Non-traditional ways to multiply multi-digit numbers. Scientific research work “Non-standard counting algorithms or fast counting without a calculator Unusual ways of multiplication

Research work in mathematics in elementary school

Brief abstract of the research paper
Each student knows how to multiply multi-digit numbers with a “column”. In this paper, the author draws attention to the existence of alternative methods of multiplication available to younger students, which can turn "tedious" calculations into a fun game.
The paper considers six non-traditional methods of multiplication multi-digit numbers used in various historical eras: Russian peasant, lattice, small castle, Chinese, Japanese, according to the table of V. Okoneshnikov.
The project is designed to develop cognitive interest in the subject being studied, to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3
Chapter 1. Alternative Ways of Multiplication 4
1.1. A bit of history 4
1.2. Russian peasant way of multiplying 4
1.3. Multiplication using the "Little Castle" method 5
1.4. Multiplication of numbers by the method of "jealousy" or "lattice multiplication" 5
1.5. Chinese multiplication method 5
1.6. Japanese multiplication method 6
1.7. Table Okoneshnikov 6
1.8. Multiplication by a column. 7
Chapter 2. Practical part 7
2.1. Peasant way 7
2.2. Little castle 7
2.3. Multiplication of numbers by the method of "jealousy" or "lattice multiplication" 7
2.4. Chinese way 8
2.5. Japanese way 8
2.6. Table Okoneshnikov 8
2.7. Questionnaire 8
Conclusion 9
Annex 10

"The subject of mathematics is so serious that it is useful to take opportunities to make it a little entertaining."
B. Pascal
Introduction
Man in Everyday life impossible to do without calculations. Therefore, in mathematics lessons, we are first of all taught to perform operations on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways for everyone that are studied at school. The question arose: are there any other alternative ways of computing? I wanted to study them in more detail. In order to answer these questions, this study was carried out.
The purpose of the study: to identify non-traditional methods of multiplication in order to study the possibility of their application.
In accordance with the goal, we formulated the following tasks:
- Find as many unusual ways of multiplication as possible.
- Learn to apply them.
- Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting.
- Check in practice the multiplication of multi-digit numbers.
- Conduct a survey of 4th grade students
Object of study: various non-standard multi-digit multiplication algorithms
Subject of research: mathematical action "multiplication"
Hypothesis: If there are standard ways to multiply multi-digit numbers, perhaps there are alternative ways.
Relevance: dissemination of knowledge about alternative methods of multiplication.
Practical significance. In the course of the work, many examples were solved and an album was created, which includes examples with various algorithms for multiplying multi-valued numbers in several alternative ways. This may interest classmates to expand their mathematical horizons and serve as the beginning of new experiments.

Chapter 1

1.1. A bit of history
The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower methods were used. And if a modern schoolboy could go back five hundred years, he would amaze everyone with the speed and accuracy of his calculations. The rumor about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful counters of that era, and people would come from all over to study with the new great master.
The operations of multiplication and division were especially difficult in the old days.
In V. Bellyustin’s book “How People Gradually Came to True Arithmetic”, 27 methods of multiplication are outlined, and the author notes: “It is quite possible that there are more methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.” And all these methods of multiplication competed with each other and were assimilated with great difficulty.
Consider the most interesting and simple ways multiplication.
1.2. Russian peasant way of multiplication
In Russia, 2-3 centuries ago, a method was spread among the peasants of some provinces that did not require knowledge of the entire multiplication table. It was only necessary to be able to multiply and divide by 2. This method was called the peasant method.
To multiply two numbers, they were written side by side, and then the left number was divided by 2, and the right one was multiplied by 2. Record the results in a column until 1 remains on the left. The remainder is discarded. We cross out those lines in which there are even numbers on the left. The remaining numbers in the right column are added.
1.3. Multiplication using the "Little Castle" method
The Italian mathematician Luca Pacioli in his treatise "The sum of knowledge in arithmetic, ratios and proportionality" (1494) gives eight different methods of multiplication. The first of them is called "Little Castle".
The advantage of the “Little Castle” multiplication method is that the digits of the highest digits are determined from the very beginning, and this can be important if you need to quickly estimate the value.
The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.
1.4. Multiplying numbers using the "jealousy" or "lattice multiplication" method
Luca Pacioli's second method is called "jealousy" or "lattice multiplication".
First, a rectangle is drawn, divided into squares. Then the square cells are divided diagonally and “... it turns out a picture that looks like lattice shutters, blinds,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing passers-by from seeing the ladies and nuns sitting at the windows.”
Multiplying each digit of the first factor with each digit of the second, the products are written in the corresponding cells, placing tens above the diagonal, and units below it. The digits of the product are obtained by adding the digits in oblique stripes. The results of additions are recorded under the table, as well as to the right of it.
1.5. Chinese multiplication method
Now let's imagine a method of multiplication, heatedly discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of lines are considered, which correspond to the number of digits of each digit of both factors.
1.6. Japanese multiplication method
The Japanese multiplication method is a graphical method using circles and lines. No less funny and interesting than Chinese. Even something like him.
1.7. Okoneshnikov's table
Candidate of Philosophy Vasily Okoneshnikov, part-time inventor new system of mental counting, believes that schoolchildren will be able to learn to verbally add and multiply millions, billions and even sextillions with quadrillions. According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.
According to the scientist, before becoming a computing "computer", it is necessary to memorize the table he created.
The table is divided into 9 parts. They are arranged according to the principle of a mini calculator: on the left in the lower corner "1", on the right in the upper corner "9". Each part is a multiplication table of numbers from 1 to 9 (according to the same "push-button" system). In order to multiply any number, for example, by 8, we find a large square corresponding to the number 8 and write out from this square the numbers corresponding to the digits of the multi-valued multiplier. We add the resulting numbers especially: the first digit remains unchanged, and all the rest are added in pairs. The resulting number will be the result of the multiplication.
If the addition of two digits results in a number greater than nine, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.
The new methodology was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed the publication of a new multiplication table in squared notebooks along with the usual Pythagorean table - so far just for acquaintance.
1.8. Column multiplication.
Not many people know that Adam Rize should be considered the author of our usual method of multiplying a multi-digit number by a multi-digit number by a column (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical part
Mastering the listed methods of multiplication, many examples were solved, an album with samples of various calculation algorithms was designed. (Appendix). Let's consider the calculation algorithm with examples.
2.1. peasant way
Multiply 47 by 35 (Appendix 1),
-write the numbers on one line, draw a vertical line between them;
-we will divide the left number by 2, multiply the right number by 2 (if a remainder occurs during division, then we discard the remainder);
- division ends when a unit appears on the left;
- cross out those lines in which there are even numbers on the left;
We add the numbers remaining on the right - this is the result.
35 + 70 + 140 + 280 + 1120 = 1645.
Output. The method is convenient because it is enough to know the table only by 2. However, when working with large numbers, it is very cumbersome. Convenient for working with two-digit numbers.
2.2. small castle
(Appendix 2). Output. The method is very similar to our modern "column". Moreover, the numbers of the highest ranks are immediately determined. This is important if you need to quickly estimate the value.
2.3. Multiplying numbers using the "jealousy" or "lattice multiplication" method
Let's multiply, for example, the numbers 6827 and 345 (Appendix 3):
1. We draw a square grid and write one of the multipliers above the columns, and the second - in height.
2. Multiply the number of each row sequentially by the numbers of each column. We successively multiply 3 by 6, by 8, by 2 and by 7, etc.
4. Add up the numbers following the diagonal stripes. If the sum of one diagonal contains tens, then we add them to the next diagonal.
From the results of adding the numbers along the diagonals, the number 2355315 is compiled, which is the product of the numbers 6827 and 345, that is, 6827 ∙ 345 = 2355315.
Output. The "lattice multiplication" method is no worse than the conventional one. It is even simpler, since numbers are entered into the cells of the table directly from the multiplication table without simultaneous addition, which is present in the standard method.
2.4. Chinese way
Suppose you need to multiply 12 by 321 (Appendix 4). On a sheet of paper, alternately draw lines, the number of which is determined from this example.
We draw the first number - 12. To do this, from top to bottom, from left to right, we draw:
one green stick (1)
and two orange (2).
We draw the second number - 321, from bottom to top, from left to right:
three blue sticks (3);
two red (2);
one lilac (1).
Now we separate the intersection points with a simple pencil and proceed to count them. We move from right to left (clockwise): 2, 5, 8, 3.
Read the result from left to right - 3852
Output. An interesting way, but to draw 9 straight lines when multiplied by 9 is somehow long and uninteresting, and then count more intersection points. Without skill, it is difficult to understand the division of a number into digits. In general, you can’t do without a multiplication table!
2.5. Japanese way
Multiply 12 by 34 (Appendix 5). Since the second factor is a two-digit number, and the first digit of the first factor is 1, we build two single circles in the top row and two binary circles in the bottom row, since the second digit of the first factor is 2.
Since the first digit of the second factor is 3, and the second is 4, we divide the circles of the first column into three parts, the second column into four parts.
The number of parts into which the circles are divided is the answer, that is, 12 x 34 = 408.
Output. The method is very similar to Chinese graphic. Only straight lines are replaced by circles. It is easier to determine the digits of a number, but drawing circles is less convenient.
2.6. Okoneshnikov's table
It is required to multiply 15647 x 5. We immediately recall the large “button” 5 (it is in the middle) and on it we mentally find small buttons 1, 5, 6, 4, 7 (they are also located, as on a calculator). They correspond to the numbers 05, 25, 30, 20, 35. We add the resulting numbers: the first digit is 0 (remains unchanged), 5 is mentally added to 2, we get 7 - this is the second digit of the result, 5 is added to 3, we get the third digit - 8 , 0+2=2, 0+3=3 and the last digit of the product remains - 5. The result is 78,235.
Output. The method is very convenient, but you need to learn by heart or always have a table at hand.
2.7. Student survey
A survey was conducted among fourth-graders. 26 people took part (Appendix 8). Based on the survey, it was revealed that all respondents know how to multiply in the traditional way. But most guys do not know about non-traditional methods of multiplication. And there are those who want to get to know them.
After the initial survey, extracurricular activity"Fun multiplication", where the guys got acquainted with alternative multiplication algorithms. After that, a survey was conducted in order to identify the most liked methods. The undisputed leader was the most modern method of Vasily Okoneshnikov. (Annex 9)
Conclusion
Having learned to count in all the presented ways, I believe that the most convenient method multiplication is the "Little Castle" method - after all, it is so similar to our current one!
Of all the unusual counting methods I found, the “Japanese” method seemed more interesting. The simplest method seemed to me to be the “doubling and splitting” method used by Russian peasants. I use it when multiplying not too large numbers. It is very convenient to use it when multiplying two-digit numbers.
Thus, I achieved the goal of my research - I studied and learned how to apply non-traditional ways of multiplying multi-digit numbers. My hypothesis was confirmed - I mastered six alternative methods and found out that these are not all possible algorithms.
The unconventional multiplication methods I studied are very interesting and have the right to exist. And in some cases, they are even easier to use. I think that you can talk about the existence of these methods at school, at home and surprise your friends and acquaintances.
So far, we have only studied and analyzed the already known methods of multiplication. But who knows, perhaps in the future we ourselves will be able to discover new ways of multiplying. Also, I do not want to stop there and continue to study non-traditional methods of multiplication.
List of information sources
1. List of references
1.1. Harutyunyan E., Levitas G. Entertaining mathematics. - M.: AST - PRESS, 1999. - 368 p.
1.2. Belyustina V. How people gradually came to real arithmetic. - LKI, 2012.-208 p.
1.3. Depman I. Stories about mathematics. - Leningrad.: Education, 1954. - 140 p.
1.4. Likum A. Everything about everything. T. 2. - M .: Philological Society "Word", 1993. - 512 p.
1.5. Olechnik S. N., Nesterenko Yu. V., Potapov M. K. entertaining tasks. – M.: Science. Main edition of physical and mathematical literature, 1985. - 160 p.
1.6. Perelman Ya.I. Entertaining arithmetic. - M.: Rusanova, 1994 - 205s.
1.7. Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L.: Lenizdat, 1941 - 12 p.
1.8. Savin A.P. Math thumbnails. Entertaining mathematics for children. - M.: Children's literature, 1998 - 175 p.
1.9. Encyclopedia for children. Maths. - M.: Avanta +, 2003. - 688 p.
1.10. I know the world: Children's Encyclopedia: Mathematics / comp. Savin A.P., Stanzo V.V., Kotova A.Yu. - M.: AST Publishing House LLC, 2000. - 480 p.
2. Other sources of information
Internet resources:
2.1. Korneev A.A. The phenomenon of Russian multiplication. History. [Electronic resource]

problem: understand the types of multiplication

Target: familiarization with various ways of multiplying natural numbers that are not used in the lessons, and their application in calculating numerical expressions.
Tasks:
1. Find and analyze various ways of multiplication.
2. Learn to demonstrate some of the methods of multiplication.
3. Talk about new methods of multiplication and teach students how to use them.
4. Develop skills independent work: search for information, selection and design of the found material.
5. Experiment "which way is faster"
Hypothesis Q: Do I need to know the multiplication table?
Relevance:IN Lately students trust gadgets more than themselves. And that's why they count only on calculators. We wanted to show that there are different ways of multiplication, so that it would be easier for students to count, and it would be interesting to learn.
INTRODUCTION
You can't do multi-digit multiplications - even two-digit multiplications - unless you memorize all the results of single-digit multiplications, that is, what is called the multiplication table.
At different times different nations mastered different ways of multiplying natural numbers.
Why now all peoples use one method of multiplying by a “column”?
Why did people abandon the old methods of multiplication in favor of the modern one?
Do the forgotten methods of multiplication have the right to exist in our time?
To answer these questions, I did the following:
1. Using the Internet, I found information about some of the multiplication methods that were used before .;
2. Studied the literature offered by the teacher;
3. I solved a couple of examples in all the ways studied in order to find out their shortcomings;
4) Identified among them the most effective;
5. Conducted an experiment;
6. Draw conclusions.
1. Find and analyze various ways of multiplication.
Finger multiplication.

The ancient Russian method of multiplying on fingers is one of the most common methods that Russian merchants have successfully used for many centuries. They learned to multiply single-digit numbers from 6 to 9 on their fingers. At the same time, it was enough to master the initial skills of finger counting in “ones”, “pairs”, “triples”, “fours”, “fives” and “tens”. The fingers here served as an auxiliary computing device.

To do this, on one hand they extended as many fingers as the first factor exceeds the number 5, and on the second they did the same for the second factor. The rest of the fingers were bent. Then the number (total) of outstretched fingers was taken and multiplied by 10, then the numbers were multiplied showing how many fingers were bent on the hands, and the results were added up.

For example, let's multiply 7 by 8. In the considered example, 2 and 3 fingers will be bent. If we add the number of bent fingers (2+3=5) and multiply the number of not bent fingers (2 3=6), then we will get the numbers of tens and units of the desired product, respectively 56 . So you can calculate the product of any single-digit numbers greater than 5.

Ways to multiply numbers in different countries Oh

Multiply by 9.

Multiplication for the number 9 - 9 1, 9 2 ... 9 10 - is easier to fade from memory and more difficult to manually recalculate by addition, but it is for the number 9 that multiplication is easily reproduced "on the fingers". Spread your fingers on both hands and turn your palms away from you. Mentally assign numbers from 1 to 10 to the fingers, starting with the little finger of the left hand and ending with the little finger of the right hand (this is shown in the figure).

Who invented finger multiplication

Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you need to bend the finger with number 6. The number of fingers to the left of the bent finger shows us the number of tens in the answer, the number of fingers to the right - the number of units. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54. The figure below shows in detail the whole principle of "calculation".

Multiplication in an unusual way

Another example: you need to calculate 9 8=?. Along the way, we will say that fingers may not necessarily act as a "calculating machine". Take, for example, 10 cells in a notebook. We cross out the 8th cell. There are 7 cells on the left, 2 cells on the right. So 9 8=72. Everything is very simple.

7 cells 2 cells.

Indian way of multiplication.

The most valuable contribution to the treasury of mathematical knowledge was made in India. The Hindus proposed the way we use to write numbers using ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The basis of this method is the idea that the same digit stands for units, tens, hundreds or thousands, depending on where this figure occupies. The place occupied, in the absence of any digits, is determined by zeros assigned to the numbers.

The Indians thought well. They came up with a very simple way to multiply. They performed multiplication, starting with the highest order, and wrote down incomplete products just above the multiplicand, bit by bit. At the same time, the senior rank was immediately visible complete work and, in addition, the omission of any figure was excluded. The multiplication sign was not yet known, so they left a small distance between the factors. For example, let's multiply them in the way 537 by 6:

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

6
Multiplication using the "LITTLE CASTLE" method.

Multiplication of numbers is now studied in the first grade of the school. But in the Middle Ages, very few mastered the art of multiplication. A rare aristocrat could boast of knowing the multiplication table, even if he graduated from a European university.

Over the millennia of the development of mathematics, many ways to multiply numbers have been invented. The Italian mathematician Luca Pacioli in his treatise "The sum of knowledge in arithmetic, ratios and proportionality" (1494) gives eight different methods of multiplication. The first of them is called "Little Castle", and the second is no less romantic called "Jealousy or Lattice Multiplication".

The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.

Ways to multiply numbers in different countries

Multiplying numbers using the "jealousy" method.

"Methods of multiplication The second method is romantically called jealousy", or "lattice multiplication".

First, a rectangle is drawn, divided into squares, and the dimensions of the sides of the rectangle correspond to the number of decimal places for the multiplier and multiplier. Then the square cells are divided diagonally, and “... it turns out a picture that looks like lattice shutters, blinds,” writes Pacioli. “Such shutters were hung on the windows of Venetian houses, preventing passers-by from seeing the ladies and nuns sitting at the windows.”

Let's multiply 347 by 29 in this way. Let's draw a table, write the number 347 above it, and the number 29 on the right.

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 the amount is less than 10, then we write it under the bottom number of the band. 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. As a result, we get the desired product 10063.

Peasant way of multiplication.

The most, in my opinion, "native" and easy way of multiplication is the method used by Russian peasants. This technique generally does not require knowledge of the multiplication table beyond the number 2. Its essence is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half while doubling another number. Bisection continues until the quotient is 1, while doubling another number in parallel. The last doubled number gives the desired result.

In the case of an odd number, one must discard the unit and divide the remainder in half; but on the other hand, to the last number of the right column it will be necessary to add all those numbers of this column that are against the odd numbers of the left column: the sum will be the desired product

The product of all pairs of corresponding numbers is the same, so

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both numbers are odd, proceed as follows:

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408
A new way to multiply.

interesting new way multiplication, which has recently been reported. Vasily Okoneshnikov, the inventor of the new mental counting system, claims that a person is able to memorize a huge amount of information, the main thing is how to arrange this information. According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.

It is very easy to count according to such a table. For example, let's multiply the number 15647 by 5. In the part of the table corresponding to the five, we select the numbers corresponding to the digits of the number in order: one, five, six, four and seven. We get: 05 25 30 20 35

The left digit (in our example, zero) is left unchanged, and the following numbers are added in pairs: five with two, five with three, zero with two, zero with three. The last digit is also unchanged.

As a result, we get: 078235. The number 78235 is the result of multiplication.

If, when adding two digits, a number exceeding nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.

Conclusion.

While working on this topic, I learned that there are about 30 different, fun and interesting ways to multiply. Some are still in use today in various countries. I chose some interesting ways for myself. But not all methods are convenient to use, especially when multiplying multi-digit numbers.

Multiplication methods

MBOU "Secondary school with. Volnoe, Kharabalinsky District, Astrakhan Region

Project on:

« Unusual ways to multiplyand I»

Work done:

5th grade students :

Tulesheva Amina,

Sultanov Samat,

Kuyanguzova Rasita.

R project manager:

mathematic teacher

Fateeva T.V.

Volnoe 201 6 year .

"Everything is a number" Pythagoras

Introduction

In the 21st century, it is impossible to imagine the life of a person who does not make calculations: these are salespeople, and accountants, and ordinary schoolchildren.

The study of almost any subject in school requires a good knowledge of mathematics, and without it it is impossible to master these subjects. Two elements dominate mathematics - numbers and figures with their infinite variety of properties and actions with them.

We wanted to learn more about the history of mathematical operations. Now, when computing technology is rapidly developing, many do not want to bother with mental arithmetic. Therefore, we decided to show not only that the process of performing actions can be interesting, but also that, having mastered the methods of fast counting well, one can argue with a computer.

The relevance of this topic lies in the fact that the use of non-standard methods in the formation of computational skills enhances students' interest in mathematics and contributes to the development of mathematical abilities.

Objective:

ANDlearn some non-standard multiplication tricks and show that their application makes the calculation process rational and interestingand for the calculation of which, mental counting or the use of a pencil, pen and paper is sufficient.

Hypothesis:

EIf our ancestors knew how to multiply in the old ways, then if, having studied the literature on this problem, will a modern student be able to learn this, or are some supernatural abilities needed.

Tasks:

1. Find unusual ways to multiply.

2. Learn to apply them.

3. Choose for yourself the most interesting or easier ones than those offered at school, and use them when counting.

4. Teach classmates to apply newewaysmultiplication.

Object of study: math operation multiplication

Subject of study: multiplication methods

Research methods:

Search method using scientific and educational literature, the Internet;

Research method in determining the methods of multiplication;

Practical method in solving examples;

- - questioning of respondents about knowledge of non-standard methods of multiplication.

History reference

There are people with extraordinary abilities who, in the speed of mental calculations, can compete with computers. They are called "miracle counters". And there are many such people.

It is said that Gauss's father, when paying his workers at the end of the week, added pay to each day's overtime earnings. One day, after Gauss's father had finished his calculations, a 3-year-old child who was watching his father's operations exclaimed: “Dad, the calculation is wrong! This is how much it should be!” The calculations were repeated and were surprised to see that the boy had indicated the correct amount.

In Russia at the beginning of the 20th century, the "wizard of calculations" Roman Semenovich Levitan, known under the pseudonym Arrago, shone with his skills. Unique abilities began to manifest themselves in the boy at an early age. In a few seconds, he squared and cubed ten-digit numbers, extracted roots of varying degrees. He seemed to do all this with extraordinary ease. But this lightness was deceptive and demanded great work brain.

In 2007, Mark Vishnya, who was then 2.5 years old, amazed the whole country with his intellectual abilities. The young participant in the show "Minute of Glory" easily counted multi-digit numbers in his mind, ahead of his parents and the jury who used calculators in the calculations. At the age of two, he mastered the table of cosines and sines, as well as some logarithms.

Computer and human competitions were held at the Institute of Cybernetics of the Ukrainian Academy of Sciences. A young counter-phenomenon Igor Shelushkov and ZVM "Mir" participated in the competition. The machine performed many complex operations in a few seconds, but Igor Shelushkov turned out to be the winner.

The University of Sydney in India also hosted a human-machine competition. Shakuntala Devi was also ahead of the computer.

Most of these people have an excellent memory and have talent. But some of them have no special abilities for mathematics. They know the secret! And this secret is that they have mastered the techniques of quick counting, memorized several special formulas. This means that we, too, can, using these techniques, quickly and accurately count.

The methods of calculation that we use now were not always so simple and convenient. In the old days, more cumbersome and slower methods were used. And if a schoolboy of the 21st century could travel back five centuries, he would impress our ancestors with the speed and accuracy of his calculations. The rumor about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful counters of that era, and people would come from all over to study with the new great master.

The operations of multiplication and division were especially difficult in the old days. At that time, there was no single technique worked out by practice for each action.

On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - methods one more intricate than the other, which a person of average ability could not remember. Each calculus teacher kept to his favorite method, each "master of division" (there were such specialists) praised his own way of performing this action.

In V. Bellyustin’s book “How People Gradually Came to True Arithmetic”, 27 methods of multiplication are outlined, and the author notes: “It is quite possible that there are more methods hidden in the recesses of book depositories, scattered in numerous, mainly handwritten collections.”

And all these methods of multiplication - "chess or organ", "bending", "cross", "lattice", "back to front", "diamond" and others competed with each other and were assimilated with great difficulty.

Let's look at the most interesting and simple ways of multiplication.

Old Russian method of multiplication on fingers

This is one of the most common methods that Russian merchants have successfully used for many centuries.

The principle of this method: multiplication on the fingers of single-digit numbers from 6 to 9. The fingers here served as an auxiliary computing device.

It is very easy to reproduce "on the fingers" multiplication for the number 9

Rastarsthosefingers on both hands and turn your hands palms away from you. Mentally assign the numbers from 1 to 10 to the fingers in sequence, starting with the little finger of the left hand and ending with the little finger of the right hand. Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you need to bend the finger with number 6. The number of fingers to the left of the bent finger shows us the number of tens in the answer, the number of fingers to the right - the number of units. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54.

Multiplying by 9 using notebook cells

Take, for example, 10 cells in a notebook. We cross out the 8th cell. There are 7 cells on the left, 2 cells on the right. So 9 8=72. Everything is very simple!

7 2

Multiplication Method "Little Castle"

The advantage of the “Little Castle” multiplication method is that the digits of the highest digits are determined from the very beginning, and this can be important if you need to quickly estimate the value.The digits of the upper number, starting from the most significant digit, are alternately multiplied by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up.

"lattice multiplication"

First, a rectangle is drawn, divided into squares, and the dimensions of the sides of the rectangle correspond to the number of decimal places for the multiplier and multiplier.

Then the square cells are divided diagonally, and “... you get a picture that looks like lattice shutters. Such shutters were hung on the windows of Venetian houses ... "

"Russian Peasant Way"

In Russia, a method was common among peasants that did not require knowledge of the entire multiplication table. All you need is the ability to multiply and divide numbers by 2.

Let's write one number on the left and another on the right on one line. We will divide the left number by 2, and multiply the right number by 2 and write the results in a column.

If a remainder occurs during division, then it is discarded. Multiplication and division by 2 continue until 1 remains on the left.

Then we cross out those lines from the column in which there are even numbers. Now let's add the remaining numbers in the right column.

This method of multiplication is much simpler than the methods of multiplication discussed earlier. But it is also very bulky.

"Multiplication with a cross"

The ancient Greeks and Hindus in the old days called the method of cross multiplication "the method of lightning" or "multiplication by a cross."

24 and 32

2 4

3 2

4x2=8 - the last digit of the result;

2x2=4; 4x3=12; 4+12=16 ; 6 - the penultimate digit of the result, remember the unit;

2x3=6 and even the number kept in mind, we have 7 - this is the first digit of the result.

We get all the digits of the product: 7,6,8. Answer:768.

Indian way of multiplication

546 7

5 7=35 35

350+ 4 7=378 378

3780 + 6 7=3822 3822

546 7= 3822

AtWe start the multiplication from the highest order, and write down the incomplete products just above the multiplicand, bit by bit. In this case, the most significant digit of the complete product is immediately visible and, in addition, the omission of any digit is excluded. The multiplication sign was not yet known, so a small distance was left between the factors

Chinese (drawing) way of multiplying

Example #1: 12 × 321 = 3852
We drawfirst number top to bottom, left to right: one green stick (1 ); two orange sticks (2 ). 12 painted
We drawsecond number from bottom to top, left to right: three blue sticks (3 ); two red ones2 ); one lilac (1 ). 321 painted

Now, with a simple pencil, we will walk along the drawing, divide the points of intersection of the stick numbers into parts and proceed to counting the points. Moving from right to left (clockwise):2 , 5 , 8 , 3 . number-result we will “collect” from left to right (counterclockwise) we got3852

Example #2: 24 × 34 = 816
There are nuances in this example ;-) When counting the points in the first part, it turned out16 . One send-add to the points of the second part (20 + 1 )…

Example #3: 215 × 741 = 159315

During the work on the project, we conducted a survey. The students answered the following questions.

1. Does a modern person need an oral account?

YesNot

2. Do you know other ways of multiplication besides multiplication by a column?

YesNot

3. Do you use them?

YesNot

4. Would you like to know other ways to multiply?

Not really

We interviewed students in grades 5-10.

This survey showed that modern schoolchildren do not know other ways to perform actions, since they rarely turn to material that is outside the school curriculum.

Output:

There are many interesting events and discoveries in the history of mathematics, unfortunately not all of this information reaches us, modern students.

With this work, we wanted to at least a little bit fill this gap and convey to our peers information about the ancient methods of multiplication.

In the course of the robots, we learned about the origin of the multiplication action. In the old days, it was not an easy thing to master this action; then, as now, there was not yet one method worked out by practice. On the contrary, almost a dozen different methods of multiplication were in use at the same time - methods one another more intricate, firm, which a person of average ability could not remember. Each calculus teacher kept to his favorite method, each "master" (there were such specialists) praised his own way of performing this action. It was even recognized that in order to master the art of fast and error-free multiplication of multi-digit numbers, one needs a special natural talent, exceptional abilities; this wisdom is not available to ordinary people.

With our work, we have proved that our hypothesis is correct, you do not need to have supernatural abilities to be able to use the ancient methods of multiplication. And we also learned how to select material, process it, that is, highlight the main thing and systematize it.

Having learned to count in all the ways presented, we came to the conclusion that the simplest ways are those that we study at school, or maybe we just got used to them.

The modern way of multiplication is simple and accessible to everyone.

But, we think that our method of multiplication in a column is not perfect and we can come up with even faster and more reliable methods.

It is possible that the first time many will not be able to quickly, on the move, perform these or other calculations.

No problem. Constant computational training is needed. It will help you develop useful mental counting skills!

Bibliography

1. Glazer, G. I. History of mathematics at school ⁄ G. I. Glazer ⁄⁄ History of mathematics at school: a guide for teachers ⁄ edited by V. N. Molodshiy. - M .: Education, 1964. - S. 376.

Perelman Ya. I. Entertaining arithmetic: Riddles and curiosities in the world of numbers. - M .: Rusanov Publishing House, 1994. - S. 142.

Encyclopedia for children. T. 11. Mathematics / Chapter. ed. M. D. Aksenova. - M .: Avata +, 2003. - S. 130.

Journal "Mathematics" №15 2011

Internet resources.

MOU "Kurovskaya average comprehensive school No. 6"

ABSTRACT ON MATHEMATICS ON THE TOPIC:

« UNUSUAL MULTIPLICATION WAYS».

Completed by a student of 6 "b" class

Krestnikov Vasily.

Supervisor:

Smirnova Tatyana Vladimirovna

Introduction…………………………………………………………………………2

Main part. Unusual ways of multiplication…………………………3

2.1. A bit of history………………………………………………………………..3

2.2. Multiplication on fingers……………………………………………………………4

2.3. Multiplication by 9………………………………………………………………………5

2.4. Indian way of multiplication……………………………………………….6

2.5. Multiplication by the “Little Castle” method……………………………………………………7

2.6. Multiplication by the “Jealousy” method………………………………………………8

2.7. Peasant way of multiplication……………………………………………..9

2.8 New way…………………………………………………………………..10

Conclusion………………………………………………………………………… 11

References…………………………………………………………….1 2

I. Introduction.

It is impossible for a person to do without calculations in everyday life. Therefore, in mathematics lessons, we are first of all taught to perform operations on numbers, that is, to count. We multiply, divide, add and subtract in the usual ways for everyone that are studied at school.

Once I accidentally came across a book by S. N. Olekhnika, Yu. V. Nesterenko and M. K. Potapov "Old entertaining problems." Leafing through this book, my attention was drawn to a page called "Multiplication on the fingers." It turned out that you can multiply not only as they offer us in mathematics textbooks. I was wondering if there are any other ways to calculate. After all, the ability to quickly make calculations is frankly surprising.

The constant use of modern computing technology leads to the fact that students find it difficult to make any calculations without having tables or a calculating machine at their disposal. Knowledge of simplified calculation techniques makes it possible not only to quickly perform simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computational skills develops memory, increases the level of mathematical culture of thinking, helps to fully assimilate the subjects of the physical and mathematical cycle.

Objective:

Show unusualmultiplication methods.

Tasks:

Find as many as possibleunusual ways of computing.

Learn to apply them.

Choose for yourself the most interesting or easier than those thatofferedat school, and use them when counting.

II. Main part. Unusual ways of multiplication.

2.1. A bit of history.

The operations of multiplication and division were especially difficult in the old days. At that time, there was no one technique developed by practice for each action. On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - methods one more complicated than the other, which a person of average ability could not remember. Each calculus teacher kept to his favorite method, each "master of division" (there were such specialists) praised his own way of performing this action.

And all these methods of multiplication - "chess or organ", "bending", "cross", "lattice", "back to front", "diamond" and others competed with each other and were assimilated with great difficulty.

Let's look at the most interesting and simple ways of multiplication.

2.2. Finger multiplication.

2.3. Multiply by 9.

Multiplication for the number 9- 9 1, 9 2 ... 9 10 - is easier to fade from memory and more difficult to manually recalculate by addition, but it is for the number 9 that multiplication is easily reproduced “on the fingers”. Spread your fingers on both hands and turn your palms away from you. Mentally assign numbers from 1 to 10 to the fingers, starting with the little finger of the left hand and ending with the little finger of the right hand (this is shown in the figure).

Another example: you need to calculate 9 8=?. Along the way, we will say that fingers may not necessarily act as a “calculating machine”. Take, for example, 10 cells in a notebook. We cross out the 8th cell. There are 7 cells on the left, 2 cells on the right. So 9 8=72. Everything is very simple.

7 cells 2 cells.

2.4. Indian way of multiplication.

The most valuable contribution to the treasury of mathematical knowledge was made in India. The Hindus proposed the way we use to write numbers using ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The Indians thought well. They came up with a very simple way to multiply. They performed multiplication, starting with the highest order, and wrote down incomplete products just above the multiplicand, bit by bit. At the same time, the senior digit of the complete product was immediately visible and, in addition, the omission of any digit was excluded. The multiplication sign was not yet known, so they left a small distance between the factors. For example, let's multiply them in the way 537 by 6:

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

2.5 . multiplication way"LITTLE CASTLE".

2.6. Number multiplicationjealousy method.

The second method is romantically called "jealousy", or "lattice multiplication."

Let's multiply 347 by 29 in this way. Let's draw a table, write the number 347 above it, and the number 29 on the right.

2.7. TOrustic way of multiplication.

The product of all pairs of corresponding numbers is the same, so

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both numbers are odd, proceed as follows:

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408

2.8 . A new way to multiply.

interesting a new way of multiplication that has recently been reported. Vasily Okoneshnikov, the inventor of the new mental counting system, claims that a person is able to memorize a huge amount of information, the main thing is how to arrange this information. According to the scientist himself, the nine-decimal system is the most advantageous in this regard - all data is simply placed in nine cells arranged like buttons on a calculator.

As a result, we get: 078235. The number 78235 is the result of multiplication.

If, when adding two digits, a number exceeding nine is obtained, then its first digit is added to the previous digit of the result, and the second is written in its “own” place.

III. Conclusion.

Of all the unusual counting methods I found, the method of “lattice multiplication or jealousy” seemed to be the most interesting. I showed it to my classmates and they also liked it very much.

The simplest method seemed to me to be the “doubling and splitting” method used by Russian peasants. I use it when multiplying not too large numbers (it is very convenient to use it when multiplying two-digit numbers).

I was interested in a new way of multiplication, because it allows you to "turn" huge numbers in your mind.

I think that our method of multiplying by a column is not perfect either, and we can come up with even faster and more reliable methods.

Literature.

Depman I. "Stories about Mathematics". - Leningrad.: Education, 1954. - 140 p.

Korneev A.A. The phenomenon of Russian multiplication. History. http://numbernautics.ru/

Olekhnik S. N., Nesterenko Yu. V., Potapov M. K. "Old entertaining problems." – M.: Science. Main edition of physical and mathematical literature, 1985. - 160 p.

Perelman Ya.I. Quick account. Thirty simple methods of mental counting. L., 1941 - 12 p.

Perelman Ya.I. Entertaining arithmetic. M.Rusanova, 1994–205p.

Encyclopedia “I know the world. Maths". – M.: Astrel Ermak, 2004.

Encyclopedia for children. "Maths". - M.: Avanta +, 2003. - 688 p.

The world of mathematics is very large, but I have always been interested in ways of multiplying. Working on this topic, I learned a lot of interesting things, learned to select the material I needed from what I read. Learned how to solve individual entertaining problems, puzzles and examples of multiplication in different ways, as well as what arithmetic tricks and intensive calculation techniques are based on.

ABOUT MULTIPLICATION

What remains in the mind of most people from what they once studied in school? Of course, different people have different things, but everyone, for sure, has a multiplication table. In addition to the efforts made to "crush" it, let's recall hundreds (if not thousands) of tasks that we solved with its help. Three hundred years ago in England, a person who knew the multiplication table was already considered a learned person.

There are many ways to multiply. The Italian mathematician of the end of the 15th - beginning of the 16th century, Luca Pacioli, in his treatise on arithmetic, gives 8 different ways of multiplication. In the first, which is called the "little castle", the digits of the upper number, starting from the highest, are multiplied in turn by the lower number and written in a column with the addition of the required number of zeros. Then the results are added up. The advantage of this method over the usual one is that the numbers of the highest digits are determined from the very beginning, and this can be important in estimating calculations.

The second method has the no less romantic name "jealousy" (or lattice multiplication). A grid is drawn, into which the results of intermediate calculations are then entered, more precisely, numbers from the multiplication table. The grid is a rectangle divided into square cells, which, in turn, are divided in half by diagonals. On the left (from top to bottom) the first multiplier was written, and at the top - the second. At the intersection of the corresponding row and column, the product of the numbers in them was written. Then the resulting numbers were added along the drawn diagonals, and the result was recorded at the end of such a column. The result was read along the bottom and right sides rectangle. “Such a lattice,” writes Luca Pacioli, “reminiscent of the lattice shutters that were hung on Venetian windows, preventing passers-by from seeing the ladies and nuns sitting at the windows.”

All the multiplication methods described in Luca Pacioli's book used the multiplication table. However, Russian peasants knew how to multiply without a table. Their method of multiplication used only multiplication and division by 2. To multiply two numbers, they were written side by side, and then the left number was divided by 2, and the right one was multiplied by 2. If the division resulted in a remainder, then it was discarded. Then those lines in the left column were crossed out, in which there are even numbers. The remaining numbers in the right column were added up. The result was the product of the original numbers. Check on several pairs of numbers that this is indeed the case. The proof of this method is shown using the binary number system.

Old Russian way of multiplication.

From ancient times and almost until the eighteenth century, Russian people dispensed with multiplication and division in their calculations: they used only two arithmetic operations - addition and subtraction, and even the so-called "doubling" and "doubling". The essence of the Russian old method of multiplication is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half (sequential, bifurcation) while doubling another number. If in a product, for example, 24 X 5, the multiplier is reduced by 2 times (“double”), and the multiplier is increased by 2 times

(“double”), then the product will not change: 24 x 5 \u003d 12 X 10 \u003d 120. Example:

Dividing the multiplicand in half continues until the quotient is 1, while doubling the factor. The last doubled number i- gives the desired result. So 32 X 17 = 1 X 544 = 544.

In those ancient times, doubling and bifurcation were taken even for special arithmetic operations. Just how special are they? actions? After all, for example, doubling a number is not a special action, but only the addition of a given number to itself.

Note that numbers are divisible by 2 all the time without a remainder. But what if the multiplicand is divisible by 2 with a remainder? Example:

If the multiplicand is not divisible by 2, then one is first subtracted from it, and then division by 2 is already performed. Lines with even multipliers are deleted, and the right parts of lines with odd multipliers are added.

21 X 17 = (20 + 1) X 17 = 20 X 17+17.

Let's remember the number 17 (the first line is not crossed out!), and replace the product 20 X 17 with its equal product 10 X 34. But the product 10 X 34, in turn, can be replaced with its equal product 5 X 68; so the second line is crossed out:

5 X 68 = (4 + 1) X 68 = 4 X 68 + 68.

Remember the number 68 (the third line is not crossed out!), and replace the product 4 X 68 with its equal product 2 X 136. But the product 2 X 136 can be replaced with its equal product 1 X 272; so the fourth line is crossed out. So, to calculate the product 21 X 17, you need to add the numbers 17, 68, 272 - the right parts of the lines with odd multipliers. Products with even multiplicands can always be replaced by dividing the multiplicand and doubling the factor by products equal to them; therefore, such strings are excluded from the calculation of the final product.

I tried to multiply the old way myself. I took the numbers 39 and 247, I got this

The columns will turn out to be even longer than mine if we take the multiplier more than 39. Then I decided, the same example in a modern way:

It turns out that our school way of multiplying numbers is much simpler and more economical than the old Russian way!

Only we must first of all know the multiplication table, and our ancestors did not know it. In addition, we should know well the very rule of multiplication, they only knew how to double and split numbers. As you can see, you can multiply much better and faster than the most famous calculator in the world. ancient Russia. By the way, several thousand years ago the Egyptians performed multiplication almost exactly the same way as the Russian people did in the old days.

It's great that people from different countries multiplied in the same way.

Not so long ago, only about a hundred years ago, memorizing the multiplication table was a very difficult task for students. To convince students of the need to know the tables by heart, the authors of mathematical books have long resorted to. to the verses.

Here are a few lines from a book unfamiliar to us: “But for multiplication, you need to have the following table, just have it firmly in your memory, such and such a number, multiplying with each, without any delay, say, or write, also 2-zhd 2 is 4 , or 2 times 3 is 6, and 3 times 3 is 9 and so on.

If someone does not repeat And in all the science of the table and is proud, not free from torment,

Can’t know Koliko doesn’t teach by number that multiplying tune is depressing

True, not everything is clear in this passage and verses: it is written somehow not quite in Russian, because all this was written more than 250 years ago, in 1703, by Leonty Filippovich Magnitsky, a wonderful Russian teacher, and since then the Russian language has changed markedly .

L. F. Magnitsky wrote and published the first printed arithmetic textbook in Russia; before him there were only handwritten mathematical books. The great Russian scientist M.V. Lomonosov, as well as many other prominent Russian scientists of the eighteenth century, studied according to L. F. Magnitsky’s Arithmetic.

And how did they multiply in those days, in the time of Lomonosov? Let's see an example.

As we understood, the operation of multiplication was then written almost the same way as in our time. Only the multiplier was called “echelichestvo”, and the product was called “product”, and, moreover, they did not write the multiplication sign.

How then was multiplication explained?

It is known that M. V. Lomonosov knew by heart the entire “Arithmetic” of Magnitsky. In accordance with this textbook, little Misha Lomonosov would explain the multiplication of 48 by 8 as follows: “8 is 8 is 64, I write 4 under the line, against 8, and I have 6 decimals in my mind. And then 8 times 4 is 32, and I keep 3 in my mind, and I will add 6 decimals to 2, and it will be 8. And I will write this 8 next to 4, in a row to my left hand, and 3 while the essence is in my mind, I will write in a row near 8, to the left hand. And there will be a product of 384 from the multiplication of 48 with 8.

Yes, and we explain almost the same, only we speak in a modern way, and not in an old way, and, in addition, we name the discharges. For example, 3 should be written in third place because it will be hundreds, and not just "in a row next to 8, to the left hand."

The story "Masha - "magician"".

I can guess not only the birthday, as Pavlik did last time, but also the year of birth, - Masha began.

Multiply the number of the month you were born in by 100, then add your birthday. , multiply the result by 2. , add 2 to the resulting number; multiply the result by 5, add 1 to the resulting number, add zero to the result. , add another 1 to the resulting number. and, finally, add the number of your years.

Done, I got 20721. - I say.

*That's right, I confirmed.

And I got 81321, - says Vitya, a third-grade student.

You, Masha, must have been mistaken, - Petya doubted. - How does it happen: Vitya is from the third grade, and he was also born in 1949, like Sasha.

No, Masha guessed correctly, - Vitya confirms. Only I was ill for a long time for one year and therefore went to the second grade twice.

* And I got 111521, - says Pavlik.

How is it, - Vasya asks, - Pavlik is also 10 years old, like Sasha, and he was born in 1948. Why not in 1949?

But because September is coming, and Pavlik was born in November, and he is still only 10 years old, although he was born in 1948, Masha explained.

She guessed the date of birth of three or four more students, and then explained how she did it. It turns out that she subtracts 111 from the last number, and then leaves two digits on three faces from right to left. The middle two digits are the birthday, the first two or one is the month, and the last two digits are the years. Knowing how old a person is, it is not difficult to determine the year of birth. For example, I got the number 20721. If you subtract 111 from it, you get 20610. So now I am 10 years old, and I was born on February 6th. Since it is now September 1959, it means that I was born in 1949.

And why is it necessary to subtract exactly 111, and not some other number? we asked. -And why is the birthday, number of the month and number of years distributed in this way?

But look, - Masha explained. - For example, Pavlik, fulfilling my requirements, solved the following examples:

1) 11 X 100 = 1100; 2) 1100 + J4 = 1114; 3) 1114 X 2 =

2228; 4) 2228 + 2 = 2230; 57 2230 X 5 = 11150; 6) 11150 1 = 11151; 7) 11151 X 10 = 111510

8)111510 1 1-111511; 9)111511 + 10=111521.

As you can see, he multiplied the number of the month (11) by 100, then by 2, then by another 5 and, finally, by another 10 (assigned a sack), and only by 100 X 2 X 5 X 10, that is, by 10000. So , 11 became tens of thousands, that is, they make up the third face, if you count from right to left, two digits each. This will tell you the number of the month in which you were born. Birthday (14) he multiplied by 2, then by 5 and, finally, by 10, and only by 2 X 5 X 10, that is, by 100. So, the birthday must be looked for among hundreds, in the second face, but here there are extraneous hundreds. Look: he added the number 2, which he multiplied by 5 and by 10. So, he got an extra 2x5x10=100 - 1 hundred. I subtract this 1 hundred from 15 hundreds in the number 111521, it turns out 14 hundreds. That's how I know my birthday. The number of years (10) was not multiplied by anything. This means that this number must be sought among the units, in the first face, but there are extraneous units here. Look: he added the number 1, which he multiplied by 10, and then added another 1. So, he got a total of extra 1 x TO + 1 = 11 units. I subtract these 11 units from 21 units in the number 111521, it turns out 10. So I find out the number of years. And in total, as you can see, I subtracted 100 + 11 = 111 from the number 111521. When I subtracted 111 from the number 111521, then it turned out PNU. Means,

Pavlik was born on November 14 and is 10 years old. Now the year is 1959, but I subtracted 10 not from 1959, but from 1958, since Pavlik turned 10 last year, in November.

Of course, you won’t remember such an explanation right away, but I tried to understand it with my own example:

1) 2 X 100 = 200; 2) 200 + 6 = 206; 3) 206 X 2 = 412;

4) 412 + 2 = 414; 5) 414 X 5 = 2070; 6) 2070 + 1 = 2071; 7) 2071 X 10 = 20710; 8) 20710 + 1 = 20711; 9) 20711 + + 10 = 20721; 20721 - 111 \u003d 2 "Obto; 1959 - 10 \u003d 1949;

Puzzle.

First task: At noon, a passenger steamer leaves Stalingrad for Kuibyshev. An hour later, a freight-passenger steamer leaves Kuibyshev for Stalingrad, moving slower than the first steamer. When the ships meet, which one will be further from Stalingrad?

This is not an ordinary arithmetic problem, but a joke! The steamboats will be at the same distance from Stalingrad, as well as from Kuibyshev.

And here is the second task. Last Sunday, our detachment and the detachment of the fifth class were planting trees along Bolshaya Pionerskaya Street. The detachments were to plant an equal number of trees, an equal number on each side of the street. As you remember, our team came to work early, and before the arrival of the fifth graders, we managed to plant 8 trees, but, as it turned out, not on our side of the street: we got excited and started work in the wrong place. Then we worked on our side of the street. Fifth graders finished work early. However, they did not remain indebted to us: they went over to our side and planted first 8 trees (“repaid their debt”), and then 5 more trees, and the work was completed by us.

The question is, how many more trees were planted by fifth-graders than we?

: Of course, the fifth-graders only planted 5 more trees than we did: when they planted 8 trees on our side, they repaid the debt; and when they planted 5 more trees, they sort of loaned us 5 trees. So it turns out that they planted only 5 more trees than we did.

No, the argument is wrong. It is true that the 5th graders did us a favor by planting 5 trees for us. But then, in order to get the right answer, you need to reason like this: we under-fulfilled our task by 5 trees, while the fifth-graders overfulfilled theirs by 5 trees. So it turns out that the difference between the number of trees planted by fifth graders and the number of trees planted by us is not 5, but 10 trees!

And here is the last puzzle task, Playing the ball, 16 students were placed on the sides of a square area so that there were 4 people on each side. Then 2 students left. The rest moved so that there were again 4 people on each side of the square. Finally, 2 more students left, but the rest settled in such a way that there were still 4 people on each side of the square. How could this happen? Decide.

Two quick multiplication tricks

One day the teacher gave his students the following example: 84 X 84. One boy quickly answered: 7056. "What did you count?" the teacher asked the student. - "I took 50 X 144 and threw out 144," he replied. Well, let's explain how the student thought.

84 x 84 \u003d 7 X 12 X 7 X 12 \u003d 7 X 7 X 12 X 12 \u003d 49 X 144 \u003d (50 - 1) X 144 \u003d 50 X 144 - 144, and 144 fifty is 72 hundreds, which means 84 X 84 = 7200 - 144 =

And now let's count in the same way how much will be 56 X 56.

56 X 56 \u003d 7 X 8 X 7 X 8 \u003d 49 X 64 \u003d 50 X 64 - 64, that is, 64 fifty, or 32 hundreds (3200), without 64, i.e., to multiply a number by 49, you need this number multiply by 50 (fifty), and subtract this number from the resulting product.

And here are examples for a different calculation method, 92 X 96, 94 X 98.

Answers: 8832 and 9212. Example, 93 X 95. Answer: 8835. Our calculations gave the same number.

You can count so quickly only when the numbers are close to 100. We find the additions to 100 to these numbers: for 93 it will be 7, and for 95 it will be 5, we subtract the addition of the second from the first given number: 93 - 5 \u003d 88 - so much will be in the product hundreds, we multiply the additions: 7 X 5 \u003d 3 5 - so much will be in the product of units. So, 93 X 95 = 8835. And why it is necessary to do this is not difficult to explain.

For example, 93 is 100 minus 7, and 95 is 100 minus 5. 95 X 93 = (100 - 5) x 93 = 93 X 100 - 93 x 5.

To subtract 5 times 93, you can subtract 5 times 100, but add 5 times 7. Then it turns out:

95 x 93 \u003d 93 x 100 - 5 x 100 + 5 x 7 \u003d 93 cells. - 5 hundred. + 5 X 7 \u003d (93 - 5) cells. + 5 x 7 = 8800 + 35 = = 8835.

97 X 94 = (97 - 6) X 100 + 3 X 6 = 9100 + 18 = 9118, 91 X 95 = (91 - 5) x 100 + 9 x 5 = 8600 + 45 = 8645.

Multiplication in. dominoes.

With the help of dominoes, it is easy to depict some cases of multiplication of multi-digit numbers by a single-digit number. For example:

402 X 3 and 2663 X 4

The winner will be the one who, in a certain time, will be able to use the largest number of dominoes, making up examples for multiplying three-, four-digit numbers by a single-digit number.

Examples for multiplying four-digit numbers by one-digit.

2234 x 6; 2425 X 6; 2336 X 1; 526 X 6.

As you can see, only 20 dominoes were used. Examples have been compiled for multiplying not only four-digit numbers by a single-digit number, but also three-, five-, and six-digit numbers by a single-digit number. 25 bones were used and the following examples were compiled:

However, all 28 bones can still be used.

Stories about how well old Hottabych knew arithmetic.

The story "I get "5" by arithmetic."

As soon as the next day I went to Misha, he immediately asked: “What was new, interesting in the circle class?” I showed Misha and his friends how smart the Russian people were in the old days. Then I asked them to mentally calculate how much 97 X 95, 42 X 42 and 98 X 93 would be. They, of course, could not do this without a pencil and paper and were very surprised when I almost immediately gave the correct answers to these examples. Finally, we all together solved this homework problem. It turns out that it is very important how the dots are located on a sheet of paper. Depending on this, it is possible to draw one, four, and six straight lines through four points, but no more.

Then I invited the children to make multiplication examples from domino bones as it was done on the mug. We managed to use 20, 24, and even 27 bones, but out of all 28 we could not make examples, although we sat at this lesson for a long time.

Misha recalled that the movie "Old Man Hottabych" was being shown in the cinema today. We quickly finished doing arithmetic and ran to the cinema.

Here is the picture! Although a fairy tale, it is still interesting: it tells about us, boys, about school life, as well as about an eccentric sage - genie Hottabych. And Hottabych made a great mistake, prompting Volka about geography! As you can see, in bygone times even Indian wise men - gins - knew geography very, very poorly. Probably Hottabych did not know arithmetic properly either.

Indian way of multiplication.

Suppose you need to multiply 468 by 7. On the left we write the multiplier, on the right the multiplier:

The Indians did not have a multiplication sign.

Now I multiply 4 by 7, it turns out 28. We write this number over the number 4.

Now we multiply 8 by 7, we get 56. We add 5 to 28, we get 33; Erase 28, and write down 33, write 6 over the number 8:

It turned out very interesting.

Now we multiply 6 by 7, we get 42, we add 4 to 36, we get 40; We will erase 36, and write down 40; We write 2 over the number 6. So, multiply 486 by 7, we get 3402:

Correctly decided, but not very quickly and conveniently! This is exactly how the most famous calculators of that time multiplied.

As you can see, old Hottabych knew arithmetic quite well. However, he recorded actions differently than we do.

A long, long time ago, more than 1300 years ago, the Indians were the best calculators. However, they did not yet have paper, and all calculations were made on a small black board, making notes on it with a reed pen and using a very liquid white paint, which left marks that were easily erased.

When we write with chalk on a blackboard, it is a bit like the Indian way of writing: white characters appear on a black background, which are easy to erase and correct.

The Indians also made calculations on a white board sprinkled with red powder, on which they wrote signs with a small stick, so that white signs appeared on a red field. Approximately the same picture is obtained when we write with chalk on a red or brown board - linoleum.

The sign of multiplication did not yet exist at that time, and only a certain gap was left between the multiplier and the multiplier. In the Indian way, one could multiply starting from units. However, the Indians themselves performed multiplication starting from the highest digit, and wrote down incomplete products just above the multiplicand, bit by bit. At the same time, the senior digit of the complete product was immediately visible and, in addition, the omission of any digit was excluded.

An example of Indian multiplication.

Arabic multiplication.

Well, how, in the date itself, to carry out multiplication in the Indian way, if written down on paper?

The Arabs adapted this multiplication technique for writing on paper. The famous Uzbek scientist Muhammad ibn Musa Alkhvariz-mi (Mohammed son of Musa from Khorezm, a city located on the territory of the modern Uzbek SSR) more than a thousand years ago performed multiplication on parchment as follows:

As you can see, he did not erase unnecessary numbers (it is already inconvenient to do this on paper), but crossed them out; he wrote down the new numbers above the crossed out ones, of course, bit by bit.

An example of multiplication in the same way, making notes in a notebook.

So, 7264 X 8 \u003d 58112. But how to multiply by a two-digit number, by a multi-digit number?

The multiplication technique remains the same, but the recording becomes much more complicated. For example, you need to multiply 746 by 64. First, they multiplied by 3 tens, it turned out

So 746 X 34 = 25364.

As you can see, deleting unnecessary digits and replacing them with new digits when multiplying even by a two-digit number leads to too cumbersome notation. And what happens if you multiply by a three-, four-digit number ?!

Yes, the Arabic way of multiplication is not very convenient.

This method of multiplication was kept in Europe until the eighteenth century, for a whole thousand years. It was called the ways of the cross, or chiasm, since the Greek letter X (chi) was placed between the multiplied numbers, gradually replaced by an oblique cross. Now we can clearly see that our modern method of multiplication is the simplest and most convenient, probably the best of all possible methods of multiplication.

Yes, our school way of multiplying multi-digit numbers is very good. However, multiplication can be written in another way. Perhaps it would be best to do this, for example, like this:

This method is really good: multiplication starts from the highest digit of the multiplier, the lowest digit of incomplete products is written under the corresponding digit of the multiplier, which eliminates the possibility of an error when zero occurs in any digit of the multiplier. This is how Czechoslovak schoolchildren write the multiplication of multi-digit numbers. That's interesting. And we thought that arithmetic operations can be written only in the way it is customary with us.

A few more puzzles.

Here is the first, simple task for you: A tourist can walk 5 km in an hour. How many miles will he cover in 100 hours?

Answer: 500 kilometers.

And this is more big question! You need to know more exactly how the tourist walked these 100 hours: without rest or with respite. In other words, you need to know: 100 hours is the time of the tourist's movement or just the time of his stay on the road. A person is probably not able to be on the move for 100 hours in a row: this is more than four days; and the speed of movement would decrease all the time. Another thing is if a tourist went with breaks for lunch, sleep, etc. Then, in 100 hours of movement, he can cover all 500 km; only on the way it should be no longer four days, but about twelve days (if it passes an average of 40 km per day). If he was on the road for 100 hours, then he could only walk about 160-180 km.

Various answers. This means that something must be added to the condition of the problem, otherwise it is impossible to give an answer.

Now let's solve the following problem: 10 chickens eat 1 kg of grain in 10 days. How many kilograms of grain will 100 chickens eat in 100 days?

Solution: 10 chickens eat 1 kg of grain in 10 days, which means that 1 chicken eats 10 times less in the same 10 days, that is, 1000 g: 10 \u003d 100 g.

In one day, a chicken eats another 10 times less, that is, 100 g: 10 = 10 g. Now we know that 1 chicken eats 10 g of grain in 1 day. So 100 chickens a day eat 100 times more, that is

10 g X 100 = 1000 g = 1 kg. In 100 days, they will eat 100 times more, that is, 1 kg X 100 = 100 kg = 1 centner. This means that 100 chickens eat a whole centner of grain in 100 days.

There is a faster solution: there are 10 times more chickens and you need to feed them 10 times longer, which means that you need 100 times more grain, that is, 100 kg. However, in all these arguments there is one omission. Let's think and find an error in reasoning.

: - Let's pay attention to the last reasoning: “100 chickens eat 1 kg of grain in one day, and in 100 days they will eat 100 times more. »

After all, in 100 days (that's more than three months!) Chickens will grow noticeably and will eat not 10 g of grain per day, but 40-50 grams, since an ordinary chicken eats about 100 g of grain per day. This means that in 100 days 100 chickens will eat not 1 centner of grain, but much more: two or three centners.

And here is the last puzzle problem for you about tying a knot: “There is a piece of rope on the table, stretched out in a straight line. It is necessary to take it with one hand for one, with the other hand for the other end and, without releasing the ends of the rope from your hands, tie a knot. » It's a well-known fact that some problems are easy to parse, going from the data to the question of the problem, and others, on the contrary, going from the question of the problem to the data.

Well, here we tried to parse this problem, going from the question to the data. Let the knot on the rope already exist, and its ends are in the hands and are not released. Let's try to return from the solved problem to its data, to the initial position: the rope lies stretched out on the table, and its ends are not released from the hands.

It turns out that if you straighten the rope without letting go of its ends, then the left hand, going under the extended rope and above the right hand, holds the right end of the rope; and the right hand, going over the rope and under the left hand, holds the left end of the rope

I think after such an analysis of the problem, it became clear to everyone how to tie a knot on a rope, you need to do everything in the reverse order.

Two more quick multiplication tricks.

I will show you how to quickly multiply numbers like 24 and 26, 63 and 67, 84 and 86, etc. n., that is, when the factors of tens "s are equal, and the units make up exactly 10 together. Set examples.

* 34 and 36, 53 and 57, 72 and 78,

* Get 1224, 3021, 5616.

For example, you need to multiply 53 by 57. I multiply 5 by 6 (1 more than 5), it turns out 30 - so many hundreds in the product; I multiply 3 by 7, it turns out 21 - so many units in the product. So 53 X 57 = 3021.

* How can I explain this?

(50 + 3) X 57 = 50 X 57 + 3 X 57 = 50 X (50 + 7) +3 X (50 + 7) = 50 X 50 + 7 X 50 + 3 x 50 + 3 X 7 = 2500 + + 50 X (7 + 3) + 3 X 7 = 2500 + 50 X 10 + 3 X 7 = =: 25 cells. + 5 hundred. +3 X 7 = 30 cells. + 3 X 7 = 5 X 6 cells. + 21.

Let's see how you can quickly multiply two-digit numbers within 20. For example, to multiply 14 by 17, you need to add units 4 and 7, you get 11 - there will be tens in the product (that is, 10 units). Then you need to multiply 4 by 7, you get 28 - there will be so many units in the product. In addition, exactly 100 must be added to the resulting numbers 110 and 28. Hence, 14 X 17 \u003d 100 + 110 + 28 \u003d 238. Indeed:

14 X 17 = 14 X (10 + 7) = 14 X 10 + 14 X 7 = (10 + + 4) X 10 + (10 + 4) X 7 = 10 X 10 + 4 X 10 + 10 X 7 + 4 X 7 \u003d 100 + (4 + 7) X 10 + 4 X 7 \u003d 100 + 110 + + 28.

After that, we solved more such examples: 13 x 16 \u003d 100 + (3 + 6) X 10 + 3 x 6 \u003d 100 + 90 + + 18 \u003d 208; 14 X 18 = 100 + 120 + 32 = 252.

Multiplication in accounts

Here are a few tricks by which anyone who can quickly add on the accounts will be able to quickly perform multiplication examples that occur in practice.

Multiplying by 2 and by 3 is replaced by double and triple addition.

When multiplying by 4, first multiply by 2 and add this result to itself.

Multiplying a number by 5 is done on the abacus like this: they transfer the entire number one wire higher, that is, multiply it by 10, and then divide this 10-fold number in half (how to divide by 2 using abacus.

Instead of multiplying by 6, multiply by 5 and add the multiplied.

Instead of multiplying by 7, multiply by 10 and subtract the multiplied three times.

Multiplication by 8 is replaced by multiplication by 10 minus two multiplied.

In the same way, multiply by 9: replace by multiplying by 10 minus one multiplied.

When multiplying by 10, as we have already said, all numbers are transferred one wire higher.

The reader will probably figure out for himself how to proceed when multiplying by numbers greater than 10, and what kind of substitutions will be most convenient here. The factor 11 must, of course, be replaced by 10 + 1. The factor 12 is replaced by 10 + 2, or practically by 2 + 10, that is, the double number is first set aside, and then the tenfold is added. The factor 13 is replaced by 10 + 3, and so on.

Consider a few special cases for factors of the first hundred:

It is easy to see, by the way, that with the help of accounts it is very convenient to multiply by numbers such as 22, 33, 44, 55, etc.; therefore, we must strive to break down the factors to use similar numbers with the same digits.

Similar tricks are resorted to when multiplying by numbers greater than 100. If such artificial tricks are tedious, then we can always, of course, multiply by counting by general rule, multiplying each digit of the multiplier and writing down partial products - this still gives some reduction in time.

"Russian" way of multiplication

You cannot perform multiplications of multi-digit numbers - even two-digit ones - if you do not remember by heart all the results of multiplication of single-digit numbers, that is, what is called the multiplication table. In the old "Arithmetic" of Magnitsky, which we have already mentioned, the need for a solid knowledge of the multiplication table is sung in such verses (alien to modern hearing):

If someone does not repeat tables and is proud, He cannot know by number what to multiply

And in all sciences, not free from torment, Koliko does not teach tuna to depress

And it will not be in favor if I forget.

The author of these verses obviously did not know or overlooked that there is a way to multiply numbers without knowing the multiplication table. This method, similar to our school methods, was used in everyday life by Russian peasants and inherited by them from ancient times.

Its essence is that the multiplication of any two numbers is reduced to a series of successive divisions of one number in half while doubling the other number. Here is an example:

Dividing in half continues until then), the pitch in the quotient does not turn out to be 1, while doubling another number in parallel. The last doubled number gives the desired result. It is not difficult to understand what this method is based on: the product does not change if one factor is halved, and the other is doubled. It is therefore clear that as a result of repeated repetition of this operation, the desired product is obtained.

However, what to do if at the same time nrikh. Is it okay to divide an odd number in half?

The popular way easily gets out of this difficulty. It is necessary, the rule says, in the case of an odd number, throw one and divide the remainder in half; but on the other hand, to the last number of the right column it will be necessary to add all those numbers of this column that are against the odd numbers of the left column - the sum will be sought? l work. In practice, this is done in such a way that all lines with even left numbers are crossed out; only those that contain an odd number to the left remain.

Here is an example (asterisks indicate that this line should be crossed out):

Adding the numbers that have not been crossed out, we get a completely correct result: 17 + 34 + 272 = 32 What is this technique based on?

The correctness of the reception will become clear if we take into account that

19X 17 \u003d (18 + 1) X 17 \u003d 18X17 + 17, 9X34 \u003d (8 + 1) X34 \u003d; 8X34 + 34, etc.

It is clear that the numbers 17, 34, etc., lost when dividing an odd number in half, must be added to the result of the last multiplication in order to obtain a product.

Accelerated Multiplication Examples

We mentioned earlier that there are also convenient ways to perform those individual multiplication operations into which each of the above tricks breaks down. Some of them are very simple and conveniently applicable; they facilitate calculations so much that it does not interfere with memorizing them at all in order to use them in ordinary calculations.

Such, for example, is the technique of cross multiplication, which is very convenient when working with two-digit numbers. The method is not new; it goes back to the Greeks and Hindus and in the old days was called the “lightning method”, or “multiplication by a cross”. Now he is forgotten, and it does not hurt to recall him.

Let it be required to multiply 24X32. Mentally arrange the number according to the following scheme, one under the other:

Now we perform the following steps in sequence:

1)4X2 = 8 is the last digit of the result.

2)2X2 = 4; 4X3=12; 4+12=16; 6 - penultimate digit of the result; 1 remember.

3) 2X3 \u003d 6, and even the unit kept in mind, we have

7 is the first digit of the result.

We get all the digits of the product: 7, 6, 8 - 768.

After a short exercise, this technique is absorbed very easily.

Another method, consisting in the use of so-called "additions", is conveniently used in cases where the multiplied numbers are close to 100.

Suppose we want to multiply 92X96. "Addition" for 92 to 100 will be 8, for 96 - 4. The action is carried out according to the following scheme: multipliers: 92 and 96 "additions": 8 and 4.

The first two digits of the result are obtained by simply subtracting the “complement” of the multiplicand from the multiplier or vice versa; i.e. subtract 4 from 92 or subtract 8 from 96.

In both cases we have 88; the product of "additions" is attributed to this number: 8X4 \u003d 32. We get the result 8832.

That the result obtained must be correct is clearly seen from the following transformations:

92x9b = 88X96 = 88(100-4) = 88 X 100-88X4

1 4X96= 4 (88 + 8)= 4X 8 + 88X4 92x96 8832+0

Another example. It is required to multiply 78 by 77: factors: 78 and 77 "additions": 22 and 23.

78 - 23 \u003d 55, 22 X 23 \u003d 506, 5500 + 506 \u003d 6006.

Third example. Multiply 99 X 9.

multipliers: 99 and 98 "additions": 1 and 2.

99-2 = 97, 1X2= 2.

In this case, remember that 97 means here the number of hundreds. So we add up.