time, warm up the brain: mentally count out loud (required!) through numbers to one hundred and back. In the morning, while standing in the shower, or while preparing breakfast, count: 2.. 4.. 6.. 100... 98.. 96. You can count in three, in eight - the main thing is to do it out loud. After just a couple of weeks of regular practice, you will be surprised how much EASIER it will become to handle numbers.