Interesting methods of multiplying multivalued numbers. Project on the topic: "Unusual methods of multiplication"

Gassales Vasily

Work theme " Unusual ways Calculations "are interesting and relevant, since students constantly perform arithmetic actions on numbers, and the ability to quickly calculate, improves success in school and develops the flexibility of the mind.

Vasily managed to clearly state the reasons for his appeal to this topic, correctly formulated the goal and task of work. Having studied various sources of information, found interesting and unusual methods of multiplication and learned to apply them in practice. The student considered the pros and cons of each method and made the right conclusion. The reliability of the output confirms a new way of multiplication. At the same time, the student skillfully uses special terminology and knowledge out school program mathematics. The theme of the work corresponds to the content, the material is stated clearly and accessible.

Work results are practical value And they can be interesting to a wide range of people.

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MOU "Kurovskaya average comprehensive school №6 "

Abstract for mathematics on the topic:

"Unusual methods of multiplication."

Fulfilled the student 6 "b" class

Cancer Vasily.

Leader:

Smirnova Tatiana Vladimirovna.

2011

Introduction .............................................................................. ....... 2
Main part. Unusual methods of multiplication ........................... ... 3

2.1. A little story ........................................................................ ..3

2.2. Multiplication on the fingers ............................................................... ... 4

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

2.4. Indian method of multiplication ...................................................... .6

2.5. Multiplying by the way "Little Castle" ....................................... 7

2.6. Multiplication by the way "Jealousy" .................................................. ... 8

2.7. Peasant Method of Multiplication ....................................................... 9

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

Conclusion .............................................................................. ... 11
List of references ......................................................................12

I. Entry.

Man B. everyday life It is impossible to do without computing. Therefore, in the lessons of mathematics, we are primarily taught to perform actions on numbers, that is, to count. We multiply, divide, fold and deduct we are familiar to all ways that are studied at school.

Once I accidentally came across the book S. N. Ololand, Yu. V. Nesterenko and M. K. Potapova "Ancient entertaining tasks". List through this book, my attention attracted a page called "Multiplication on the fingers". It turned out that you can multiply not only because they offer us in mathematics textbooks. It became interesting to me, and whether there are some other calculations. After all, the ability to quickly make calculations causes frank surprise.

The continuous use of modern computing equipment leads to the fact that students find it difficult to produce any calculations without having a table or counting machine at their disposal. Knowing simplified calculation techniques makes it possible not only to quickly produce simple calculations In the mind, but also control, evaluate, find and correct errors as a result of mechanized calculations. In addition, the development of computing skills develops memory, increases the level of mathematical culture of thinking, helps to fully absorb objects of the physico-mathematical cycle.

Purpose of work:

Show unusual methods of multiplication.

Tasks:

Find as many unusual calculation methods as possible.
Learn to apply them.
Choose for yourself the most interesting or lighter than those are offered at school, and use them with the score.

II. Main part. Unusual methods of multiplication.

2.1. A little story.

Those methods of calculations we use now were not always so simple and comfortable. In the old days enjoyed more cumbersome and slow techniques. And if the 21st century schoolboy could be transferred to five centuries ago, he would have struck our ancestors to the speed and error of his calculations. The surrounding schools and monasteries would fly about it about him, eclipsed by the glory of the most scene counters of that era, and from all sides would come to learn from the New Great Master.

Especially difficult in the old days were the actions of multiplication and division. Then there was no one generated admission practice for each action. On the contrary, in the go was at the same time almost a dozen different ways Multiplication and divisions - receptions one of the other confusing, remember that there was no power of medium abilities. Each teacher of the Accounts was held by his favorite reception, each "Master of Denilation" (there were such specialists) praised his own way of doing this action.

In the book of V. Bellyustin "As people gradually reached the real arithmetic" set out 27 methods of multiplication, and the author notes: "It is very possible that there are still methods hidden in the caches of books, scattered in numerous, mainly handwritten collections."

And all these multiplication techniques are "chess or organizing", "bending", "cross", "lattice", "backward", "diamond" and others competed with each other and assimilated with great difficulty.

Let's consider the most interesting and simple ways Multiplication.

2.2. Multiplication on the fingers.

Ancient Russian method of multiplication on the fingers is one of the most common methods that Russian merchants have successfully used for many centuries. They learned to multiply on the fingers of unambiguous numbers from 6 to 9. At the same time, it was enough to own the initial skills of the finger account "units", "couples", "three", "fours", "fives" and "dozens". The fingers of the hands here served as auxiliary computing device.

For this, so many fingers pulled out on one hand, as far as the first factor exceeds the number 5, and on the second they did the same for the second factor. The remaining fingers were fucked. Then the number (total) elongated fingers was taken and was multiplied by 10, then multiplying the numbers showing how much fingers were hung on their hands, and the results were folded.

For example, multiply 7 on 8. In the considered example, 2 and 3 fingers will be replaced. If you fold the quantities of the bent fingers (2 + 3 \u003d 5) and multiply the amounts of non-bent (2 3 \u003d 6), then the number of tens and units of the desired work 56 is obtained. So you can calculate the product of any unambiguous numbers, More than 5.

2.3. Multiplication by 9.

Multiplication for number 9 - 9 · 1, 9 · 2 ... 9 · 10 - it is easier to eat out of memory and it is more difficult to manually by the method of addition, but it is for the number of 9 multiplication that "on the fingers" is easily reproduced. Pour your fingers on both hands and turn your hands with your palms from ourselves. Mentally presate the fingers sequentially numbers from 1 to 10, starting with the mother's maiden and ending with the little finger of the right hand (this is shown in the figure).

Suppose we want to multiply 9 on 6. Craw your finger with the number, equal numberwhich we will multiply nine. In our example, you need to bend a finger with number 6. The number of fingers to the left of the bent finger shows us the number of dozens in the answer, the number of fingers on the right is the number of units. On the left we have 5 fingers are not reducing, on the right - 4 fingers. Thus, 9 · 6 \u003d 54. Below in the figure, the entire principle of "calculations" is shown in detail.

Another example: need to calculate 9 · 8 \u003d?. In the course of the matter, let's say that the fingers of the hands may not necessarily act as a "counting machine". Take, for example, 10 cells in the notebook. Excrying the 8th cell. On the left there are 7 cells left, on the right - 2 cells. So 9 · 8 \u003d 72. Everything is very simple.

7 cells 2 cells.

2.4. Indian multiplication method.

The most valuable contribution to the treasury of mathematical knowledge was performed in India. Hindus offered the method of recording numbers used by us with ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The basis of this method is the idea that one and the same figure denotes units, dozens, hundreds or thousands, depending on what place this figure takes. The place occupied, in the absence of any discharges, is determined by zeros attributed to the numbers.

Hindus considered great. They came up with a very simple way of multiplication. They performed multiplied, starting with the older discharge, and recorded incomplete works just above the multiple, blessing. At the same time, the senior discharge was immediately visible. full work And, in addition, there was a pass of any digit. The multiplication sign has not yet been known, so they left a small distance between the multipliers. For example, multiply in the way 537 to 6:

537 6

(5 ∙ 6 =30) 30

537 6

(300 + 3 ∙ 6 = 318) 318

537 6

(3180 +7 ∙ 6 = 3222) 3222

2.5. Multiplication by the method of "Little Castle".

Multiplication of numbers is now studying in the first class school. But in the Middle Ages, very few people owned the art of multiplication. A rare aristocrat could boast of knowledge of the multiplication table, even if he graduated from European University.

For the millennium, the development of mathematics was invented many ways to multiply numbers. Italian Mathematics of Luke Pachet in his treatise "The sum of knowledge of arithmetic, relationships and proportionality" (1494) leads eight different multiplication methods. The first of them is called "Little Castle", and the second no less romantic name "jealousy or lattice multiplication".

The advantage of the method of multiplying the "Little Castle" is that from the very beginning the numbers of high-level digits are determined, and this is important if it is required to quickly appreciate the value.

The numbers of the top number, starting with the older discharge, are multiplied alternately to the lower number and are written in a column with adding the desired number zeros. Then the results fold.

2.6. Multiplication of numbers by the "Jealousy" method.

The second method wears the romantic name "Jealousy", or "lattice multiplication".

First, the rectangle is drawn, separated into squares, and the sizes of the sides of the rectangle correspond to the number of decimal signs in the multiplier and multiplier. Then the square cells are divided according to the diagonal, and "... it turns out a picture similar to the lattice shutters-blinds," writes Pacheti. "Such shutters were hanging on the windows of Venetian houses, preventing street passers-by to see the windows sitting at the windows and nuns."

Multiply in this way 347 to 29. Note the table, write down the number 347 above it, and on the right number 29.

In each line, we write the work of numbers standing on this cell and to the right of it, with the number of tens of works, we write above the oblique feature, and the numbers are units under it. Now we add numbers in each oblique strip, performing this operation, right to left. If the amount is less than 10, then it is writing under the bottom of the strip. If it is more than 10, then we write only the number of units of the amount, and the figure of tens add to the next amount. As a result, we get the desired work 10063.

3 4 7

10 0 6 3

2.7. Peasant method of multiplication.

Most, in my opinion, "native" and easy way Multiplication is a way that Russian peasants consumed. This reception does not require knowledge of the multiplication table on the number 2. The essence of it is that the multiplication of any two numbers is reduced to a row of sequential divisions of one number in half while rejection of another number. The division in half is continued until 1, in parallel, doubles another number. The last tweed number and gives a desired result.

In the case of an odd number, it is necessary to learn a unit and divide the residue in half; But it will be necessary to add all those numbers of this column to the last number of the right column, which are against the odd numbers of the left column: the amount and will be the desired work

37……….32

74……….16

148……….8

296……….4

592……….2

1184……….1

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 odd, we do as follows:

24 ∙ 17

24 ∙ 16 =

48 ∙ 8 =

96 ∙ 4 =

192 ∙ 2 =

384 ∙ 1 = 384

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

2.8. New way of multiplication.

Interesting a new way of multiplication, which recently appeared messages. The inventor of the new oral account system Candidate of Philosophical Sciences Vasily Okneshovnikov claims that a person is able to memorize a huge supply of information, the main thing - how to place this information. According to the scientist himself, the most advantageous in this regard is a nine-sized system - all data is simply placed in nine cells located like buttons on the calculator.

It is very simple to count on such a table. For example, multiply the number 15647 by 5. In terms of the table corresponding to the top selected, select the numbers corresponding to the numbers of the number in order: a unit, a five, six, fourth and seven. We get: 05 25 30 20 35

Left digit (in our example - zero), we leave unchanged, and the following numbers fold in pairs: a twin five, a top five, zero with a twos, zero with a triple. The last digit is also unchanged.

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

If, when folding two digits, the number exceeding nine, its first digit is added to the previous figure of the result, and the second is written to "its" place.

III. Conclusion.

Of all the unusual ways found by me, the method of "lattice multiplication or jealousy" seemed more interesting. I showed it to my classmates, and he also really liked.

The simplest method of "doubling and split" seemed to me, which Russian peasants used. I use it when you multiply 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" with huge numbers in the mind.

I think that our method of multiplication in the column is not perfect and you can come up with even faster and more reliable ways.

Literature.

Depima I. "Stories about mathematics." - Leningrad: Education, 1954. - 140 s.
Koreev A.A. The phenomenon of Russian multiplication. History. http://numbernautics.ru/
Olochnik S. N., Nesterenko Yu. V., Potapov M. K. "Ancient entertaining tasks." - M.: Science. The main editorial office of physico-mathematical literature, 1985. - 160 p.
Perelman Ya.I. Quick account. Thirty simple techniques oral account. L., 1941 - 12 s.
Perelman Ya.I. Entertaining arithmetic. M. Russanova, 1994--205c.https://accounts.google.com.
Signatures for slides:
The work performed a student of 6 "b" class of the gods of Vasily. Leader: Smirnova Tatyana Vladimirovna Unusual methods of multiplication
Objective: show unusual methods of multiplication. Tasks: Find unusual methods of multiplication. Learn to apply them. Choose for yourself the most interesting or lighter and use them with the score.
Multiplication on the fingers.
Multiplication by 9.
The Italian Mathematics of Luke Pacioli was born in 1445.
Multiplying in the way "Little Castle"
Multiplication by the "Jealousy" method
Multiplying the grid meter. 3 4 7 2 9 6 8 1 4 3 6 6 3 7 2 3 6 0 10 347 29 \u003d 10063
Russian peasant method 37 32 37 ..........32 74 ..........16 148 ..........8 296 ......... .4 592 ..........2 1184 ......... 1 37 32 \u003d 1184
Thanks for attention

problem : figure out the types of multiplication

purpose: Acquaintance with various methods of multiplication natural numbersnot used in lessons, and their use in calculations of numerical expressions.
Tasks:
1. Find and disassemble various methods of multiplication.
2. Learn to demonstrate some methods of multiplication.
3. To tell about new methods of multiplication and teach them to use students.
4. Split skills independent work: Search for information, selection and design of the found material.
5. Experiment "What is the way faster"
Hypothesis: Do I need to know the multiplication table?
Relevance: Recently, students trust gadgets more than themselves. And about this is considered only on calculators. We wanted to show that there are different ways to multiply, that it would be easier for disciples, and it is interesting to learn.
Introduction
You will not be able to perform multiplications multivalued numbers - At least even double-digit - if not remember, by hearting all the results of multiplying unambiguous numbers, i.e. what is called multiplication table.
At various times, different peoples owned various ways of multiplying natural numbers.
Why now all nations use one way of multiplying a "column"?
Why did people refuse the old ways to multiply in favor of modern?
Do you have forgotten ways to multiply the right to exist in our time?
To answer these questions I did the following job:
1. With the help of the Internet, I found information about some multiplication methods that were used earlier.;
2. He studied the literature proposed by the teacher;
3. Solved a couple of examples by all the studied ways to learn their shortcomings;
4) revealed among them the most effective;
5. He conducted an experiment;
6. Made the conclusions.
1. Find and disassemble various methods of multiplication.
Multiplication on the fingers.

Methods of multiplication of numbers in different countriesoh

Multiplication by 9..

The multiplication for the number 9 - 9 · 1, 9 · 2 ... 9 · 10 - it is easier to eat out of memory and it is more difficult to manually by the method of addition, however, for the number 9, multiplication is easily reproduced "on the fingers". Pour your fingers on both hands and turn your hands with your palms from ourselves. Mentally presate the fingers sequentially numbers from 1 to 10, starting with the mother's maiden and ending with the little finger of the right hand (this is shown in the figure).

Who invented multiplication on the fingers

Suppose, we want to multiply 9 on 6. Beaging your finger with a number equal to the number that we will multiply nine. In our example, you need to bend a finger with number 6. The number of fingers to the left of the bent finger shows us the number of dozens in the answer, the number of fingers on the right is the number of units. On the left we have 5 fingers are not reducing, on the right - 4 fingers. Thus, 9 · 6 \u003d 54. Below in the figure, the entire principle of "calculations" is shown in detail.

Multiplying in an unusual way

Another example: need to calculate 9 · 8 \u003d?. In the course of the matter, let's say that the fingers of the hands may not necessarily be as a "counting machine". Take, for example, 10 cells in the notebook. Excrying the 8th cell. On the left there are 7 cells left, on the right - 2 cells. So 9 · 8 \u003d 72. Everything is very simple.

7 cells 2 cells.

Indian multiplication method.

The most valuable contribution to the treasury of mathematical knowledge was performed in India. Hindus offered the method of recording numbers used by us with ten signs: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

Hindus considered great. They came up with a very simple way of multiplication. They performed multiplied, starting with the older discharge, and recorded incomplete works just above the multiple, blessing. At the same time, the senior discharge of a complete work was immediately visible and, moreover, a pass of any number was excluded. The multiplication sign has not yet been known, so they left a small distance between the multipliers. For example, multiply in the way 537 to 6:

(5 ∙ 6 =30) 30

(300 + 3 ∙ 6 = 318) 318

(3180 +7 ∙ 6 = 3222) 3222

6
Multiplication by the method of "Little Castle".

The top numbers, starting with the older discharge, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold.

Methods of multiplication of numbers in different countries

Multiplication of numbers by the "Jealousy" method.

"Multiplication methods The second method wears the romantic title of jealousy", or "lattice multiplication".

Multiply in this way 347 to 29. Note the table, write down the number 347 above it, and on the right number 29.

Peasant method of multiplication.

The most, in my opinion, the "native" and light way of multiplication is a way that Russian peasants consumed. This reception does not require knowledge of the multiplication table on the number 2. The essence of it is that the multiplication of any two numbers is reduced to a row of sequential divisions of one number in half while rejection of another number. The division in half is continued until 1, in parallel, doubles another number. The last tweed number and gives a desired result.

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 odd, we do as follows:

384 ∙ 1 = 384

24 ∙ 17 = 24∙(16+1)=24 ∙ 16 + 24 = 384 + 24 = 408
New way of multiplication.

Left digit (in our example - zero), we leave unchanged, and the following numbers fold in pairs: a twin five, a top five, zero with a twos, zero with a triple. The last digit is also unchanged.

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

If, when folding two digits, the number exceeding nine, its first digit is added to the previous figure of the result, and the second is written to "its" place.

Consection.

Working on this topic, I learned that there are about 30 different, funny and interesting ways to multiply. Some in different countries still use so far. I chose some interesting ways for myself. But not all the ways are easy to use, especially when multiplying multivalued numbers.

Methods of multiplication

the second way of multiplication:

In Russia, the peasants did not apply the multiplication tables, but perfectly considered the work of multivalued numbers.

In Russia, starting with deep antiquity and almost up to eighteenthcentury, Russian people in their calculations did without multiplication anddivision. They used only two arithmetic actions - addition andsubtraction. Yes, the so-called "doubling" and "split". Butcommercial and other activities required to producemultiplication of sufficiently large numbers, both double-digit and three-digit.To do this, there was a special way of multiplying such numbers.

The essence of the ancient Russian method of multiplication is thatmultiplication of any two numbers was reduced to a number of consecutive divisions.one number in half (sequential split) while simultaneouslydoubling another number.

For example, if in the work 24 ∙ 5 multiply 24 reduce in twotimes (split), and multiply increased twice (double), i.e. takeproduction 12 ∙ 10, then the work remains equal to the number 120. Thisthe property of the work noticed our distant ancestors and learnedapply it when multiplying numbers by its special old Russianmode of multiplication.

Multiply in this way 32 ∙ 17 ..
32 ∙ 17
16 ∙ 34
8 ∙ 68
4 ∙ 136
2 ∙ 272
1 ∙ 544 Answer: 32 ∙ 17 \u003d 544.

In the disassembled example, division into two - "split" occurswithout residue. And what if the multiplier is not divided into two without a residue? ANDit seemed on the shoulder ancient calculations. In this case, they received this:
21 ∙ 17
10 ∙ 34
5 ∙ 68
2 ∙ 136
1 ∙ 272
357 Answer: 357.

From the example it is clear that if the multiplier is not divided into two, then from itfirst took the unit, then the result was separated by the result "and so5 to the end. Then all the lines with even the numbers were deleted (2nd, 4th,6th, etc.), and all right parts of the remaining lines folded and receivedthe desired work.

How did an ancient calculations aroused, justifying their waycalculations? That's how:21 ∙ 17 = 20 ∙ 17 + 17.
Number 17 is remembered, and the product 20 ∙ 17 \u003d 10 ∙ 34 (split -dutch) and write. Production 10 ∙ 34 \u003d 5 ∙ 68 (split -we double), but no matter how unnecessary work 10 ∙ 34 is crossing out. As 5 * 34\u003d 4 ∙ 68 + 68, then the number 68 is remembered, i.e. The third line does not strike, but4 ∙ 68 \u003d 2 ∙ 136 \u003d 1 ∙ 272 (split - double), while the fourtha string containing as if unnecessary work 2 ∙ 136 is crossed out, andthe number 272 is remembered. So it turns out that to multiply 21 at 17,it is necessary to add numbers 17, 68 and 272 - it is just equal parts of the rowsit is with odd multiple.
Russian way of multiplication and elegant and extravagant at the same time

I bring to your attention three examples in color pictures (in the upper right corner checking).

Example number 1: 12 × 321 = 3852
Draw first number from top to bottom, left to right: one green wand ( 1 ); Two orange sticks ( 2 ). 12 Drew.
Draw second number bottom up, to the left: Three blue wands ( 3 ); Two red ( 2 ); one lilac ( 1 ). 321 Drew.

Now, a simple pencil in drawing stroll, the points of intersection of numbers-sticks on the parts split and proceed to the counting of dots. Moving to right left (clockwise): 2 , 5 , 8 , 3 . Number-result We will "collect" from left to right (counterclockwise) and ... voila, got 3852

Example number 2: 24 × 34 = 816
In this example there are nuances. When counting dots in the first part it turned out 16 . Send-add to the dots of the second part ( 20 + 1 )…

Example number 3: 215 × 741 = 159315
No comments

At first, it seemed to me somewhat funeral, but at the same time intriguing and surprisingly harmonious. On the fifth example caught herself on the thought that multiplication goes in the fly and works in autopilot mode: Draw, point dots, i don't remember about the multiplication table, it seems like we do not know it at all.

To be honest, then checking a drawing method of multiplication And referring to the multiplication of a column, and more than once, and not two to their shame noted some slow motion, testifying that the multiplication table was rushing in some places and it is not worth forgetting it. When working with more "serious" numbers a drawing way of multiplication became too cumbersome, and multiplication of column Gone to joy.

P.S.: Glory and praise the native column!
In terms of building a way for unassuming and compact, very high speed, memory trains - not allowed to forget the multiplication table.

And therefore, I strongly recommend both yourself and you, if possible, forget about calculators in phones and on computers; and periodically indulge yourself with a multiplication of a column. And then not even an hour and the plot from the film "Rebells of the Machines" will unfold not on the cinema screen, but on our kitchen or the lawn next to the house ...

Three times through the left shoulder ..., knock on the tree ... ... and most importantly do not forget about gymnastics for the mind!

Learn the multiplication table !!!

Research work in mathematics in elementary school

Brief Abstract Research
Each schoolboy can multiply multivalued numbers "Stumpy." In this paper, the author draws attention to the existence of alternative methods of multiplication, affordable to younger schoolchildren who can "tedious" calculations to turn into a merry game.
The paper discusses six non-traditional methods of multiplying multivalued numbers used in various historical era: Russian peasant, lattice, small castle, Chinese, Japanese, according to Table V.Okonheshnikova.
The project is intended for the development of cognitive interest in the subject studied, to deepen knowledge in the field of mathematics.
Table of contents
Introduction 3.
Chapter 1. Alternative Methods of Multiplication 4
1.1. A little story 4.
1.2. Russian peasant method of multiplication 4
1.3. Multiplying in the way "Little Castle" 5
1.4. Multiplication of numbers by the "jealousy" or "lattice multiplication" 5
1.5. Chinese Method of Multiplication 5
1.6. Japanese multiplication method 6
1.7. Table Okneshikov 6.
1.8.Motion by the Stage. 7.
Chapter 2. Practical Part 7
2.1. Peasant Method 7.
2.2. Little Castle 7.
2.3. Multiplication of numbers by the "jealousy" or "lattice multiplication" 7
2.4. Chinese method 8.
2.5. Japanese method 8.
2.6. Table Okneshikov 8.
2.7. Questioning 8.
Conclusion 9.
Appendix 10.

"The subject of mathematics is so serious that it is useful not to lose cases of doing it a little entertaining."
B. Pascal
Introduction
It's impossible to do without computing a person in everyday life. Therefore, in the lessons of mathematics, we are primarily taught to perform actions on numbers, that is, to count. We multiply, divide, fold and deduct we are familiar to all ways that are studied at school. The question arose: Are there any other alternative methods of calculations? I wanted to explore them in more detail. In search of a response to the questions, this study was carried out.
The purpose of the study: identification of non-traditional multiplication methods to explore the possibility of their use.
In accordance with the purpose of the goal, we have formulated the following tasks:
- Find as many unusual multiplication methods as possible.
- Learn to apply them.
- Choose for yourself the most interesting or lighter than those offered at school, and use them with the score.
- Check in practice multiplication of multivalued numbers.
- Conduct the survey of students in 4th grades
Object of study: Various non-standard multiplication algorithms multiplying numbers
Subject: Mathematical action "Multiplication"
Hypothesis: If there are standard methods for multiplying multi-valued numbers, there may be alternative ways.
Relevance: Dissemination of knowledge about alternative multiplication methods.
Practical significance. In the course of the work, many examples were solved and the album was created, which includes examples with different algorithms multiplying multi-valued numbers by several alternative methods. It may be interested in classmates to expand the mathematical outlook and will serve as the beginning of new experiments.

Chapter 1. Alternative Methods of Multiplication

1.1. A bit of history
Those methods of calculations we use now were not always so simple and comfortable. In the old days enjoyed more cumbersome and slow techniques. And if a modern schoolboy could go for five hundred years ago, he would have struck all the speed and error of his calculations. The surrounding schools and monasteries would fly about it about him, eclipsed by the glory of the most scene counters of that era, and from all sides would come to learn from the New Great Master.
Especially difficult in the old days were the actions of multiplication and division.
In the book of V. Bellyustin "As people gradually reached the real arithmetic" set out 27 methods of multiplication, and the author notes: "It is very possible that there are still methods hidden in the caches of books, scattered in numerous, mainly handwritten collections." And all these techniques of multiplication competed with each other and digested with great difficulty.
Consider the most interesting and simple methods of multiplication.
1.2. Russian peasant method of multiplication
In Russia, 2-3 centuries ago, a method was distributed among the peasants of some provinces that did not require knowledge of the entire multiplication table. It was necessary only to be able to multiply and divide on 2. This method was called the peasant.
To multiply two numbers, they were recorded near, and then the left number was divided into 2, and the right was multiplied by 2. The results are recorded in the column until the left will remain 1. The residue is discarded. We highlight the lines in which there are even numbers. The remaining numbers in the right column are folded.
1.3. Multiplication of the way "Little Castle"
The Italian Mathematics of Luke Pachet in his treatise "The amount of knowledge of arithmetic, relationships and proportionality" (1494) leads eight different multiplication methods. The first of them is called "Little Castle".
The advantage of the method of multiplying the "Little Castle" is that from the very beginning the numbers of high-level digits are determined, and this is important if it is required to quickly appreciate the value.
The top numbers, starting with the older discharge, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold.
1.4. Multiplication of numbers by the "jealousy" or "lattice multiplication"
The second method of Luke Pachet is called "Jealousy" or "Detergent Multiplication".
First draws a rectangle, separated into squares. Then the square cells are divided diagonally and "... It turns out a picture similar to the lattice shutters," Pachet writes. "Such shutters were hanging on the windows of Venetian houses, preventing street passers-by to see the windows sitting at the windows and nuns."
Multipling each figure of the first factor with each number of the second, the works are written to the corresponding cells, there are tens of diagonal, and units under it. The figures of the works are obtained by adding numbers in oblique bands. The results of the additions are recorded under the table, as well as to the right.
1.5. Chinese way multiplication
Now imagine the multiplication method, the rapidly discussed on the Internet, which is called Chinese. When multiplying numbers, the points of intersection of direct, which correspond to the number of numbers of each discharge of both multipliers are considered.
1.6. Japanese way multiplication
A Japanese multiplication method is a graphic method using circles and lines. No less fun and interesting than Chinese. Even something like him.
1.7. Table Okoneshikov
Candidate of Philosophical Sciences Vasily Okneshnikov, part-time inventor of a new oral account system, believes that schoolchildren will be able to learn to master and multiply millions, billions and even sextillion with quadrillion. According to the scientist himself, the most advantageous in this regard is a nine-sized system - all data is simply placed in nine cells located like buttons on the calculator.
According to the thoughts, before becoming a computing "computer", you need to send the table created by it.
The table is divided into 9 parts. They are located on the principle of mini calculator: on the left in the lower corner "1", on the right in the upper corner of "9". Each part is the multiplication table of numbers from 1 to 9 (along the same "key" system). In order to multiply any number, for example, on 8, we find big squarecorresponding to the number 8 and write numbers from this square corresponding to the numbers of a multi-valued multivariate multi-factor. The numbers obtained are specifically: the first digit remains unchanged, and all the rest are folded pairwise. The resulting number will be the result of multiplication.
If when two digits are addition, it turns out the number superior to nine, then its first digit is added to the previous figure of the result, and the second is written to "its" place.
The new technique was tested in several Russian schools and universities. The Ministry of Education of the Russian Federation allowed publishing in notebooks into the cells along with the usual Pythagore table a new multiplication table - so far just for dating.
1.8. Multiplication of a column.
Not many know that the author of our usual way of multiplying a multi-valued number to multi-equity should be considered Adam Riza (Appendix 7). This algorithm is considered the most convenient.
Chapter 2. Practical Part
Mastering the listed methods of multiplication, a variety of examples were solved, an album was decorated with samples of various calculation algorithms. (Application). Consider the calculation algorithm on the examples.
2.1. Peasant fashion
Multiply 47 on 35 (Appendix 1),
- Purchased numbers on one line, carry out a vertical line between them;
- by 2, we will divide 2, right - multiplied by 2 (if the residue occurs during the division, then the residue is discarding);
- ending when one appears on the left;
-The strings in which there are left numbers;
-The appropriate numbers 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 on 2. However, when working with large numbers it is very cumbersome. It is convenient for working with double-digit numbers.
2.2. Little castle
(Appendix 2). Output. The method is very similar to our modern "column". Yes, and immediately define the numbers of senior discharges. This is important if you need to quickly appreciate the value.
2.3. Multiplication of numbers by the "jealousy" or "lattice multiplication"
Multiply, for example, numbers 6827 and 345 (Appendix 3):
1. Draw a square grid and write one of the multipliers over the columns, and the second is height.
2. Multiply the number of each row sequentially in the number of each column. Consistently multiply 3 by 6, by 8, 2 and 7, etc.
4. We fold the numbers by following diagonal stripes. If the sum of one diagonal contains dozens, then add them to the next diagonal.
From the results of the addition of figures on the diagonals, the number 2355315 is composed, which is the product of Numbers 6827 and 345, that is, 6827 ∙ 345 \u003d 2355315.
Output. The "lattice multiplication" method is not worse than the generally accepted. It is even simpler because there are numbers directly from the multiplication table without simultaneous addition, which is present in the standard method.
2.4. Chinese fashion
Suppose you need to multiply 12 to 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, to the left, we draw:
one green wand (1)
and two orange (2).
We draw the second number - 321, from the bottom up, to the left to the right:
Three blue sticks (3);
two red (2);
one lilac (1).
Now a simple pencil separating the intersection points and proceed to their calculation. Moving right left (clockwise): 2, 5, 8, 3.
Received result Read from left to right - 3852
Output. An interesting way, but spend 9 direct when multiplying 9 somehow for a long time and uninteresting, and then another point of intersection count. Without skill it is difficult to understand the division of the number on the discharge. In general, no multiplication table do not do!
2.5. Japanese fashion
Multiply 12 to 34 (Appendix 5). Since the second multiplier is a two-digit number, and the first figure of the first factor 1, we build two single circles in the upper line and two binary circles in the bottom line, since the second figure of the first factor is 2.
Since the first digit of the second multiplier 3, and the second 4, divide the circles of the first column into three parts, the second column into four parts.
The number of parts on which circles were divided and is the answer, that is, 12 x 34 \u003d 408.
Output. The method is very similar to Chinese graphic. Only direct are replaced with circles. It is easier to define discharges in the number, however draw circles less convenient.
2.6. Table Okoneshikov
It is required to multiply 15647 x 5. Immediately remember the large "button" 5 (it is in the middle) and we mentally find small buttons 1, 5, 6, 4, 7 (they are also located, as on the calculator). They correspond to numbers 05, 25, 30, 20, 35. The obtained numbers fold: the first digit 0 (remains unchanged), 5 mentally add from 2, we get 7 - this is the second digit of the result, 5 fold with 3, we get the third digit - 8 0 + 2 \u003d 2, 0 + 3 \u003d 3 and the last digit of the work remains - 5. As a result, it turned out 78,235.
Output. The method is very convenient, but you need to learn by heart or always have a table at hand.
2.7. Questioning of students
Quart term books were conducted. 26 people took part (Appendix 8). On the basis of the survey, it was revealed that all respondents can multiply in a traditional way. But about the unconventional methods of multiplication, most guys do not know. And there are wishing to meet them.
After the primary survey was carried out extracurricular occupation "Multiplication with hobby", on which the guys got acquainted with alternative multiplication algorithms. After that, a survey was conducted to identify the most likely ways. Unconditional leader became the most modern method Vasily Okheneshikov. (Appendix 9)
Conclusion
Having learned to count by all the presented ways, I believe that the most convenient multiplication method is the "Little Castle" method - because it looks like this now!
From all those found by me of unusual ways of account, the Japanese method seemed more interesting. The simplest method of "doubling and split" seemed to me, which Russian peasants used. I use it when multiplying is not too large numbers. It is very convenient to use it when multiplying two-digit numbers.
Thus, I reached my research goals - I studied and learned to apply non-traditional methods for multiplying multivalued numbers. My hypothesis was confirmed - I took possession of six alternative ways and found out that this is not all possible algorithms.
Memo studied unconventional methods Multiplications are very interesting and have the right to exist. And in some cases they even easier to use. I believe that the existence of these methods can be told at school, at home and surprise your friends and acquaintances.
While we just studied and analyzed the already known methods of multiplication. But who knows, perhaps, in the future, we will be able to open new ways of multiplication. I also do not want to stop at reached and continue the study of non-traditional multiplication methods.
List of sources of information
1. List of references
1.1. Harutyunyan E., Levitas. Entertaining mathematics. - M.: AST - Press, 1999. - 368 p.
1.2. Bellyustina V. How gradually reached people to real arithmetic. - LKI, 2012.-208 p.
1.3. Depman I. Stories about mathematics. - Leningrad: Education, 1954. - 140 s.
1.4. Likum A. All about everything. T. 2. - M.: Philological Society "Word", 1993. - 512 p.
1.5. Olochnik S. N., Nesterenko Yu. V., Potapov M. K .. Vintage entertaining tasks. - M.: Science. The main editorial office of physico-mathematical literature, 1985. - 160 p.
1.6. Perelman Ya.I. Entertaining arithmetic. - M.: Rusanova, 1994 - 205c.
1.7. Perelman Ya.I. Quick account. Thirty simple oral receptions. L.: Lenzdat, 1941 - 12 p.
1.8. Savin A.P. Mathematical miniatures. Entertaining mathematics for children. - M.: Children's literature, 1998 - 175 p.
1.9. Encyclopedia for children. Mathematics. - M.: Avanta +, 2003. - 688 p.
1.10. I will know the world: Children's Encyclopedia: Mathematics / Sost. Savin A.P., Stozo V.V., Kotova A.Yu. - M.: LLC "Publisher AST", 2000. - 480 p.
2. Other sources of information
Internet resources:
2.1. Koreev A.A. The phenomenon of Russian multiplication. History. [Electronic resource]

The world of mathematics is very large, but I have always been interested in multiplication methods. Working on this topic, I learned a lot of interesting things, learned how to pick up the material I needed from the read. He learned how individual entertaining tasks, puzzles and examples of multiplying in various ways are solved, as well as what arithmetic focuses and intensive computing techniques are based.

About multiplication

What remains from most people in the head from the fact that they were once studied at school? Of course, W. different people - Miscellaneous, but everyone is probably a multiplication table. In addition to the efforts attached to her "ascall" will recall hundreds (if not thousands) tasks solved by us with its help. Three hundred years ago in England, a person who knows the multiplication table was already considered a scientist man.

Multiplication methods were invented a lot. Italian mathematician of the end of the XV - the beginning of the XVI century onion of Pacioli in the treatise on arithmetic leads 8 different methods of multiplication. In the first one, which is called the "Little Castle", the numbers of the top number, starting with the older, alternately multiply on the lower number and are recorded in the column with the addition of the desired number of zeros. Then the results fold. The advantage of this method before ordinary is that from the very beginning the numbers of high-level digits are determined, and this is important in capex calculations.

The second method is of no less romantic name "Jealousy" (or lattice multiplication). A grill is drawn into which the results of intermediate calculations enter, more precisely, the number from the multiplication table. The grille is a rectangle divided into square cells, which, in turn, are separated by half-diagonals. On the left (from top to bottom) was written by the first factor, and at the top - the second. At the intersection of the corresponding line and column, the product of the numbers standing in them was written. Then the obtained numbers were folded along the diagonals spent, and the result was recorded at the end of this column. The result was read along the lower and right sides of the rectangle. "Such a grill," writes Luka Pacioli, "reminds the lattice shutters-blinds, which were hung on the Venetian windows, preventing passersby to see the windows sitting in the windows and nuns."

All methods of multiplication described in the Book Book of Pacioli used the multiplication table. However, the Russian peasants were able to multiply without a table. Their method of multiplication used only multiplication and division on 2. To multiply two numbers, they were recorded near, and then the left number was divided by 2, and the right was multiplied by 2. If the balance was obtained, then it was discarded. Then they were drawn out those lines in the left column, in which there are even numbers. The remaining numbers in the right column were evolved. As a result, the work of the initial numbers was obtained. Check out on several pairs of numbers, that this is true. Proof of the justice of this method is shown using a binary number system.

Old russian multiplication method.

With deep antiquity and almost until the eighteenth century, Russian people in their computations did without multiplication and division: they used only two arithmetic actions - addition and subtraction, and even the so-called "doubling" and "split". The essence of the Russian antique multiplication method is that multiplication of any two numbers is reduced to a row of sequential divisions of one number in half (sequential, split) with simultaneous doubling of another number. If in the work, for example 24 x 5, multiply reduce 2 times ("split"), and the multiplier increases 2 times

("Double"), then the work will not change: 24 x 5 \u003d 12 x 10 \u003d 120. Example:

The division of the multiple in half is continued until 1 is in private, while at the same time double the multiplier. The last twice number is the desired result. So, 32 x 17 \u003d 1 x 544 \u003d 544.

In those long-standing times, doubling and split was taken even for special arithmetic action. Just what kind of special. actions? After all, for example, the doubling of the number is not a special action, but only the addition of this number with itself.

Note the numbers share Pa 2 all the time without a residue. But what if the multiplier is divided into 2 with the remnant? Example:

If the multiplier is not divided into 2, then it first takes away the unit, and then the division is already being divided into 2. The lines with a self-intelligence are highlighted, and the right parts of the lines with odd multiple are folded.

21 x 17 \u003d (20 + 1) x 17 \u003d 20 x 17 + 17.

Number 17 We will remember (the first line is not triggered!), And the product 20 x 17 will be replaced with an equal to it 10 x 34. But the product 10 x 34, in turn, can be replaced with an equal to the product 5 x 68; Therefore, the second line is highlighted:

5 x 68 \u003d (4 + 1) x 68 \u003d 4 x 68 + 68.

The number 68 is remembered (the third line is not triggered!), And the product 4 x 68 will be replaced by an equal to it with a piece of 2 x 136. But the product 2 x 136 can be replaced with an equal to the product 1 x 272; Therefore, the fourth line is highlighted. So in order to calculate the work 21 x 17, you need to add numbers 17, 68, 272 - the right parts of the lines with odd multiple. The works with even intelligence can always be replaced with the help of splitting the multiplier and doubling the multiplier with their works; Therefore, such lines are excluded from the calculation of the final work.

I tried to multiply myself an old way. I took the number 39 and 247, I got such

The columns will turn out even longer than I have if you take a multiplier more than 39. Then I decided that the same example is in modern one:

It turns out that our school method of multiplication of numbers is much easier and more economical than an old Russian way!

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

That's great that people from different countries have multiplied by the same way.

Not so long ago, just about a hundred years ago, to learn the multiplication table was very difficult for students. In order to convince students in the need to know the tables, the authors of mathematical books have long been resorted. To poems.

Here are a few lines from unfamiliar books: "But the multiplication is required to have a subsequent table, only in memory of Having, Tako, yes, I am a number, with which I am smart, without herself, speech say, or writing, Butter 2 there is 2 , or 2-Wa in 3 there are 6, and 3 years 3 have 9 and so on. "

Anyone who does not feel and in all the science of the table and is progressing, non-free from flour,

I can not know, I do not take into account that many Tuna will depress

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

L. F. Magnitsky wrote and published the first arithmetic textbook in Russia; There were only handwritten mathematical books before him. According to the "arithmetic" L. F. Magnitsky studied the Great Russian scientist M. V. Lomonosov, as well as many other prominent Russian scientists of the eighteenth century.

And how was multiplied in those days, during the time of Lomonosov?. Let's see an example.

As we understood, the action of multiplication was then recorded almost as in our time. Only the factory called "Etlehood", and the product is "product" and, moreover, did not write a sign of multiplication.

And then how explained the multiplication?

It is known that M. V. Lomonosov knew by heart all the "arithmetic" of Magnitsky. In accordance with this textbook, a small Misha Lomonosov multiplying 48 to 8 would explain like this: "8-Wa 54 There are 64, I am writing under the lard, against 8, and I have 6 decimals in your mind. And further 8-Wa in 4 there are 32, and I hold 3 in your mind, and I will put 6 deciments, and it will be 8. And this 8 will write 4, in a row to the left hand, and 3 in the mind there is an essence, I will write in a row Follow 8, to the left hand. And it will be from multiplication 48 with 8 work 384 ".

And we almost also explain, only we speak in modern, and not an old and, moreover, call the discharge. For example, 3 must write in third place because it will be hundreds, and not just "in a row of 8, to the left hand."

The story "Masha -" Focusnitsa "."

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

The number of the month in which you were born, multiply by 100., then add a birthday. , multiply the result to 2., add 2 to the resulting number 2; The result multiply to 5, add 1 to the resulting number 1, add zero to the result. , Add to the resulting number 1. And finally, add the number of your years.

Finish, I got 20721. - I say.

* Right, - I confirmed.

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

You, Masha probably mistaken, - Petya doubted. - How does it work: Vitya from the third class, and born, too, in 1949, like Sasha.

No, Masha faithfully guessed, "confirms Vitya. Only one year I had a long time and therefore he went twice into the second class.

* And I got 111521, "Pavlik reports.

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

And because now September is now, and Pavlik was born in November, and he was still 10 years old, although he was born in 1948, "Masha explained.

She guessed the date of birth of another three-four students, and then explained how she did it. It turns out that it takes 111 from the last number, and then the residue passes by three marks to the right of two digits to the right. Middle two figures denote birthday, the first two plays one - number of the month, and the last two digits number of years. Knowing how much a person is, it is not difficult to determine the year of birth. For example, I got the number 20721. If it takes 111 from it, then it turns out 20610. So, now I am 10 years old, but I was born on February 6th. Since September 1959 is now being coming, then I was born in 1949.

And why should I take away 111, and not any other number? We asked. -And why exactly are the birthday, the number of month and the number of years?

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

1) 11 x 100 \u003d 1100; 2) 1100 + j4 \u003d 1114; 3) 1114 x 2 \u003d

2228; 4) 2228 + 2 \u003d 2230; 57 2230 x 5 \u003d 11150; 6) 11150 1 \u003d 11151; 7) 11151 x 10 \u003d 111510

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

As can be seen, the number of the month (11) was multiplied by 100, then 2, then another 5 and, finally, another 10 (attributed to Kul), and only 100 x 2 x 5 x 10, that is, 10,000. So , 11 became tens of thousands, that is, they make up the third facet, if you count on the right left two digits. So learn the number of the month in which you were born. Birthday (14) He was multiplied by 2, then on 5 and finally, another 10, and only 2 x 5 x 10, that is, at 100. So, the birthday must be found among hundreds, in the second face, but here There are extraneous hundreds. See: He added the number 2, which was multiplied by 5 and 10. So, it turned out excess 2x5x10 \u003d 100 - 1 hundred. This 1 hundred I and take away from 15 hundred apartments 1,11521, it turns out 14 hundred. So I recognize my birthday. The number of years (10) has not been multiplied by anything. So, this number should be found among units, in the first face, but there are extraneous units here. See: He added the number 1, which was multiplied by 10, and then added 1. It means that it turned out all extra 1 x + 1 \u003d 11 units. These 11 units I and take away from 21 units. Among 111521, it turns out 10. So I recognize the number of 111521. I took 100+ 11 \u003d 111. When I took 111 from the number 111521, then It turned out. It means

Pavlik was born on November 14, and he was 10 years old. Now there is 1959th year, but I did not take 10 from 1959, and from 1958, since 10 years Pavlik turned last year in November.

Of course, such an explanation immediately do not remember, but I tried to understand it on my example:

1) 2 x 100 \u003d 200; 2) 200 + 6 \u003d 206; 3) 206 x 2 \u003d 412;

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

Puzzle.

The first task: at noon, a passenger steamer comes from Stalingrad to Kuibyshev. An hour later from Kuibyshev to Stalingrad comes out the goods-passenger steamer, which moves slower than the first steamer. When the steamers will meet, which one will be further from Stalingrad?

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

But the second task, in the past Sunday, our squad and a detachment of the fifth grade put trees along a large pioneer street. The detachments were supposed to sit row of trees, on an equal number on each side of the street. As you remember, our detachment came to work early, and before the arrival of the five-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 not where it was necessary. Then we worked on our side of the street. Fifth-graders finished work earlier. However, they did not remain in debt to us: they switched to our side and first put 8 trees first ("gave the debt"), and then 5 more trees, and the work was completed by us.

It is asked how many trees were planted for five-graders, what are we?

: Of course, the fifth graders were planted only on 5 trees more than we: when they planted on our side of 8 trees, thereby gave a debt; And when they planted 5 more trees, then as if they gave us 5 trees. So it turns out that they were planted only on 5 trees more than we.

No reasoning is incorrect. It is true that the fifth graders made us a favor, putting 5 trees for us. But then, in order to get a sure answer, it is necessary to reason this: we have not fulfilled our task on 5 trees, the five-graders exceeded their 5 trees. So it turns out that the difference between the number of trees planted with fifth graders, and the number of trees planted by us, is not 5, and 10 trees!

But the last puzzle task, playing the ball, 16 students are located on the sides of the square site so that there were 4 people on each side. Then 2 student left the rest moved so that on each side of the square was again 4 people. Finally, 2 more student left, but the rest were located in such a way that on each side of the square was still 4 people. How could this happen? Decide.

Two rapid multiplication

Once the teacher proposed such an example to his students: 84 x 84. One boy answered quickly: 7056. "How did you think?" - asked the teacher's student. "I took 50 x 144 and threw out 144," the one replied. Well, explain how the student believed.

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 hundred, it means 84 x 84 \u003d 7200 - 144 \u003d

And now we count in the same way as 56 x 56 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 fifters, or 32 hundred (3200), without 64, i.e. to multiply the number on 49, this number is needed. Multiply 50 (fifty), and from the resulting product to subtract this number.

But examples on another method of calculation, 92 x 96, 94 x 98.

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

So quickly can be considered only when the numbers are close to 100. We find add-ons to 100 to these numbers: for 93 there will be 7, and for 95 will be 5, from the first given number, we take a second supplement: 93 - 5 \u003d 88 - so much will be in the work hundreds, replacing additions: 7 x 5 \u003d 3 5 - So much will be in the work of units. So, 93 x 95 \u003d 8835. And why it is necessary to do that, it is not difficult to explain.

For example, 93 is 100 without 7, and 95 is 100 without 5. 95 x 93 \u003d (100 - 5) x 93 \u003d 93 x 100 - 93 x 5.

To take away 5 times 93, you can take 100 times from 100 times, but then add 5 times to 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) honeycomb. + 5 x 7 \u003d 8800 + 35 \u003d 8835.

97 x 94 \u003d (97 - 6) x 100 + 3 x 6 \u003d 9100 + 18 \u003d 9118, 91 x 95 \u003d (91 - 5) x 100 + 9 x 5 \u003d 8600 + 45 \u003d 8645.

Multiplication in. Domino.

With the help of Domino bones easily portray some cases of multiplying multivalued numbers per unambiguous number. For example:

402 x 3 and 2663 x 4

Winner will be recognized by the one who for a certain time will be able to use the greatest number Domino bones, making up examples on multiplication of three-, four-digit numbers per unambiguous number.

Examples for multiplying four-digit numbers to unambiguous.

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

As can be seen, only 20 Domino bones are used. Examples are made to multiply not only four-digit numbers per unambiguous number, but also three-, and five, and six-digit numbers per unambiguous number. 25 bones were used and such examples are compiled:

However, all 28 bones can still be used.

Stories about whether Old Man Hottabych knew the arithmetic.

The story "I get on arithmetic" 5 ".

As soon as the next day I went to Misha, he immediately asked: "What's new, interesting was in the circle?" I showed Mishe and his friends, how cleverly taught Russian people in the old days. Then I suggested in mind to count how much it will be 97 x 95, 42 x 42 and 98 x 93. They, of course, without a pencil and paper could not do this and were very surprised when I almost instantly gave these examples to these examples. Finally, we all decided that the task was given to the house. It turns out, it is very important how points are located on a sheet of paper. Depending on this, you can spend one and four, and six straight lines, but not more.

Then I suggested the guys to make examples of multiplication of Domino bones as it was done on a circle. We managed to use 20, 24, and even 27 bones, but from in C E X 28, we could not create examples, although we sat for a long time.

Misha remembered that today the movie "Old Man Hottabych" is demonstrated in the cinema. We quickly ended the arithmetic and ran to the movies.

This is a picture! Although the fairy tale, but still interesting: Talk about us, boys, o school Life, as well as about the eccentric sage - Gina Hottabich. And greatly sounded Hottabych, suggesting the halter in geography! As can be seen, in the long time, even the Indian wise men - Gina - very, very poorly knew geography, I wonder, but how did the old man of Hottabych become "to prompt, if the Washa handed over the arithmetic exam? Probably hottabych and arithmetic did not know.

Indian multiplication method.

Let you need to unvevy 468 to 7. On the left you write the multiplier, the right multiplier:

Indians had no sign of multiplication.

Now I will multiply on 7, it will turn out to be 28. This number is written by Supprand 4.

Now 8 is multiplied by 7, it will turn out 56. 5 by the increase to 28, it turns out 33; 28 hundred, and 33 We write, 6 write over the number 8:

It turned out very interesting.

Now 6 is multiplied by 7, it will turn out to be 42, 4 increments to 36, it will turn out 40; 36 hundred, and 40 write; 2 pointed over the number 6. So 486 multiplied by 7, it turns out 3402:

It is true, but only no penalty is quick and convenient! This is what the most famous computers are multiplied.

As you can see, the old man hottabych arithmetic knew not badly. However, he made a record of actions not as we do.

For a long time, more than a thousand three years ago, the Indians were the best computers. However, they had no more papers, and all the calculations were made on a small black board, making on it with a cane pen and applying a very liquid white paint that left signs easily.

When we write with chalk on a blackboard, then this is a little resembling an Indian writing method: on a black background there are white signs that are easy to erase and correct.

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

The sign of multiplication at that time did not yet exist, and only some interval was left between the multiplier and the multiplier. The Indian way could be multiplied by and from units. However, the Indians themselves were performed since the older discharge, and recorded incomplete works just above the multiple, blessingly. At the same time, the senior discharge of a complete work was immediately visible and, moreover, a pass of any number was excluded.

An example of multiplication by the Indian way.

Arabic multiplication method.

Well, what about, in the date, do the multiplication of the Indian way, if you write on paper?.

This technique for writing on paper adapted Arabs, the famous scientist of antiquity of Uzbek Muhammed Ibn Musa Alwariz-Mi (Muhammed Son Musa from Khorezmaya, which was located on the territory of the modern Uzbek SSR) more than a thousand years ago performed multiplication on parchment so:

As can be seen, he did not erase unnecessary numbers (on paper it is already inconvenient), but shouted them; He recorded the new numbers to be crucified, of course, is frozen.

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

Therefore, 7264 x 8 \u003d 58112. But how to multiply on a two-digit number, to multivalued?.

The reception of multiplication remains the same, but the recording is significantly complicated. For example, you need to multiply 746 on 64. First multiplied by 3 dozen, it turned out

So, 746 x 34 \u003d 25364.

As you can see, highlighting unnecessary digits and replacing them with new numbers when multiplying even on a two-digit number leads to too cumbersome recording. And what will happen if multiplied by three-, a four-digit number?!

Yes, arab method Multiplication is not very convenient.

This method of multiplication was kept in Europe until the eighteenth century, as many as a thousand years. It was called the crossing methods, or chiam, since the Greek letter X (hee) was put between the variable numbers), gradually replaced by the oblique cross. Now we see well that our modern method of multiplication is the easiest and most convenient, probably the best of all possible methods Multiplication.

Yes, our School way of multiplying multivalued numbers is very good. However, multiplication recording can be done differently. Perhaps it would be best to do it, for example, like this:

This method is actually good: multiplication begins with the older discharge of the multiplier, the lowest discharge of incomplete works is recorded under the corresponding discharge of the multiplier, which eliminates the possibility of an error in the case when zero is found in any discharge of the multiplier. Approximately the multiplication of multivalued numbers Czechoslovak schoolchildren. That's interesting. And we thought that arithmetic actions can only be recorded as it was customary.

A few more puzzles.

Here is the first, simple task: the tourist can go through the hour 5 km. How many kilometers will it pass for 100 hours?

Answer: 500 kilometers.

And this is another big question! It is necessary to know more accurately as the tourist walked these 100 hours: without rest or with the gear. 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 way. Being in a consecutive movement 100 hours is probably not able to: it's more than four days; Yes, and the speed of movement would decrease all the time. Another thing, if the tourist walked with the transes for lunch, for sleep, etc., then it can pass and all 500 km; Only on the way, it should no longer be four days, but about twelve days (if it goes the day on average 40 km). If he was 100 hours on the way, it could be about only 160-180 km.

Different answers. So in the task condition, it is necessary to add something to something, otherwise the answer is impossible.

We now decide such a task: 10 chickens in 10 days eaten 1 kg of grain. How many kilograms of grain will eat 100 chickens in 100 days?

Solution: 10 chickens of 10 days eaten 1 kg of grain, it means that 1 chicken for the same 10 days eaten 10 times less, that is, 1000 g: 10 \u003d 100 g.

In one day, the chick eats another 10 times less, that is, 100 g: 10 \u003d 10 g. Now we know that 1 chicken in 1 day eats 10 g of grain. It means 100 chicks a day eaten 100 times more, that is

10 g x 100 \u003d 1000 g \u003d 1 kg. In the same periods, they will eat another 100 times more, that is, 1 kg x 100 \u003d 100 kg \u003d 1 c. So, 100 chickens in 100 days eaten a whole centner of grain.

There is a faster solution: the chickens are 10 times more and breeded longer than 10 times, it means that all the grains should be 100 times more than 100 times, that is, 100 kg. However, in all these arguments there is one omission. We think and find a mistake in reasoning.

: - We look at the last reasoning: "100 chickens in one day are eaten 1 kg of grain, and in 100 days they will eat 100 times more. "

After all, for 100 days (this is more than three months!) Chickens will noticeably grow up and on the day they will not eat 10 g of grain, and grams of 40 - 50, since the ordinary chicken eats about 100 g of grain per day. So, for 100 days, 100 chickens will be eaten not 1 C grain, but much more: two or three centners.

But you have the last task-puzzle about the tie of the node: "On the table lies a piece of rope, elongated in a straight line. It is necessary to take it with one hand for one, the other hand for the other end and, without the ends of the rope from the hands, tie a node. »A well-known case, one tasks are easy to disassemble, going from the data to the problem of the problem, while others, on the contrary, going from the problem of the data task.

Well, here we tried to disassemble this task, going from the question of the data. Let the knot on the rope already exist, and the ends are in their hands and are not produced. We will try to return to its data from a solved problem, to the original position: the rope lies, elongated on the table, and the ends are not produced from the hands.

It turns out that if you fix the rope, I do not produce ends of it from the hands, then the left hand, going under an elongated rope and over the right hand, keeps the right end of the rope; And the right hand, going over the rope and under the left hand, keeps the left end of the rope

I think after such a parsing task everything became clear how to tie a knot on the rope, you need to do everything in the reverse order.

Two more receivers of fast multiplication.

I will show you how to quickly multiply the numbers such as 24 and 26, 63 and 67, 84 and 86 IT. p., That is, when in the factors dozen "sideln, and the units are exactly 10 together. Enter examples.

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

* It turns out 1224, 3021, 5616.

For example, it is necessary to multiply 53 by 57. I multiply on 6 (1 more than 5), it turns out 30 - so many hundreds in the work; 3 I multiply on 7, it turns out 21 - so many units in the work. So, 53 x 57 \u003d 3021.

* How to explain it?

(50 + 3) x 57 \u003d 50 x 57 + 3 x 57 \u003d 50 x (50 + 7) +3 x (50 + 7) \u003d 50 x 50 + 7 x 50 + 3 x 50 + 3 x 7 \u003d 2500 + + 50 x (7 + 3) + 3 x 7 \u003d 2500 + 50 x 10 + 3 x 7 \u003d \u003d: 25 hundred. + 5 hundred. +3 x 7 \u003d 30 hundred. + 3 x 7 \u003d 5 x 6 cells. + 21.

Let's see how quickly multiply two-digit numbers within 20. For example, to multiply 14 to 17, it is necessary to fold the units 4 and 7, it will turn out to be at all dozens in the work (that is, 10 units). Then you need to multiply on 7, it will turn out to be 28 - so many units will be in the work. In addition, to the obtained numbers 110 and 28 it is necessary to add evenly 100. So, 14 x 17 \u003d 100 + 110 + 28 \u003d 238. In fact:

14 x 17 \u003d 14 x (10 + 7) \u003d 14 x 10 + 14 x 7 \u003d (10 + 4) x 10 + (10 + 4) x 7 \u003d 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 decided more such examples: 13 x 16 \u003d 100 + (3 + 6) x 10 + 3 x 6 \u003d 100 + 90 + + 18 \u003d 208; 14 x 18 \u003d 100 + 120 + 32 \u003d 252.

Multiplication on accounts

Here are some receptions, using anyone who knows how to quickly fold on accounts will be able to promptly perform examples of examples of u m.

Multiplication by 2 and 3 is replaced by two-time and troped addition.

At multiplication, 4 is multiplied first to 2 and fold this result with themselves.

The multiplication of the number on 5 is performed on the scores like this: tolerate all the number of one wire above, that is, it is multiplied by 10, and then divide this 10-fold number in half (how to divide on 2 using scores.

Instead of multiplication, 6 is multiplied by 5 and add multiply.

Instead of multiplication by 7, multiply by 10 and take a multiply three times.

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

In the same way, they are multiplied by 9: replace multiplication by 10 minus one multiply.

When multiplying is transferred to 10, as we have said, all numbers are one wire above.

The reader is likely to figure out how to act when multiplying the numbers, large 10, and what kind of replacement will be the most convenient. Multiplier 11 is necessary, of course, replaced by 10 + 1. The multiplier 12 is replaced by 10 + 2 or practically 2 + 10, i.e., first postpone the doubled number, and then add-on. The multiplier 13 is replaced by 10 + 3, etc.

Consider several special occasions For multipliers of the first hundreds:

It is easy to see, by the way that with the help of the scores it is very convenient to multiply on such numbers as on 22, 33, 44, 55, etc.; Therefore, it is necessary to strive when breaking multipliers to enjoy similar numbers with the same numbers.

To similar techniques are resorted to multiplication in numbers, large 100. If such artificial techniques are tedious, then we always, of course, can multiply with the help of accounts by general rule, Multiplying every digit of the multiplier and recording private works - it still gives some time reducing.

"Russian" method of multiplication

You cannot perform multiplication of multivalued numbers, - at least even double-digit - if you do not remember by hearting all the results of multiplying unambiguous numbers, i.e. what is called the multiplication table. In the old "arithmetic" of the Magnitsky, about which we have already mentioned, the need solid knowledge Multiplication tables in such (alien for modern hearing) verses:

Anyone who does not feel the table and is progressing, can not know the number that sets

And on all sciences, non-volatile from flour, he does not teach Tuna to depress

And in favor, I will not forget.

The author of these verses obviously did not know or missed that there was a method to multiply the numbers and without knowing the multiplication table. The method of this, similar to our school techniques, was used in everyday life of Russian peasants and inherited by them from deep antiquity.

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

The division in half continues until then), the pitch in private will not work 1, in parallel doubling another number. The last tweed number and gives a desired result. It is not difficult to understand what this method is based: the product does not change, if one multiplier is doubled, and the other is to double. It is clear that, as a result of a multiple repetition of this operation, a desired work is obtained.

However, how to do, if at the same time Nrich. Will you share in half the number odd?

People's way easily comes out of this difficulty. It is necessary, says the rule, in case of an odd number about kicked the unit and divide the residue in half; But it would have been necessary to add all those numbers of this column to the other than the number of this column, which are against the left column. l work. Almost this makes that all rows with even left numbers are burned; Only those that contain the left odd number remain.

We give an example (asterisks indicate that this line must be shocked):

Fasting not crossed out numbers, we get quite the right result: 17 + 34 + 272 \u003d 32 What is this reception based on?

The correctness of the reception will be 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 numbers 17, 34, etc., lost when dividing an odd number in half, must be added to the result of the last multiplication to get a product.

Examples of accelerated multiplication

We mentioned earlier that to perform those separate multiplication actions to which each of the above techniques disintegrates, there are also convenient ways. Some of them are quite simple and conveniently applicable they make it easier to calculate that it does not interfere with generally remember them to enjoy under normal calculations.

Such, for example, the reception of cross-multiplication is very convenient under action with double-digit numbers. The method is not new; He goes back to the Greeks and Hindu and in the old days was called "way of lightning", or "multiplication of a cross." Now he is forgotten, and it does not interfere with it.

Let it be required to multiply 24x32. Mentally, we have a number according to the following scheme, one under another:

Now consistently produce the following actions:

1) 4x2 \u003d 8 is the last digit of the result.

2) 2x2 \u003d 4; 4x3 \u003d 12; 4 + 12 \u003d 16; 6 - penultimate digit of the result; 1 remember.

3) 2x3 \u003d 6, and even restrained in mind the unit, we have

7 is the first digit of the result.

We get all the figures of the work: 7, 6, 8 - 768.

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

Another method consisting in the use of so-called "add-ons" is conveniently used in cases where the multiple numbers are close to 100.

Suppose that you want to multiply 92x96. "Supplement" for 92 to 100 will be 8, for 96 - 4. The action is made according to the following scheme: multipliers: 92 and 96 "add-ons": 8 and 4.

The first two digits of the result are obtained by simply subtracting from the multiplier "add-on" or vice versa; i.e., 4 or 96 are subtracted from 92.

85 and another case we have 88; This number is credited with the work of "add-ons": 8x4 \u003d 32. We obtain the result 8832.

That the result obtained should be faithful, clearly seen from the following transformations:

92x9b \u003d 88x96 \u003d 88 (100-4) \u003d 88 x 100-88x4

1 4x96 \u003d 4 (88 + 8) \u003d 4x 8 + 88x4 92x96 8832 + 0

Another example. It is required to multiply 78 to 77: multipliers: 78 and 77 "Supplements": 22 and 23.

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

Third example. Multiply 99 x 9.

farmers: 99 and 98 "Supplements": 1 and 2.

99-2 \u003d 97, 1x2 \u003d 2.

In this case, it must be remembered that 97 means here the number of hundreds. Therefore, we fold.

Interesting methods of multiplying multivalued numbers. Project on the topic: "Unusual methods of multiplication"

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