Methods for multiplying numbers in different countries. Methods for multiplying numbers

Purpose of work: Explore and show unusual ways of multiplying Objectives: Find unusual ways of multiplying. Learn to apply them. Choose for yourself the most interesting or lighter ones than those offered at the school, and use them when counting. Teach classmates to use a new way of multiplication

Methods: search method using scientific and educational literature, as well as search for the necessary information on the Internet; a practical method for performing calculations using non-standard counting algorithms; analysis of the data obtained in the course of the research The relevance of this topic lies in the fact that the use of non-standard techniques in the formation of computational skills increases students' interest in mathematics and contributes to the development of mathematical abilities

In math lessons, we learned an unusual way to multiply with a column. We liked it and we decided to find out other ways to multiply natural numbers. We asked our classmates if they knew other ways of counting? Everyone talked only about the methods that are taught in school. It turned out that all our friends know nothing about other methods. In the history of mathematics, there are about 30 methods of multiplication, which differ in the writing scheme or in the course of the calculation itself. The column multiplication method, which we learn in school, is one way. But is this the most efficient way? Let's see! Introduction

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. To do this, on one hand, they stretched out 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 the extended 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. For example, multiply 7 by 8. In this example, 2 and 3 fingers will be bent. If you add up the number of bent fingers (2 + 3 \u003d 5) and multiply the number of unbent fingers (23 \u003d 6), you get the tens and units of the desired product, respectively, 56. So you can calculate the product of any single-digit numbers greater than 5.

Multiplication for the number 9 is very easily reproduced "on the fingers". Spread your fingers on both hands and turn your hands with your palms away from you. Mentally assign the numbers from 1 to 10 to your fingers in sequence, starting with the little finger of your left hand and ending with the little finger of your right hand. Let's say we want to multiply 9 by 6. Bend the finger with the number equal to the number by which we will multiply nine. In our example, you need to bend finger number 6. The number of fingers to the left of the curled finger shows us the number of tens in the answer, the number of fingers to the right is the number of ones. On the left we have 5 fingers not bent, on the right - 4 fingers. So 9 6 \u003d 54.

Multiplication method "Small castle" The advantage of the multiplication method "Small castle" is that the most significant digits are determined from the very beginning, which is important if you need to quickly estimate the value. The digits of the upper number, starting with 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. The results are then added up.

"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 in the multiplier and the multiplier. Then the square cells are divided diagonally, and "... a picture is obtained that looks like lattice shutters, - Pacioli writes. "Such shutters were hung on the windows of Venetian houses ..."

Lattice multiplication \u003d +1 +2

The peasant way This is the way of the Great Russian peasants Its essence lies in the fact that the multiplication of any numbers is reduced to a series of consecutive divisions of one number in half, while doubling another number ……… .32 74 ……… ……… .8 296 ……… .4 592 ……… ……… 1 3732 \u003d 1184

Peasant way (odd numbers) 47 x \u003d 1645

Step 1. first number 15: Draw the first number - with one line. We draw the second number with five lines. Step 2. second number 23: Draw the first number with two lines. We draw the second number with three lines. Step 3. Count the number of points in groups. Step 4. Result - 345. Multiply two two-digit numbers: 15 * 23

Indian multiplication method (cross) 24 and X 3 2 1) 4x2 \u003d 8 - the last digit of the result; 2) 2x2 \u003d 4; 4x3 \u003d 12; 4 + 12 \u003d 16; 6- the penultimate figure of the result, we remember the unit; 3) 2x3 \u003d 6 and even a figure kept in mind, we have 7 - this is the first figure of the result. We get all the numbers of the product: 7,6,8. Answer: 768.

The Indian way of multiplying \u003d \u003d \u003d \u003d 3822 The basis of this method is the idea that the same number stands for units, tens, hundreds, or thousands, depending on where this number occupies. The occupied space, in the absence of any digits, is determined by zeros assigned to the digits. we start multiplication with the most significant bit, and write down incomplete products just over the multiplicand, bit by bit. In this case, the most significant bit 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

Reference number Multiply 18 * 19 20 (reference number) * 2 1 (18-1) * 20 \u003d Answer: 342 Short notation: 18 * 19 \u003d 20 * 17 + 2 \u003d 342

A new way to multiply X \u003d, 5 + 2, 5 + 3, 0 + 2, 0 + 3, 5

Conclusion: Having learned to count in all the presented ways, we came to the conclusion: that the simplest methods are those that we study at school, or maybe we just got used to them Of all the unusual methods of counting considered, the method of graphical multiplication seemed more interesting. We showed him to our classmates, and they also really liked him. The simplest seemed to be the method of "doubling and doubling", which was used by Russian peasants. Having worked with literature and materials on the Internet, we realized that we have considered a very small number of methods of multiplication, which means that there are many interesting things ahead of us.

Conclusion Describing the ancient methods of calculations and modern methods of fast counting, we tried to show that, both in the past and in the future, one cannot do without mathematics, a science created by the human mind. complicated due to the variety of methods and their cumbersome implementation The modern method of multiplication is simple and accessible to everyone. But, we think that our method of multiplying in a column is not perfect and we can come up with even faster and more reliable ways. It is possible that the first time many will not be able to quickly, on the move, perform these or other calculations. It doesn't matter. You need constant computational training. It will help you acquire useful verbal counting skills!

Used materials: html Encyclopedia for children. "Mathematics". - M .: Avanta +, - 688 p. Encyclopedia “I get to know the world. Mathematics". - M .: Astrel Ermak, Perelman Ya.I. Fast counting. Thirty Easy Verbal Counting Techniques. L., p.

In ancient India, two methods of multiplication were used: grids and galleys.
At first glance, they seem very difficult, but if you follow step by step in the proposed exercises, you will see that it is quite simple.
Let's multiply, for example, numbers 6827 and 345:
1. Draw a square grid and write one of the numbers above the columns, and the second in height. In the example provided, you can use one of these grids.

2. Having chosen the grid, we multiply the number of each row sequentially by the numbers of each column. In this case, we successively multiply 3 by 6, by 8, by 2 and by 7. Look at this diagram how the work is written in the corresponding cell.

3. See what the mesh looks like with all the padded cells.

4. Finally, add the numbers following the diagonal stripes. If the sum of one diagonal contains tens, then we add them to the next diagonal.

See how the number 2355315 is compiled from the results of adding the digits along the diagonals (they are highlighted in yellow), which is the product of the numbers 6827 and 345.

Municipal educational institution

Staromaksimkinskaya basic secondary school

District scientific and practical conference on mathematics

"Step into Science"

Research work

"Non-standard counting algorithms or fast counting without a calculator"

Head:,

mathematic teacher

from. Art. Maksimkino, 2010

Introduction ………………………………………………………………… .. …………… .3

Chapter 1. Account history

1.2. Miracle - counters ………………………………………………………………… ... 9

Chapter 2. Old ways of multiplication

2.1. Russian peasant way of multiplication… .. ……………. ………………. …… .. The “lattice” method ………………. …… .. …………………………… …….………..13

2.3. The Indian way of multiplication ………………………………………………… ..15

2.4. Egyptian way of multiplication ………………………………………………… .16

2.5. Multiplication on fingers …………………………………………………………… ..17

Chapter 3. Oral counting - gymnastics of the mind

3.1. Multiplication and division by 4 …………… .. ………………………. ………………… .19

3.2. Multiplication and division by 5 …………………………………… ... ………………… .19

3.3. Multiplication by 25 …………………………………………………………………… 19

3.4. Multiplication by 1.5 …………………………………………………………… ....... 20

3.5. Multiplication by 9 ………. ………………………………………………………… .20

3.6. Multiplication by 11 ……………………………………………… .. …………….… .20

3.7. Multiplying a three-digit number by 101 …………………………………………… 21

3.7. Square a number ending in 5 ……………………… 21

3.8. Square a number close to 50 ………………. ……………………… 22

3.9. Games ……………………………………………………………………………… .22

Conclusion ………………………………………………………………………….… 24

List of used literature ……………………………………………… ... 25

Introduction

Can you imagine a world without numbers? Without numbers, you won't make a purchase, you won't know the time, and you won't dial a phone number. And what about spaceships, lasers and all other technical achievements ?! They would be simply impossible if not for the science of numbers.

Two elements dominate in mathematics - numbers and figures with their infinite variety of properties and relationships. In our work, preference is given to the element of numbers and actions with them.

Now, at the stage of the rapid development of informatics and computer technology, modern schoolchildren do not want to bother with mental arithmetic. Therefore, we considered it is important to show not only that the process of performing an action itself can be interesting, but also that, having mastered the techniques of fast counting well, one can argue with a computer.

Objectstudies are counting algorithms.

Subject research favors the process of computing.

Goal:study non-standard calculation techniques and experimentally identify the reason for refusing to use these methods in teaching mathematics to modern schoolchildren.

Tasks:

To reveal the history of the account and the phenomenon of "Miracle - counters";

Describe the old methods of multiplication and experimentally identify difficulties in their use;

Consider some of the techniques of oral multiplication and show the advantages of their use with specific examples.

Hypothesis:in the old days they said: "Multiplication is my torment." This means that it used to be difficult and difficult to multiply. Is our modern multiplication method simple?

When working on a report, I used the following methods :

Ø search method using scientific and educational literature, as well as searching for the necessary information on the Internet;

Ø practical method of performing calculations using non-standard counting algorithms;

Ø analysis data obtained during the study.

Relevance This topic is that the use of non-standard techniques in the formation of computational skills increases students' interest in mathematics and contributes to the development of mathematical abilities.

The simple act of multiplication hides the secrets of the history of mathematics. The words “multiplication by the lattice”, “chess method”, which I heard by chance, intrigued. I wanted to know these and other methods of multiplication, to compare them with our today's multiplication action.

In order to find out whether modern schoolchildren know other ways of performing arithmetic operations, in addition to multiplication by a column and division by "corner" and would like to learn new ways, an oral survey was conducted. 20 students of grades 5-7 were interviewed. This survey showed that modern schoolchildren do not know other ways to perform actions, since they rarely refer to material outside the school curriculum.

Results of the survey:

(The diagrams represent the percentage of students who answered yes).

1) Is it necessary to be able to perform arithmetic operations with natural numbers for a modern person?

2) a) Do you know how to multiply, add,

b) Do you know of other ways to perform arithmetic operations?

3) would you like to know?

Chapter 1. Account history

1.1. How the numbers came to be

People learned to count objects in the ancient Stone Age - the Paleolithic, tens of thousands of years ago. How did this happen? At first, people just visually compared different quantities of the same objects. They could determine which of the two heaps had more fruit, which herd had more deer, etc. If one tribe exchanged the caught fish for stone knives made by people of the other tribe, there was no need to count how many fish were brought and how many knives. It was enough to put a knife next to each fish for the exchange between the tribes to take place.

To be successful in agriculture, you needed arithmetic knowledge. Without counting the days, it was difficult to determine when to sow the fields, when to start watering, when to expect offspring from animals. You had to know how many sheep were in the flock, how many sacks of grain were put in the barns.
And more than eight thousand years ago, ancient shepherds began to make mugs of clay - one for each sheep. To find out if at least one sheep had disappeared during the day, the shepherd put aside a mug each time another animal entered the pen. And only after making sure that the sheep returned as many as there were circles, he calmly went to bed. But not only sheep were in his flock - he grazed cows, goats, and donkeys. Therefore, other figurines had to disappear from clay. And farmers, with the help of clay figurines, kept records of the harvested harvest, noting how many bags of grain were put in the barn, how many jugs of oil were squeezed out of olives, how many pieces of linen were woven. If the sheep bore offspring, the shepherd added new ones to the circles, and if some of the sheep went for meat, several circles had to be removed. So, not yet knowing how to count, the ancient people were engaged in arithmetic.

Then numerals appeared in the human language, and people were able to name the number of objects, animals, days. Usually such numbers were few. For example, the Murray River tribe in Australia had two simple numerals: Enea (1) and Petcheval (2). They expressed other numbers with composite numerals: 3 \u003d "petcheval-ea", 4 "petcheval-petcheval", etc. Another Australian tribe, the Kamiloroi, had simple numerals mal (1), bulan (2), guliba (3). And here other numbers were obtained by adding less: 4 \u003d "bulan - bulan", 5 \u003d "bulan - guliba", 6 \u003d "guliba - guliba", etc.

For many peoples, the name of the number depended on the objects being counted. If the inhabitants of the Fiji Islands counted boats, then the number 10 was called "bolo"; if they counted coconuts, the number 10 was called "karo". The Nivkhs living on Sakhalin and on the shores of the Amur did the same. Even in the last century, they called the same number in different words if they counted people, fish, boats, nets, stars, sticks.

We still use different indefinite numbers with the meaning "many": "crowd", "herd", "flock", "heap", "bunch" and others.

With the development of production and trade exchange, people began to better understand what three boats and three axes, ten arrows and ten nuts have in common. The tribes often exchanged item for item; for example, they traded 5 edible roots for 5 fish. It became clear that 5 is the same for the roots and for the fish; hence, it can be called in one word.

Similar methods of counting were used by other peoples. This is how numbering arose, based on the count of fives, tens, twenty.

So far, we have talked about oral counting. How were the numbers recorded? At first, even before the advent of writing, they used notches on sticks, notches on bones, knots on ropes. A wolf bone found in Dolny Vestonice (Czechoslovakia) had 55 incisions made over 25,000 years ago.

When writing appeared, numbers also appeared to write numbers. At first, the numbers resembled notches on sticks: in Egypt and Babylon, in Etruria and Dates, in India and China, small numbers were written with sticks or dashes. For example, the number 5 was written with five sticks. The Asteka and Maya Indians used dots instead of sticks. Then there were special characters for some numbers, such as 5 and 10.

At that time, almost all numbering was not positional, but similar to Roman numbering. Only one Babylonian sexagesimal numbering was positional. But even in it there was no zero for a long time, as well as a comma separating the whole part from the fractional part. Therefore, the same number could mean 1, 60, or 3600. The meaning of the number had to be guessed according to the meaning of the problem.

Several centuries before the new era, a new way of writing numbers was invented, in which the letters of the ordinary alphabet served as numbers. The first 9 letters denoted tens 10, 20,…, 90, and another 9 letters denoted hundreds. This alphabetical numbering was used until the 17th century. To distinguish "real" letters from numbers, a dash was placed over the letters-numbers (in Russia this dash was called "titlo").

In all these numberings, it was very difficult to perform arithmetic operations. Therefore, the invention in the 6th century. Indians of decimal positional numbering are rightfully considered one of the greatest achievements of mankind. Indian numbering and Indian numerals became known in Europe from the Arabs, and are usually called Arabic.

When writing fractions for a long time, the whole part was written in a new, decimal numbering, and the fractional - in sixagesimal. But at the beginning of the 15th century. Samarkand mathematician and astronomer al - Kashi began to use decimal fractions in his calculations.

The numbers we work with are positive and negative numbers. But it turns out that these are not all numbers that are used in mathematics and other sciences. And you can learn about them without waiting for high school, but much earlier, if you study the history of the emergence of numbers in mathematics.

1.2 "Miracle - counters"

He understands everything at a glance and immediately formulates a conclusion, to which an ordinary person, perhaps, will come through long and painful meditation. He absorbs books with incredible speed, and in the first place on his short list of bestsellers is a textbook on entertaining mathematics. At the moment of solving the most difficult and unusual tasks, the fire of inspiration burns in his eyes. Requests to go to the store or wash the dishes are ignored or met with great dissatisfaction. The best reward is a visit to the lecture hall, and the most valuable gift is a book. He is as practical as possible and in his actions basically obeys reason and logic. He has a cold attitude towards the people around him and prefers roller-skating to a chess game with a computer. As a child, he is aware of his own shortcomings beyond his years, is characterized by increased emotional stability and adaptability to external circumstances.

This portrait was by no means painted with a CIA analyst.
This, according to psychologists, looks like a human calculator, an individual with unique mathematical abilities that allow him to make the most complex calculations in his mind in the blink of an eye.

Beyond the threshold of consciousness, a miracle - bookkeepers who are able to perform unimaginably complex arithmetic operations without a calculator, have unique memory features that distinguish them from other people. As a rule, in addition to huge lines of formulas and calculations, these people (scientists call them mnemonics - from the Greek word mnemonika, meaning "the art of memorization") keep in mind lists of addresses not only of friends, but also of casual acquaintances, as well as of numerous organizations where they once had to be.

In the laboratory of the Research Institute of Psychotechnology, where they decided to study the phenomenon, they conducted such an experiment. They invited a unique person - an employee of the Central State Archives of St. Petersburg. He was offered various words and numbers for memorization. He had to repeat them. In just a couple of minutes, he could fix up to seventy elements in his memory. Dozens of words and numbers were literally "loaded" into Alexander's memory. When the number of elements exceeded two hundred, we decided to test its capabilities. To the surprise of the participants in the experiment, the mega-memory did not give a single failure. Moving his lips for a second, he began to reproduce the whole series of elements with amazing accuracy, as if reading.

Another, for example, one scientist - researcher conducted an experiment with Mademoiselle Osaka. The subject was asked to square 97 to get the tenth power of that number. She did it instantly.

Aron Chikashvili lives in the Van region of western Georgia. He quickly and accurately performs complex calculations in his mind. Somehow friends decided to test the capabilities of the "miracle counter". The task was difficult: how many words and letters will the announcer say when commenting on the second half of the football match "Spartak" (Moscow) - "Dynamo" (Tbilisi). The tape recorder was switched on at the same time. The answer came as soon as the announcer said the last word: 17427 letters, 1835 words. It took… .5 hours to check. The answer turned out to be correct.

It is said that Gauss's father used to pay his workers at the end of the week, adding to each day's wages for overtime. One day after Gauss the father finished the calculations, the child, who was three years old, was following the father's operations, exclaimed: “Dad, the count is not correct! This should be the amount. " The calculations were repeated and were surprised to make sure that the kid indicated the correct amount.

Interestingly, many "miracle counters" have no idea at all how they count. “We count, that's all! And as we think, God knows him. " Some “counters” were completely uneducated people. The Englishman Buxton, a "virtuoso counter", never learned to read; American "negro counter" Thomas Fuller died illiterate at the age of 80 years.

Competitions were held at the Institute of Cybernetics of the Ukrainian Academy of Sciences. The young "counter-phenomenon" Igor Shelushkov and the "Mir" computer took part in the competition. The machine did many complex mathematical operations in a few seconds. Igor Shelushkov became the winner in this competition.

Most of these people have excellent memories and gifts. But some of them have no mathematics whatsoever. They know the secret! And this secret is that they have mastered the fast counting techniques well, memorized several special formulas. But a Belgian employee, who, in 30 seconds, according to the multi-digit number proposed to him, obtained from multiplying a certain number by itself 47 times, calls this number (he extracts the root of the 47th

degree from a multiple-digit number), has achieved such tremendous success in the account as a result of many years of training.

So, many "phenomenon counters" use special quick counting techniques and special formulas. This means that we can also use some of these techniques.

ChapterII ... Old ways of multiplication.

2.1. Russian peasant way of multiplication.

In Russia 2-3 centuries ago, among the peasants of some provinces, a method was widespread 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 peasant (there is an opinion that it originates from the Egyptian).

Example: multiply 47 by 35,

Let's write the numbers on one line, draw a vertical line between them;

The left number will be divided by 2, the right number will be multiplied by 2 (if a remainder occurs during division, then we discard the remainder);

Division ends when one appears on the left;

Cross out those lines in which there are even numbers on the left;

35 + 70 + 140 + 280 + 1120 = 1645.

2.2. Lattice method.

1). Prominent Arab mathematician and astronomer Abu Moussa al - Khorezmi lived and worked in Baghdad. "Al - Khorezmi" literally means "from Khorezm", that is, he was born in the city of Khorezm (now part of Uzbekistan). The scientist worked in the House of Wisdom, where there was a library and an observatory; almost all major Arab scholars worked here.

There is very little information about the life and work of Muhammad al - Khorezmi. Only two of his works have survived - on algebra and arithmetic. The last of these books gives four rules for arithmetic operations, much the same as those used today.

2). In its "Book of Indian Accounts" the scientist described a method invented in ancient India, and later called "Lattice method" (he is "jealousy"). This method is even simpler than the one used today.

Let's multiply 25 and 63.

Let's draw a table in which two cells are in length and two in width, we write down one number in length and another in width. In the cells, we write down the result of multiplying these numbers, at their intersection we separate tens and ones by a diagonal. We add the resulting numbers diagonally, and the result can be read along the arrow (down and to the right).

We have considered a simple example, however, this method can be used to multiply any multi-digit numbers.

Let's consider another example: multiply 987 and 12:

Draw a 3 by 2 rectangle (according to the number of decimal places for each multiplier);

Then we divide the square cells diagonally;

At the top of the table, write the number 987;

On the left of the table is the number 12 (see figure);

Now in each square we write the product of numbers - factors located in one line and in one column with this square, tens above the diagonal, ones below;

After filling in all the triangles, the numbers in them are added along each diagonal;

We write the result on the right and at the bottom of the table (see figure);

987 ∙ 12=11844

This algorithm for multiplying two natural numbers was common in the Middle Ages in the East and Italy.

We noted the inconvenience of this method in the laboriousness of preparing a rectangular table, although the calculation process itself is interesting and filling the table resembles a game.

2.3 Indian multiplication method

Some experienced teachers in the last century believed that this method should replace the generally accepted method of multiplication in our school.

The Americans liked it so much that they even called it “The American Way”. However, it was used by the inhabitants of India back in the 6th century. n. e., and it is more correct to call it "the Indian way." Multiply any two two-digit numbers, say 23 by 12. I immediately write what happens.

You see: the answer was received very quickly. But how is it obtained?

The first step: x23 I say: "2 x 3 \u003d 6"

The second step: x23 I say: "2 x 2 + 1 x 3 \u003d 7"

Third step: x23 I say: "1 x 2 \u003d 2".

12 I write 2 to the left of the number 7

276 we get 276.

We got acquainted with this method on a very simple example without crossing the discharge. However, our research has shown that it can also be used when multiplying numbers with a transition through a digit, as well as when multiplying multi-digit numbers. Here are some examples:

x528 x24 x15 x18 x317

123 30 13 19 12

In Russia, this method was known as the cross-stitch multiplication method.

In this "cross" lies the inconvenience of multiplication, it is easy to get confused, moreover, it is difficult to keep in mind all the intermediate products, the results of which must then be added.

2.4. Egyptian way of multiplication

The designations of numbers that were used in antiquity were more or less suitable for recording the result of counting. But it was very difficult to perform arithmetic operations with their help, especially with regard to the operation of multiplication (try, multiply: ξφß * τδ). The Egyptians found a way out of this situation, so the method was called egyptian. They replaced multiplication by any number by doubling, that is, adding a number to itself.

Example: 34 ∙ 5 \u003d 34 ∙ (1 + 4) \u003d 34 ∙ (1 + 2 ∙ 2) \u003d 34 ∙ 1+ 34 ∙ 4.

Since 5 \u003d 4 + 1, then to get the answer, it remained to add the numbers in the right column against the numbers 4 and 1, i.e. 136 + 34 \u003d 170.

2.5. Multiplication on fingers

The ancient Egyptians were very religious and believed that the soul of the deceased in the afterlife was subjected to a finger counting test. This already speaks of the importance that the ancients attached to this method of performing multiplication of natural numbers (it received the name finger counting).

Single-digit numbers from 6 to 9 were multiplied on the fingers. To do this, on one hand, they stretched out as many fingers as the first factor exceeded the number 5, and on the second, they did the same for the second factor. The rest of the fingers were bent. After that, they took as many tens as the extended fingers on both hands, and added to this number the product of bent fingers on the first and second hands.

Example: 8 ∙ 9 \u003d 72

Later, finger counting was improved - they learned to show numbers up to 10,000 with their fingers.

Finger movement

And here is another way to help memory: with the help of your fingers, memorize the multiplication table by 9. Putting both hands side by side on the table, we number the fingers of both hands in order as follows: the first finger on the left we denote 1, the second behind it we denote the number 2, then 3 , 4… up to the tenth finger, which means 10. If you need to multiply any of the first nine numbers by 9, then for this, without moving your hands from the table, you need to lift up that finger, whose number means the number by which nine is multiplied; then the number of fingers lying to the left of the raised finger determines the number of tens, and the number of fingers lying to the right of the raised finger indicates the number of units of the resulting product.

Example. Let you find a 4x9 product.

With both hands on the table, raise the fourth finger, counting from left to right. Then there are three fingers (tens) before the raised finger, and 6 fingers (ones) after the raised finger. The product of 4 by 9, therefore, is 36.

Another example:

Suppose you want to multiply 3 * 9.

From left to right, find the third finger, that finger will be straightened 2 fingers, they will mean 2 dozen.

To the right of the bent finger, 7 fingers will be straightened, they mean 7 units. Add 2 tens and 7 units to 27.

The fingers themselves showed this number.

// // /////

So, the old methods of multiplication we have considered show that the algorithm used in the school for multiplying natural numbers is not the only one and it was not always known.

However, it is fast enough and the most convenient.

Chapter 3. Oral counting - gymnastics of the mind

3.1. Multiplication and division by 4.

To multiply a number by 4, you double it twice.

For instance,

214 * 4 = (214 * 2) * 2 = 428 * 2 = 856

537 * 4 = (537 * 2) * 2 = 1074 * 2 = 2148

To divide a number by 4, divide it twice by 2.

For instance,

124: 4 = (124: 2) : 2 = 62: 2 = 31

2648: 4 = (2648: 2) : 2 = 1324: 2 = 662

3.2. Multiplication and division by 5.

To multiply a number by 5, you need to multiply it by 10/2, that is, multiply by 10 and divide by 2.

For instance,

138 * 5 = (138 * 10) : 2 = 1380: 2 = 690

548 * 5 (548 * 10) : 2 = 5480: 2 = 2740

To divide the number by 5, you need to multiply it by 0.2, that is, in the doubled original number, separate the last digit with a comma.

For instance,

345: 5 = 345 * 0,2 = 69,0

51: 5 = 51 * 0,2 = 10,2

3.3. Multiplication by 25.

To multiply a number by 25, you need to multiply it by 100/4, that is, multiply by 100 and divide by 4.

For instance,

348 * 25 = (348 * 100) : 4 = (34800: 2) : 2 = 17400: 2 = 8700

3.4. Multiplication by 1.5.

To multiply a number by 1.5, you need to add half of it to the original number.

For instance,

26 * 1,5 = 26 + 13 = 39

228 * 1,5 = 228 + 114 = 342

127 * 1,5 = 127 + 63,5 = 190,5

3.5. Multiplication by 9.

To multiply a number by 9, 0 is assigned to it and the original number is subtracted. For instance,

241 * 9 = 2410 – 241 = 2169

847 * 9 = 8470 – 847 = 7623

3.6. Multiplication by 11.

1 way... To multiply the number by 11, 0 is assigned to it and the original number is added. For instance:

47 * 11 = 470 + 47 = 517

243 * 11 = 2430 + 243 = 2673

Method 2. If you want to multiply a number by 11, then do this: write down the number you want to multiply by 11, and insert the sum of these numbers between the digits of the original number. If the sum is a two-digit number, then 1 is added to the first digit of the original number. For instance:

45 * 11 = * 11 = 967

This method is only suitable for multiplying two-digit numbers.

3.7. Multiply a three-digit number by 101.

For example 125 * 101 \u003d 12625

(we increase the first factor by the number of its hundreds and assign the last two digits of the first factor to it on the right)

125 + 1 = 126 12625

Children learn this technique easily when writing down the calculation in a column.

x x125
101
+ 125
125 _
12625

x x348
101
+348
348 _
35148

Another example: 527 * 101 = (527+5)27 = 53227

3.8. Square a number ending in 5.

To square a number ending in the digit 5 \u200b\u200b(for example, 65), multiply the number of its tens (6) by the number of tens increased by 1 (by 6 + 1 \u003d 7), and assign 25 to the resulting number

(6 * 7 \u003d 42 Answer: 4225)

For instance:

3.8. Squaring a number close to 50.

If you want to square a number close to 50, but greater than 50, then do this:

1) subtract 25 from this number;

2) add to the result with two digits the square of the excess of the given number over 50.

Explanation: 58 - 25 \u003d 33, 82 \u003d 64, 582 \u003d 3364.

Explanation: 67 - 25 \u003d 42, 67 - 50 \u003d 17, 172 \u003d 289,

672 = 4200 + 289 = 4489.

If you want to square a number close to 50, but less than 50, then do this:

1) subtract 25 from this number;

2) add the square of the lack of this number to 50 to the result with two digits.

Explanation: 48 - 25 \u003d 23, 50 - 48 \u003d 2, 22 \u003d 4, 482 \u003d 2304.

Explanation: 37 - 25 \u003d 12, \u003d 13, 132 \u003d 169,

372 = 1200 + 169 = 1369.

3.9. Games

Guessing the resulting number.

1. Think of a number. Add 11 to it; multiply the resulting amount by 2; subtract 20 from this work; Multiply the resulting difference by 5 and subtract from the new product 10 times the number you intended.

I guess you got 10. Right?

2. Think of a number. Morning it. Subtract from the received 1. Received multiply by 5. To the received add 20. Divide received by 15. Subtract the conceived from the received.

You get 1.

3. Think of a number. Multiply it by 6. Subtract 3. Multiply by 2. Add 26. Subtract twice what you intended. Divide by 10. Subtract your plan.

You get 2.

4. Think of a number. Triple it. Subtract 2. Multiply by 5. Add 5. Divide by 5. Add 1. Divide by your plan. You get 3.

5. Think of a number, double it. Add 3. Multiply by 4. Subtract 12. Divide by your plan.

You get 8.

Guessing the intended numbers.

Invite your friends to think of any numbers. Let everyone add 5 to their intended number.

Let the resulting sum be multiplied by 3.

Let him subtract 7 from the work.

Let him subtract 8 more from the result.

Let everyone give you the final result sheet. Looking at the piece of paper, you immediately tell everyone what number they have in mind.

(To guess the planned number, divide the result written on a piece of paper or spoken to you orally by 3)

Conclusion

We have entered the new millennium! Great discoveries and achievements of mankind. We know a lot, we can do a lot. It seems to be something supernatural that with the help of numbers and formulas one can calculate the flight of a spaceship, the "economic situation" in the country, the weather for "tomorrow", and describe the sound of notes in a melody. We know the statement of the ancient Greek mathematician, philosopher who lived in the 4th century BC - Pythagoras - "Everything is number!"

According to the philosophical view of this scientist and his followers, numbers control not only measure and weight, but also all phenomena that occur in nature, and are the essence of harmony that reigns in the world, the soul of the cosmos.

Describing ancient methods of calculations and modern methods of fast counting, we tried to show that, both in the past and in the future, one cannot do without mathematics, a science created by the human mind.

The study of ancient methods of multiplication showed that this arithmetic operation was difficult and complex due to the variety of methods and their cumbersome implementation.

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

Upon acquaintance with the scientific literature, they discovered faster and more reliable methods of multiplication. Therefore, the study of the action of multiplication is a promising topic.

It is possible that the first time many will not be able to quickly carry out these or other calculations on the fly. Let the technique shown in the work fail at first. No problem. You need constant computational training. From lesson to lesson, from year to year. It will help you acquire useful oral counting skills.

List of used literature

1. Wangqiang: Textbook for grade 5. - Samara: Publishing house

"Fedorov", 1999.

2., Ahadov's world of numbers: A student's book, - M. Enlightenment, 1986.

3. "From play to knowledge", M., "Education" 1982.

4. Svechnikov, figures, tasks M., Enlightenment, 1977.

5.http: // matsievsky. ***** / sys-schi / file15.htm

6.http: // ***** / mod / 1/6506 / hystory. html

Indian way of multiplication

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

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

The Hindus thought very well. They came up with a very simple way to multiply. They performed the multiplication, starting from the most significant digit, and wrote down incomplete works just above the multiplicable, bit by bit. In this case, the most significant digit of the complete product was immediately visible and, in addition, the omission of any digit was excluded. The sign of the multiplication was not yet known, so they left a small distance between the factors. For example, let's multiply them in the 537 way by 6:

LITTLE CASTLE multiplication

Multiplication of numbers is now being taught in the first grade of 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 development of mathematics, many ways have been invented to multiply numbers. The Italian mathematician Luca Pacioli, in his treatise The Sum of Knowledge in Arithmetic, Relations and Proportionality (1494), gives eight different methods of multiplication. The first of them is called "Little Castle", and the second is no less romantic name "Jealousy or Lattice Multiplication".

The advantage of the "Little Castle" multiplication method is that the most significant digits are determined from the very beginning, which is important if you need to quickly estimate the value.

The digits of the upper number, starting with 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. The results are then added up.

Vasily Krestnikov

The theme of the work "Unusual ways of computing" is interesting and relevant, since students constantly perform arithmetic operations on numbers, and the ability to quickly calculate, increases academic success and develops the flexibility of the mind.

Vasily was able to clearly state the reasons for his appeal to this topic, correctly formulated the goal and objectives of the work. Having studied various sources of information, I found interesting and unusual ways to multiply and learned how to apply them in practice. The student considered the pros and cons of each method and made the correct conclusion. The reliability of the conclusion is confirmed by a new method of multiplication. At the same time, the student skillfully uses special terminology and knowledge outside the school mathematics curriculum. The theme of the work corresponds to the content, the material is presented clearly and easily.

The results of the work are of practical importance and may be of interest to a wide range of people.

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MOU "Kurovskaya secondary school No. 6"

ABSTRACT ON MATHEMATICS ON THE TOPIC:

"UNUSUAL WAYS OF MULTIPLICATION".

Completed by a student of 6 "b" grade

Krestnikov Vasily.

Leader:

Smirnova Tatiana Vladimirovna.

2011

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 method ……………………………………………………………… ..10

Conclusion ………………………………………………………………… ... 11
References …………………………………………………………… .12

I. Introduction.

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

Once I accidentally came across a book by S. N. Olekhnik, Yu. V. Nesterenko and M. K. Potapov "Old entertaining tasks." Leafing through this book, my attention was attracted by a page called "Multiplication on the fingers." It turned out that it is possible to multiply not only as they suggest to us in mathematics textbooks. I wondered if there were any other ways of calculating. After all, the ability to quickly perform 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 make simple calculations in the mind, but also to control, evaluate, find and correct errors as a result of mechanized calculations. In addition, mastering computational skills develops memory, raises the level of mathematical culture of thinking, and helps to fully master the subjects of the physics and mathematics cycle.

Objective:

Show unusual ways to multiply.

Tasks:

Find as many unusual ways of computing as possible.
Learn to apply them.
Choose for yourself the most interesting or lighter ones than those offered at the school, and use them when counting.

II. Main part. Unusual ways to multiply.

2.1. A bit of history.

The methods of computing that we use now have not always been so simple and convenient. In the old days, they used more cumbersome and slow methods. And if a schoolboy of the 21st century could travel back five centuries, he would have amazed our ancestors with the speed and accuracy of his calculations. Rumors about him would have spread around the surrounding schools and monasteries, eclipsing the glory of the most skillful enumerators of that era, and people would come from all sides to learn from the new great master.

The actions of multiplication and division were especially difficult in the old days. At that time, there was no one practice-developed technique for every action. On the contrary, almost a dozen different methods of multiplication and division were in use at the same time - the methods of each other are more confusing, which a person of average abilities could not remember. Each teacher of counting adhered to his favorite technique, each "master of division" (there were such specialists) praised his own way of doing this.

In the book by V. Bellustin "How people gradually got to real arithmetic" 27 methods of multiplication are set forth, and the author notes: "it is quite possible that there are still methods hidden in the caches of book depositories, scattered in numerous, mainly manuscript 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 absorbed with great difficulty.

Let's take a look at the most interesting and simple ways to multiply.

2.2. Multiplication on the fingers.

The Old 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 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 “ones”, “pairs”, “threes”, “fours”, “fives” and “tens”. The fingers here served as an auxiliary computing device.

To do this, on one hand, they pulled out 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 the extended 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.

For example, multiply 7 by 8. In this example, 2 and 3 fingers will be bent. If you add up the number of bent fingers (2 + 3 \u003d 5) and multiply the number of unbent fingers (2 3 \u003d 6), you get the number of tens and units of the desired product 56, respectively. This way you can calculate the product of any single digit numbers greater than 5.

2.3. Multiplication by 9.

Multiplication for the number 9 - 9 · 1, 9 · 2 ... 9 · 10 - more easily disappears from memory and is more difficult to recalculate manually by the method of addition, however, 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 the numbers from 1 to 10 to your fingers in sequence, starting with the little finger of your left hand and ending with the little finger of your right hand (this is shown in the figure).

Let's say we want to multiply 9 by 6. Bend the finger with the number equal to the number by which we will multiply nine. In our example, you need to bend finger number 6. The number of fingers to the left of the curled finger shows us the number of tens in the answer, the number of fingers to the right is the number of ones. On the left we have 5 fingers not bent, on the right - 4 fingers. So 9 6 \u003d 54. The figure below shows the whole principle of "calculation" in detail.

Another example: you need to calculate 9 8 \u003d ?. Along the way, let's say that the fingers of the hands may not necessarily act as a "calculating machine". Take, for example, 10 cells in a notebook. Cross out the 8th box. There are 7 cells on the left, 2 cells on the right. So 9 8 \u003d 72. Everything is very simple.

7 cells 2 cells.

2.4. The Indian way of multiplying.

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

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 "LITTLE CASTLE" method.

The advantage of the "Little Castle" multiplication method is that the most significant digits are determined from the very beginning, which is 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. The results are then added up.

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

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 the multiplier. Then the square cells are divided diagonally, and “... a picture looks like a lattice shutter-jalousie,” Pacioli writes. "Such shutters were hung on the windows of Venetian houses, making it difficult for street passers-by to see the ladies and nuns sitting at the windows."

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

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 - below it. Now we add the numbers in each oblique strip, performing this operation, from right to left. If the amount is less than 10, then we write it under the lower number of the strip. 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.

3 4 7

10 0 6 3

2.7. Peasant way of multiplying.

The most, in my opinion, "native" and easy way of multiplication is the method used by the Russian peasants. This technique 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 consecutive divisions of one number in half while simultaneously doubling the other number. The division in half is continued 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, discard 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 stand against the odd numbers of the left column: the sum will be the desired product

37……….32

74……….16

148……….8

296……….4

592……….2

1184……….1

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

37 ∙ 32 = 1184 ∙ 1 = 1184

In the case when one of the numbers is odd or both numbers are odd, we proceed 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. A new way to multiply.

An interesting new way of multiplication, about which there were recent reports. Vasily Okoneshnikov, the inventor of the new oral counting system, Candidate of Philosophy, 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 most advantageous in this respect is the ninefold system - all the data are simply placed in nine cells, located like buttons on a calculator.

It is very easy to count from such a table. For example, let's multiply the number 15647 by 5. In the part of the table corresponding to five, 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

We leave the left digit (in our example - zero) unchanged, and add the following numbers in pairs: five with two, five with three, zero with two, zero with three. The last figure 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 "proper" place.

III. Conclusion.

Of all the unusual counting methods I found, the more interesting was the "lattice multiplication or jealousy" method. I showed it to my classmates, and they also really liked it.

The simplest method seemed to me to be the “doubling and doubling” method used by the 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 me to "move" with huge numbers in my mind.

I think that our method of long multiplication is not perfect and we can come up with even faster and more reliable methods.

Literature.

Depman I. "Stories about Mathematics". - Leningrad .: Education, 1954 .-- 140 p.
A.A. Korneev The phenomenon of Russian multiplication. History. http://numbernautics.ru/
Olekhnik S. N., Nesterenko Yu. V., Potapov M. K. "Ancient entertaining tasks". - M .: Science. Main edition of physical and mathematical literature, 1985. - 160 p.
Perelman Ya.I. Fast counting. Thirty Easy Verbal Counting Techniques. L., 1941 - 12 p.
Perelman Ya.I. Entertaining arithmetic. M. Rusanova, 1994-205s.https://accounts.google.com
Slide captions:
The work was carried out by the student of 6 "B" class Vasily Krestnikov. Head: Smirnova Tatyana Vladimirovna Unusual ways of multiplication
Purpose of work: To show unusual ways of multiplication. Objectives: Find unusual ways to multiply. Learn to apply them. Choose for yourself the most interesting or lighter ones and use them when counting.
Multiplication on the fingers.
Multiplication by 9
Italian mathematician Luca Pacioli was born in 1445.
Multiplication by the "Little Castle" method
Multiplication by the method "Jealousy"
Multiplication by the lattice method. 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 way 37 32 37 ……… .32 74 ……… .16 148 ……… .8 296 ……… .4 592 ……… .2 1184 ……… 1 37 32 \u003d 1184
Thank you for attention

Methods for multiplying numbers in different countries. Methods for multiplying numbers

Head:,

mathematic teacher

Indian way of multiplication

LITTLE CASTLE multiplication

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