How easy it is to count large numbers in your mind. How to learn to quickly count complex numbers in your head

In the age of calculators and cash registers, we rarely have to count in our heads. We completely rely on computer technology, although it can fail, or it may simply not be at hand at the right time. Unbeknownst to ourselves, we lose the skills of fast and accurate counting, and sometimes we realize very late that this is our weak point. However, the ability to quickly count in the mind is an undeniable advantage and dignity of someone who has such a skill. A person who easily operates with numbers will never be deceived in calculations. But most importantly, the ability to calculate will constantly be kept in good shape and develop his mental abilities, which is especially important for children and young people during the training period.

The sooner the child is taught to count at home and in kindergarten, the more likely it is that the process of operating with larger numerical values ​​and all mathematical operations, including multiplication and division, will go faster and will be easier for the child.

In teaching children under 4 years old, it is better to use visual material. You need to count everything that is possible. Small flocks of birds, cats basking in the sun, motorcyclists roaring past you, bright fire engines rushing to a fire - everything that attracts attention can be counted. Simultaneously with counting skills, the child will develop attention and observation. Gradually increase the difficulty of the tasks. In the morning, on the way to kindergarten, you saw two cats, and returning home, three more. Tell your child: “Well, what a lot of cats in our yard! How many cats have we seen today? Praise the baby for observation and accuracy, because these are qualities that will be very useful to him in life.

In the elementary grades, the child must absolutely freely and quickly make any calculations within the limits determined by the school curriculum. In order to learn how to count quickly, you need to constantly train. Therefore, the task of parents is to constantly encourage the child to count and make this activity interesting for the child. The more often you train your baby in counting, the easier it will be for him to make quick and accurate calculations in his mind.

How to learn to count quickly as an adult
If a child has been taught quick counting since childhood, over time he will learn to operate with large values ​​​​without much effort. But if a student or a person of a more respectable age decided to master the skills of quick counting, then a simple technique will have to be applied, the development of which, with a certain perseverance, will certainly bring positive results.

Like any training, you need to start small. If you know the multiplication table well, that's good. If you forgot, or never knew, use this method of counting. For example, you need to find out how much 9 will be multiplied by 7. We write the example in this way:

1 3
------- = 63
9 x 7

Answer 63 we got by simple calculations. Namely. Having written down the example 9x7, we draw a straight line over it and enter over each number how much is missing to 10. Over 9 we write 1, over 7 we write 3. The first digit of the answer will be the difference between the numbers of the bottom line and the top line diagonally. 9-3= 6, 7-1=6 - you can take any pair for calculation - the answer will always be the same. So, we have calculated that the first digit of the answer will be 6. Now we calculate the second digit. To do this, multiply the numbers of the top row 1x3=3. Our example is solved: 9x7=63.

Larger numerical values ​​are calculated slightly differently. For example, you need to find out how much 12x14 will be.

2 4
---------- = 160+8=168
12 x 14

In the bottom line we write an example 12x14. In the top line we write how many these numbers are greater than 10. We get 2 and 4. We add the numbers diagonally. We get 12+4=16, 14+2=16. We got 16 tens, because our original numbers are more than ten. Therefore, we multiply 16 by 10. 16x10 \u003d 160. It remains only to multiply the upper numbers 2x4 \u003d 8 and add the resulting figure to the answer.

Such calculation methods are difficult only at the very beginning. Therefore, you can start with the simplest examples, gradually complicating the tasks. But in order to learn how to count in your head, you need to completely abandon the use of notes, and make all calculations only in your head.

Children can also be taught using similar methods, but only if they fully cope with the school curriculum. Otherwise, you can not achieve results in a quick account, but harm the assimilation of school knowledge.

Having mastered the manipulation of two-digit numbers, in the future you can master the calculation of multi-digit numbers - hundreds and thousands.

Recently, a new methodology for the development of intelligence has begun to gain popularity in Russia. Instead of the usual chess sections, parents send their children to schools of mental arithmetic. How kids are taught to count in their minds, how much such classes cost and what experts say about them - in the material "AiF-Volgograd".

What is mental arithmetic?

Mental arithmetic is a Japanese technique for developing a child's intellectual abilities using calculations on special "soroban" accounts, sometimes called "abacus".

“By performing actions with numbers in the mind, children imagine these scores and mentally add, subtract, multiply and divide any numbers in a split second, even three-digit, even six-digit,” says Natalia Chaplieva, teacher of the Volga club in which children are taught according to this method.

According to her, when children are just learning all these actions, they count the numbers directly on the soroban, touching the bones with their fingers. Then they gradually move from the account to the "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 as well, starting to visualize the soroban completely.

Soban accounts. Photo: AiF / Eugene Strokan

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

How much does it cost?

The abacus has a rectangular frame containing 23-31 knitting needles, each of which has 5 bones strung together, separated by a crossbar. Above it is one knuckle, which means "five", and below it - 4 knuckles, denoting units.

It is necessary to move the bones only with two fingers - thumb and forefinger. Soroban counting starts from the very first needle on the right. It stands for units. The needle to the left of it is tens, the one following it is hundreds, and so on.

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

At the first stage, children are engaged in accounts. Photo: AiF / Eugene 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 imitates an abacus.

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

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

Tasks for students. Photo: AiF / Eugene Strokan

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

What do the experts say about it?

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

“Mental arithmetic develops well the imagination, creativity of the child, his thinking, memory, fine motor skills, attentiveness, perseverance. Its classes are aimed at ensuring that the child develops both hemispheres at the same time, which is very important, because the traditional preparation of the child for school develops only the right hemisphere of the brain, ”says teacher Natalia 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 can hardly concentrate for such a time, unless we are talking about watching a cartoon,” the expert says. - But if the lesson is built in a playful way, if the children study on the abacus, decorate something, then they will acquire knowledge while being in their natural environment - in the game. In addition, children should not be hard, do not exceed the permissible load level. For example, for 4-year-olds, classes should go no more than 30 minutes. I can say that mental arithmetic for children 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 classes, then it will be a waste of time, effort and money.

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, since having the abilities 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.

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 it is necessary 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.

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 the intersection points 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=?

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.

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.

Pure mathematics is in its way the poetry of the logical idea. Albert Einstein

In this article, we offer you a selection of simple mathematical tricks, many of which are quite relevant in life and allow you to count faster.

1. Fast interest calculation

Perhaps, in the era of loans and installments, the most relevant mathematical skill can be called a virtuoso mental calculation of interest. The fastest way to calculate a certain percentage of a number is to multiply the given percentage by this number and then discard the last two digits in the resulting result, because the percentage is nothing but one hundredth.

How much is 20% of 70? 70 × 20 = 1400. We discard two digits and get 14. When you rearrange the factors, the product does not change, and if you try to calculate 70% of 20, then the answer will also be 14.

This method is very simple in the case of round numbers, but what if you need to calculate, for example, a percentage of the number 72 or 29? In such a situation, you will have to sacrifice accuracy for the sake of speed and round the number (in our example, 72 is rounded up to 70, and 29 to 30), and then use the same trick with multiplying and discarding the last two digits.

2. Quick divisibility check

Can 408 candies be divided equally between 12 children? It is easy to answer this question without the help of a calculator, if we recall the simple signs of divisibility that we were taught back in school.

• A number is divisible by 2 if its last digit is divisible by 2.
• A number is divisible by 3 if the sum of the digits that make up the number is divisible by 3. For example, take the number 501, represent it as 5 + 0 + 1 = 6. 6 is divisible by 3, which means that the number 501 itself is divisible by 3 .
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, take 2340. The last two digits form the number 40, which is divisible by 4.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 6 if it is divisible by 2 and 3.
• A number is divisible by 9 if the sum of the digits that make up the number is divisible by 9. For example, let's take the number 6,390 and represent it as 6 + 3 + 9 + 0 = 18. 18 is divisible by 9, which means the number 6 itself 390 is divisible by 9.
• A number is divisible by 12 if it is divisible by 3 and 4.

3. Fast calculation of the square root

The square root of 4 is 2. Anyone can count that. What about the square root of 85?

For a quick approximate solution, we find the nearest square number to the given one, in this case it is 81 = 9^2.

Now find the next nearest square. In this case it is 100 = 10^2.

The square root of 85 is somewhere between 9 and 10, and since 85 is closer to 81 than it is to 100, the square root of that number is 9 something.

4. Quick calculation of the time after which a cash deposit at a certain percentage will double

Do you want to quickly find out the time it will take for your cash deposit at a certain interest rate to double? There is also no need for a calculator, it is enough to know the “rule of 72”.

We divide the number 72 by our interest rate, after which we get the approximate period after which the deposit will double.

If the deposit is made at 5% per annum, then it will take 14-odd years for it to double.

Why exactly 72 (sometimes they take 70 or 69)? How it works? These questions will be answered in detail by Wikipedia.

5. Quick calculation of the time after which a cash deposit at a certain percentage will triple

In this case, the interest rate on the deposit should become a divisor of 115.

If the deposit is made at 5% per annum, then it will take 23 years for it to triple.

6. Quick calculation of the hourly rate

Imagine that you are interviewing with two employers who do not give salaries in the usual “rubles per month” format, but talk about annual salaries and hourly pay. How to quickly calculate where they pay more? Where the annual salary is 360,000 rubles, or where they pay 200 rubles per hour?

To calculate the payment for one hour of work when voicing the annual salary, it is necessary to discard the last three characters from the named amount, and then divide the resulting number by 2.

360,000 turns into 360 ÷ 2 = 180 rubles per hour. Other things being equal, it turns out that the second proposal is better.

7. Advanced math on fingers

Your fingers are capable of much more than simple addition and subtraction.

With your fingers, you can easily multiply by 9 if you suddenly forgot the multiplication table.

Let's number the fingers on the hands from left to right from 1 to 10.

If we want to multiply 9 by 5, then we bend the fifth finger from the left.

Now let's look at the hands. It turns out four unbent fingers to bent. They represent tens. And five unbent fingers after the bent one. They represent units. Answer: 45.

If we want to multiply 9 by 6, then we bend the sixth finger from the left. We get five unbent fingers before the bent finger and four after. Answer: 54.

Thus, you can reproduce the entire column of multiplication by 9.

8. Fast multiplication by 4

There is an extremely easy way to lightning-fast multiply even large numbers by 4. To do this, it is enough to decompose the operation into two steps, multiplying the desired number by 2, and then again by 2.

See for yourself. Not everyone can multiply 1,223 immediately by 4 in their minds. And now we do 1223 × 2 = 2446 and then 2446 × 2 = 4892. This is much easier.

9. Quick determination of the required minimum

Imagine that you are taking a series of five tests, for which you need a minimum score of 92 to pass. The last test remains, and the results for the previous ones are: 81, 98, 90, 93. How to calculate the required minimum that you need to get in the last test?

To do this, we consider how many points we missed / went over in the tests already passed, denoting the shortage with negative numbers, and the results with a margin - positive.

So, 81 − 92 = −11; 98 - 92 = 6; 90 - 92 = -2; 93 - 92 = 1.

Adding these numbers, we get the adjustment for the required minimum: -11 + 6 - 2 + 1 = -6.

It turns out a deficit of 6 points, which means that the required minimum increases: 92 + 6 = 98. Things are bad. :(

10. Quick representation of the value of an ordinary fraction

The approximate value of an ordinary fraction can be very quickly represented as a decimal fraction, if you first bring it to simple and understandable ratios: 1/4, 1/3, 1/2 and 3/4.

For example, we have a fraction 28/77, which is very close to 28/84 = 1/3, but since we increased the denominator, the original number will be slightly larger, that is, slightly more than 0.33.

11. Number Guessing Trick

You can play a bit of David Blaine and surprise your friends with an interesting but very simple math trick.

1. Ask a friend to guess any whole number.
2. Let him multiply it by 2.
3. Then add 9 to the resulting number.
4. Now let's subtract 3 from the resulting number.
5. And now let him divide the resulting number in half (it will be divided without a remainder anyway).
6. Finally, ask him to subtract from the resulting number the number that he thought of at the beginning.

The answer will always be 3.

Yes, very stupid, but often the effect exceeds all expectations.

Bonus

And, of course, we could not help but insert into this post that same picture with a very cool way of multiplying.

Bibliographic description: Vladimirov A. I., Mikhailova V. V., Shmeleva S. P. Interesting ways of fast counting // Young scientist. 2016. №6.1. S. 15-17..02.2019).



Introduction

Mental counting is gymnastics for the mind. Mental counting is the oldest way of calculating. Mastering computational skills develops memory and helps to assimilate subjects of the natural and mathematical cycle.

There are many ways to simplify arithmetic operations. Knowledge of simplified calculation techniques is especially important in cases where the calculator does not have tables and a calculator at his disposal.

We want to dwell on the methods of addition, subtraction, multiplication, division, for the production of which it is enough to count or use a pen and paper.

The motivation for choosing the topic was the desire to continue the formation of computational skills, the ability to quickly and clearly find the result of mathematical operations.

The rules and techniques of calculations do not depend on whether they are performed in writing or orally. However, mastering the skills of oral calculations is of great value not because they are used more often in everyday life than written calculations. This is also important because they speed up written calculations, gain experience in rational calculations, and give a gain in computational work.

In mathematics lessons, we have to do a lot of oral calculations, and when the teacher showed us how to quickly multiply by the numbers 11, we had an idea if there were still methods of quick calculation. We set ourselves the task of finding and testing other methods of fast calculation.

b) to do well in school; (16%)

c) to decide quickly; (16%)

d) to be literate; (52%)

2. List, when studying, which school subjects you will need to count correctly ?

a) mathematics; (80%)

b) physics; (15%)

c) chemistry; (5%)

d) technology;

e) music;

3. Do you know how to count quickly?

a) yes, a lot;

b) yes, a few (85%);

c) no, I don't know (15%).

4. Do you use fast counting techniques in calculations?

b) no (85%)

5. Would you like to learn quick counting techniques to quickly count?

b) no (8%).

They say that if you want to learn how to swim, you must enter the water, and if you want to be able to solve problems, you must start solving them. But first you need to master the basics of arithmetic. You can learn to count quickly, count in your mind only with a great desire and systematic training in solving problems.

But the methods of fast mental counting have been known for a long time. The excellent mental arithmetic abilities of such brilliant mathematicians as Gauss, von Neumann, Euler or Wallis are a real delight. Much has been written about this. We want to tell and show some well-known computational secrets. And then a completely different math will open before you. Lively, useful and understandable.

1. Methods for fast multiplication

1. COUNTING ON FINGERS

A way to quickly multiply numbers within the first ten by 9.

Let's say we need to multiply 7 by 9.

Let's turn our hands with palms facing us and bend the seventh finger (starting to count from the thumb to the left).

The number of fingers to the left of the bent one will be equal to tens, and to the right - units of the desired product.

Rice. 1. Finger counting

2. MULTIPLICATION OF NUMBERS FROM 10 TO 20

It is very easy to multiply such numbers.

To one of the numbers it is necessary to add the number of units of the other, multiply by 10 and add the product of units of numbers.

Example 1. 16∙18=(16+8) ∙ 10+6 ∙ 8=288, or

Example 2. 17 ∙ 17=(17+7) ∙ 10+7 ∙ 7=289.

Task: Multiply quickly 19 ∙ 13. Answer 19 ∙13=(19+3) ∙10 +9 ∙3=247.

3. MULTIPLY BY 11

To multiply a two-digit number whose sum of digits does not exceed 10 by 11, you need to move the digits of this number apart and put the sum of these digits between them.

72 ∙ 11 = 7 (7 + 2) 2 = 792;

35 ∙ 11 = 3 (3 + 5) 5 = 385.

To multiply by 11 a two-digit number whose sum of digits is 10 or more than 10, you must mentally push the digits of this number, put the sum of these digits between them, and then add one to the first digit, and leave the second and last (third) unchanged.

Example .

94 ∙ 11 = 9 (9 + 4) 4 = 9 (13) 4 = (9 + 1) 34 = 1034.

Task: Multiply quickly 54 ∙ 11 (594)

Task: Multiply quickly 67∙ 11 (737)

4. MULTIPLYING BY 22, 33, ..., 99

To multiply a two-digit number by 22, 33, ..., 99, this multiplier must be represented as a product of a single-digit number (from 2 to 9) by 11, that is, 44 \u003d 4 11; 55 = 5 ∙ 11 etc. Then multiply the product of the first numbers by 11.

Example 1. 24 ∙ 22 = 24 ∙ 2 ∙ 11 = 48 ∙ 11 = 528

Example 2. 23 ∙ 33 = 23 ∙ 3 ∙ 11= 69 ∙ 11 = 759

Task: Multiply 18∙44

5. MULTIPLY BY 5, BY 50, BY 25, BY 125

When multiplying by these numbers, you can use the following expressions:

a ∙ 5=a ∙ 10:2 a ∙ 50=a ∙ 100:2

a ∙ 25=a ∙ 100:4 a ∙ 125=a ∙ 1000:8

Example1. 17 ∙ 5=17 ∙ 10:2=170:2=85

Example 2. 43 ∙ 50=43 ∙ 100:2=4300:2=2150

Example 3. 27 ∙ 25=27 ∙ 100:4=2700:4=675

Example 4. 96 ∙ 125=96:8 ∙ 1000=12 ∙ 1000=12000

Task: multiply 824∙25

Task: multiply 348∙50

&2. Ways to quickly divide

1. DIVISION BY 5, BY 50, BY 25

When dividing by 5, by 50, by 25, you can use the following expressions:

a:5= a ∙ 2:10 a:50=a ∙ 2:100

a:25=a ∙ 4:100

35:5=35 ∙ 2:10=70:10=7

3750:50=3750 ∙ 2:100=7500:100=75

6400:25=6400 ∙ 4:100=25600:100=256

&3. Ways to quickly add and subtract natural numbers.

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

Example. 785+963=785+(963+7)-7=785+970-7= 1748

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.

Example. 762+639=(762+8)+(639-8)=770 + 631=1401

If the subtrahend is reduced by several units and the minuend is increased by the same number of units, then the difference will not change.

Example. 529-435=(529-5)-(435+5)=524-440=84

Conclusion

There are ways to quickly add, subtract, multiply, divide, exponentiate. We have considered only a few ways to quickly count.

All the methods of mental calculation we have considered speak of the long-standing interest of scientists and ordinary people in playing with numbers. Using some of these methods in the classroom or at home, you can develop the speed of calculations, achieve success in the study of all school subjects.

Multiplication without a calculator is a training of memory and mathematical thinking. Computer technology is improving to this day, but any machine does what people put into it, and we have learned some tricks of mental counting that will help us in life.

We were interested in working on the project. So far, we have only studied and analyzed the already known methods of fast counting.

But who knows, perhaps in the future we ourselves will be able to discover new ways of fast computing.

Literature:

1. Arutyunyan E., Levitas G. Entertaining Mathematics. - M .: AST - PRESS, 1999. - 368 p.
2. Gardner M. Mathematical miracles and secrets. - M., 1978.
3. Glazer G.I. History of mathematics at school. - M., 1981.
4. "First of September" Mathematics No. 3 (15), 2007.
5. Tatarchenko T.D. Methods for quick counting in the classroom, "Mathematics at School", 2008, No. 7, p.68.
6. Oral account / Comp. P.M. Kamaev. - M .: Chistye Prudy, 2007 - Library "First of September", series "Mathematics". Issue. 3(15).
7. http://portfolio.1september.ru/subject.php