Andriyannikov Nikita

Andriyannikov Nikita studied in detail and created a presentation on the history of decimal fractions from ancient times to the present day. His work contains interesting material that can be used by teachers and students in preparing for mathematics lessons in both the 5th and 6th grades as an electronic manual, and this material can also be used for extracurricular work on the subject.

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Design and research work

Completed: 5th grade student

Andriyannikov Nikita

Head: Stolyarova T.E.

Dolgoprudny, 2012

1.Introduction____________________________________________2

2. Abstract "History of decimal fractions" _______________3-7

3.Conclusion____________________________________________8

4. Sources of information _________________________________ 9

Number expressed as a decimal point
Will be read by both German and Russian,
And the Yankees are the same.
DI. Mendeleev

Introduction.

History of fractionshas been going on since the early stages of human development.The need for fractional numbers arose as a result of human practical activity. Therefore, the history of the development of fractional numbers is closely connected with the history of the development of mankind. I was interested in the question of when and where decimal fractions appeared, who was the first to use a new form of writing ordinary fractions with denominators 10, 100, 1000. etc.

Based on this, the leader and I set the following goals and objectives.

Goals:

Find out when and in what ancient sources decimal fractions were first mentioned.
See how decimal notation has changed over the centuries.
Find out who was the first to introduce a comma into a decimal record.

Tasks:

To study and analyze the history of decimal fractions in various sources.
Collect information using Internet resources, systematize the information received.
Present the results of the study in the form of a presentation "History of decimal fractions" using the Power Point program.

4. Acquire the skills of independent work with information, be able to see the task

And outline ways to solve it.

NPOSH "Commonwealth"

Essay

"History of decimals"

Andriyannikov Nikita, 5B class

2012

Mathematics is one of the oldest sciences, and its first steps are connected with the very first steps of the human mind. It originated in the labor activity of people. Developing

mathematics more and more accurately solved those complex problems that life itself put before a person. In the 17th century, trade, all production, and the economies of countries fell into a difficult situation. Navigators needed accurate maps, merchants needed quick and correct calculations without cheating, and for the construction of machine tools, ships, temples and dwellings, drawings verified to 1mm. Production developed, and the inability to make calculations quickly and with greater accuracy literally hampered the development of science and technology. Life set before scientists the task of simplifying calculations, increasing their accuracy and speed. These requirements were met by decimal fractions.

Mathematicians came to decimal fractions at different times in Asia and in Europe. The origin and development of decimal fractions in some Asian countries was closely connected with metrology (the study of measures). Already in the II century. BC. there was a decimal system of measures of length.

(slide number 2) In ancient China, they already used the decimal system of measures,
represent fractions in words using measures of lengthchi, tsuni, shares, ordinal, hairs, the thinnest, cobwebs.

(slide number 3)

A fraction of the form 2.135436 looked like this: 2 chi, 1 cun, 3 shares, 5 ordinal, 4 hairs, 3 finest, 6 cobwebs. This is how fractions were written for two centuries, and in the 5th century, the Chinese scientist Jiu-Chun-Zhi took not chi as a unit, ah zhang \u003d 10 chi, then this fraction looked like this: 2 zhang, 1 chi, 3 cun, 5 shares, 4 ordinal, 3 hairs, 6 thinnest, 0 cobwebs.

(slide 4)

A more complete and systematic interpretation was given to decimal fractions in the works of the Central Asian scholar al-Kashi in the 20s of the 15th century.

The Central Asian city of Samarkand was in the XV century. great cultural center. There, the famous observatory, created by the prominent astronomer Ulugbek, the grandson of Tamerlane, worked in the 20s of the 15th century. great scientist of the timeJamshid Giyaseddin al-Kashi. It was he who first expounded the doctrine of decimal fractions.

In his book The Key of Arithmetic, written in 1427, al-Kashi writes:

“Astronomers use fractions whose consecutive denominators are 60 and its successive powers. By analogy, we introduced fractions in which the consecutive denominators are 10 and its successive powers.

It introduces a decimal-specific notation:the integer and fractional parts are written on the same line. To separate the first part from the fractional, he does not use

comma, but writes the whole part in blackin ink, fractional in red or separates the whole part from the fractionalvertical line.

In 1579, decimal fractions are used in the "Mathematical Canon" of the French mathematician François Vieta (1540-1603), published in Paris. In this work, which is a collection of trigonometric tables, Viet decisively spoke in favor of using, as he expressed it, thousandths and thousands, hundredths and hundreds, tenths and tens, etc. instead of the sexagesimal system of integers and fractions. When writing decimal fractions, Viet did not adhere to any one notation. Often he writes both the numerator and the denominator, sometimes he separates the numbers of the integer part from the fractional vertical line, or he depicts the numbers of the whole part in bold type, or, finally, he gives the numbers of the fractional part in smaller print and underlines. Fraction designation 2.135436 2 1579 F. Viet France

(slide number 6) The discovery of al-Kashi's decimal fractions became known in Europe only 300 years after these fractions were at the end of the 16th century. rediscovered by S. Stevin.

(slide number 7) Flemish engineer and scientist Simon Stevin (1548-1620), about 150 years after al-Kashi, expounded the doctrine of decimal fractions in Europe.

He is considered the inventor of decimal fractions.Stevin, a native of Bruges, was at first a merchant, then during the Dutch Revolution an engineer in the troops of Moritz of Orange, who headed the republic. "Astrologers, farmers, measurers of volumes, checkers of barrel capacities, stereometers in general, coin masters and all merchants - hello Simon Stevin," the inventor of decimal fractions addresses his readers in his book "The Tenth" (1585). This small work (only 7 pages) contained an explanation of the notation and rules for working with decimal fractions. In the book, he tries to convince people to use decimals, saying that when they are used,difficulties, strife, mistakes, losses and other accidents, the usual companions of calculations. "He wrote the numbers of a fractional number in one line with the numbers of an integer, while numbering them.

Stevin's decimal notation was different from ours. Here, for example, is how he wrote the number 35.912:

35 0 9 1 1 2 2 3

So, instead of a comma, zero in a circle. In other circles or above the numbers, the decimal place is indicated: 1 - tenths, 2 - hundredths, etc. Stevin pointed out the great practical importance of decimal fractions and persistently promoted them. He was the first scientist to demand the introduction of the decimal system of weights and measures.(slide number 8)

The comma in the recording of fractions was first encountered in 1592, and in 1617. Scottish mathematician John Napier proposed to separate decimal places from an integer either by a comma or a period.

The modern notation of decimal fractions i.e. separation of the integer part of the comma, proposed by Johannes Kepler (1571 - 1630). In countries where English is spoken (England, USA, Canada, etc.), a dot is written instead of a comma. Fraction notation 2.135436 2.135436 2.135436 1571 - 1630 Kepler Germany In Russia, the first systematic information about decimal fractions is found in Magnitsky's "Arithmetic" (1703). From the beginning of the 17th century, intensive penetration of decimal fractions into science and practice begins. The development of technology, industry and trade required more and more cumbersome calculations, which were easier to perform with the help of decimal fractions. Decimal fractions were widely used in the 19th century after the introduction of the metric system of measures and weights, closely related to them. For example, in agriculture and industry, decimal fractions and their special form - percentages - are used much more often than ordinary fractions.

In countries where they speakEnglish (England, USA, Canada, etc.), and now they write a dot instead of a comma, for example: 2.3 and read: two dot three.(slide number 9)

In “Arithmetic, that is, the science of numerals” (1703), the first Russian teacher-mathematician Leonty Filippovich Magnitsky (1669-1739), decimal fractions were given a separate chapter. « M. V. Lomonosov called this book the gates of his learning. The publication in 1703 of Magnitsky's book was an important fact in the history of mathematical education in Russia. For half a century, the book was the "gateway to learning" for Russian youth who aspired to education. Magnitsky came from the people, was born in 1669, died in 1739. His real name is unknown. Peter I talked with him many times about the mathematical sciences and was so delighted with his deep knowledge, which attracted people to him, that he called him a magnet and ordered to be written Magnitsky.

Information sources:.

1. http://www.referat-web.ru/content/referat/mathematics/mathematics49.php

2. http://otherreferats.allbest.ru/mathematics/00007546_0.html

5. http://tolian1999.narod.ru/mywork.html

Conclusion.

In the course of the design and research activities, I found a lot of interesting and informative information on the history of mathematics. The work of finding the right material was rewarding and exciting. In the process of research, I found answers to all the questions that my supervisor and I posed before starting work: where and when were decimal fractions invented, who invented the modern notation of these numbers. I did a little research on how the decimal notation has changed over the course of several centuries and the results are reflected in the form of a table.

Working on the project taught me how to systematize the material found, analyze the data and extract the necessary facts from a large amount of information.

But the most important thing in working on a project is that in the process I learned how to work with the Power Point program, which gives me the opportunity to further present my projects in the form of presentations.

Information sources:.

1. http://www.referat-web.ru/content/referat/mathematics/mathematics49.php

2. http://otherreferats.allbest.ru/mathematics/00007546_0.html

3. Journey into the history of mathematics or How people learned to count: A book for those who teach and study. M.: Pedagogy-Press, 1995. 168 p.

4. Depman I.Ya. History of arithmetic. M.: Enlightenment, 1965

The first fraction that people met was half. The next fraction was a third. Both the Egyptians and the Babylonians had special symbols for the fractions 1/3 and 2/3, which did not match the symbols for other fractions.

The Egyptians tried to write all fractions as sums of shares, that is, fractions of the form 1 / n. For example, instead of 8/15 they wrote 1/3 + 1/5. The only exception was, as we said, the fraction 2/3. Sometimes it was convenient. There is a task in the papyrus of Ahmes:
"To divide 7 loaves among 8 people."
If you cut each bread into 8 pieces, you will have to make 49 cuts.

And in Egyptian this problem was solved like this. The fraction 7/8 was written as shares: 1/2 + 1/4 + 1/8. This means that each person must be given half a loaf, a quarter of a loaf, and an eighth loaf; Therefore, we cut four loaves in half, two loaves - into 4 parts and one loaf - into 8 shares, after which we give each part of it.

But adding such fractions was inconvenient. After all, the same parts can enter into both terms, and then, when added, a fraction of the form 2/n will appear. And the Egyptians did not allow such fractions. Therefore, the papyrus of Ahmes begins with a table in which all fractions of this type from 2/5 to 2/99 are written as sums of shares. With the help of this table, division of numbers was also performed. For example, how 5 was divided by 21:

The Egyptians also knew how to multiply and divide fractions. But for multiplication, you had to multiply fractions by fractions, and then, perhaps, use the table again. Division was even more difficult. The Babylonians went the other way. They only worked with sexagesimal fractions. Since the denominators of such fractions are the numbers 60, 60 2, 60 3, etc., such fractions as 1/7 could not be exactly expressed through sexagesimal ones: they were expressed approximately through them. Since the Babylonians had a positional number system, they worked with sexagesimal fractions using the same tables as for natural numbers.

Sexagesimal fractions, inherited from Babylon, were used by Greek and Arabic mathematicians and astronomers. But it was inconvenient to work on natural numbers written in decimal and fractions written in sexagesimal. And it was very difficult to work with ordinary fractions. Therefore, the Dutch mathematician Simon Stevin suggested moving to decimal fractions. At first they were written very difficult, but gradually they switched to modern recording. Now computers use binary fractions, which were once used in Rus': half, four, half a quarter, half a quarter, etc.

An interesting system of fractions was in Ancient Rome. It was based on a division into 12 parts of a unit of weight, which was called ass. The twelfth of an ace was called an ounce. And the way, time and other quantities were compared with a visual thing - weight. For example, a Roman could say that he walked seven ounces of the road or read five ounces of a book. In this case, of course, it was not about weighing the path or the book. It meant that 7/12 of the way was covered or 5/12 of the book was read.

And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names. Even now it is sometimes said: "He scrupulously studied this question." This means that the issue has been studied to the end, that not a single smallest ambiguity remains. And the strange word "scrupulously" comes from the Roman name 1/288 assa - "scrupulus". There were also such names in use: "semis" - half of the ass, "sextane" - its sixth share, "semiounce" - half an ounce, that is, 1/24 of the ass, etc. In total, 18 different names of fractions were used. To work with fractions, it was necessary for these fractions to remember both the addition table and the multiplication table. Therefore, Roman merchants knew for sure that when adding a trience (1/3 ass) and sextans, a semis is obtained, and when a bes (2/3 ass) is multiplied by a sescution (3/2 ounce, that is, 1/8 ass), an ounce is obtained. To facilitate the work, special tables were compiled, some of them have come down to us.

Due to the fact that there are no fractions with denominators 10 or 100 in the duodecimal system, the Romans found it difficult to divide by 10, 100, etc. When dividing 1001 asses by 100, one Roman mathematician first received 10 asses, then split the ases into ounces, etc. e. But he did not get rid of the remainder. In order not to deal with such calculations, the Romans began to use percentages. They took interest from the debtor (that is, money in excess of what was given in debt). At the same time, they said: not "the interest will be 16 hundredths of the amount of the debt," but "for every 100 sesterces of debt you will pay 16 sesterces of interest." And the same thing is said, and there was no need to use fractions! Since the words "one hundred" sounded in Latin "about a centum", then a hundredth part was called a percentage. And although now fractions, and especially decimal fractions, are known to everyone, percentages are still used in financial calculations and in planning, that is, in various areas of human activity. And earlier they also used ppm - this is what they called thousandths (in Latin "about a mille" - per thousand). Unlike percentages, which are denoted by the sign%, ppm denotes% o.

There were no fractions in Greek writings on mathematics. Greek scientists believed that mathematics should deal only with whole numbers. They provided merchants, artisans, as well as astronomers, surveyors, mechanics and other "black people" with fractions. But the old proverb says: "Drive nature through the door - it will fly in through the window." Therefore, even in the strictly scientific writings of the Greeks, fractions penetrated "from the rear." In addition to arithmetic and geometry, Greek science included music. The Greeks called music the doctrine of harmony. This teaching was based on that part of our arithmetic, which deals with ratios and proportions. The Greeks knew that the longer the stretched string, the lower the sound it makes, and the short string makes a high sound. But every musical instrument has not one, but several strings. In order for all the strings to sound "according" when played, pleasant to the ear, the lengths of their sounding parts must be in a certain ratio. Therefore, the doctrine of ratios and fractions was used in Greek music theory.

The modern system of writing fractions with a numerator and a denominator was created in India. Only there they wrote the denominator on top, and the numerator - below and did not write a fractional line. And the Arabs began to write fractions exactly as they do now.

The word numerator, as its stress shows, could appear in the Russian literary language no earlier than the 17th century, when the Ukrainian-Polish influence brought with it new norms of word production and stress. The mathematical term numerator arises as a calque rendering of the Latin numerator (numerare - `to number', `to count'), cf. German Zähler (zählen). Thus, the history of the word numerator begins in the 17th century.

From the history of ordinary fractions.

The need for fractional numbers arose in man at a very early stage of development. Already the division of prey, which consisted of several killed animals, between the participants in the hunt, when the number of animals turned out to be not a multiple of the number of hunters, could lead primitive man to the concept of a fractional number.

Along with the need to count objects, people from ancient times have a need to measure length, area, volume, time and other quantities. It is not always possible to express the result of measurements by a natural number, and parts of the measure used must also be taken into account. Historically, fractions originated in the process of measurement.

The need for more accurate measurements led to the fact that the initial units of measure began to be divided into 2, 3 or more parts. The smaller unit of measure, which was obtained as a result of fragmentation, was given an individual name, and the values were already measured by this smaller unit.

In connection with this necessary work, people began to use the expressions: half, third, two and a half steps. From where it could be concluded that fractional numbers arose as a result of measuring quantities. The peoples went through many ways of recording fractions until they came to the modern notation.

Fractions in ancient Egypt

In ancient Egypt, architecture reached a high level of development. In order to build grandiose pyramids and temples, to calculate the lengths, areas and volumes of figures, it was necessary to know arithmetic.

From the deciphered information on the papyri, scientists learned that the Egyptians 4,000 years ago had a decimal (but not positional) number system, were able to solve many problems related to the needs of construction, trade and military affairs.

In ancient Egypt, some fractions had their own special names - namely, 1/2, 1/3, 2/3, 1/4, 3/4, 1/6 and 1/8, which often appear in practice. In addition, the Egyptians knew how to operate with the so-called aliquot fractions (from Latin aliquot - several) of the 1 / n type - therefore they are sometimes also called "Egyptian"; these fractions had their own spelling: an elongated horizontal oval and under it the designation of the denominator. As for the rest of the fractions, they should have been decomposed into the Egyptian sum. The ancient Egyptians already knew how to divide 2 objects into three, for this number - 2/3 - they had a special icon. This was the only fraction in the everyday life of Egyptian scribes, which did not have a unit in the numerator - all other fractions certainly had a unit in the numerator (the so-called basic fractions). If the Egyptian needed to use other fractions, he represented them as the sum of the basic fractions. For example, instead of 8/15 they wrote 1/3+1/5. Sometimes it was convenient. The Egyptians also knew how to multiply and divide fractions. But for multiplication, you had to multiply fractions by fractions, and then, perhaps, use the table again. Division was even more difficult. Important work on the study of Egyptian fractions was carried out by the 13th century mathematician Fibonacci.

Fractions in Ancient Greece

Egyptian fractions continued to be used in ancient Greece and subsequently by mathematicians around the world until the Middle Ages, despite the comments of ancient mathematicians (for example, Claudius Ptolemy spoke about the inconvenience of using Egyptian fractions compared to the Babylonian system). Maxim Planud Greek monk, scientist, mathematician in the 13th century introduced the name of the numerator and denominator

In Greece, along with single, "Egyptian" fractions, common ordinary

fractions. Among the various entries, the following was also used: the denominator is on top, below it is the numerator of the fraction. For example, meant three-fifths. Even 2-3 centuries before Euclid and Archimedes, the Greeks were fluent in arithmetic operations with fractions.

Fractions in India.

The modern system of writing fractions was created in India. Only there they wrote the denominator on top, and the numerator on the bottom, and did not write a fractional line. But the whole fraction was placed in a rectangular frame. Sometimes a "three-story" expression with three numbers in one frame was also used; depending on the context, this could mean an improper fraction (a + b/c) or the division of an integer a by a fraction b/c. The rules for operations with fractions did not differ much from modern ones.

Fractions of the Arabs.

Write down fractions as the Arabs have now begun. Medieval Arabs used three systems for writing fractions. First, in the Indian manner, writing the denominator under the numerator; the fractional line appeared at the end of the 12th - beginning of the 13th century. Secondly, officials, land surveyors, merchants used the calculation of aliquot fractions, similar to the Egyptian one, while fractions with denominators not exceeding 10 were used (the Arabic language has special terms only for such fractions); approximate values were often used; Arab scholars worked to improve this calculus. Thirdly, Arab scholars inherited the Babylonian-Greek sexagesimal system, in which, like the Greeks, they used alphabetic notation, extending it to whole parts.

Fractions in Babylon

The Babylonians used only two numbers. A vertical dash denoted one unit, and an angle of two lying dashes denoted ten. These lines were obtained in the form of wedges, because the Babylonians wrote with a sharp stick on damp clay tablets, which were then dried and fired.

In ancient Babylon, a constant denominator of 60 was preferred. Sexagesimal fractions, inherited from Babylon, were used by Greek and Arabic mathematicians and astronomers. Researchers explain the appearance of the sexagesimal number system among the Babylonians in different ways. Most likely, the base 60 was taken into account here, which is a multiple of 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, which greatly simplifies all sorts of calculations.

But it was inconvenient to work on natural numbers written in decimal and fractions written in sexagesimal. And it was already quite difficult to work with ordinary fractions. Therefore, the Dutch mathematician Simon Stevin suggested moving to decimal fractions.

Fractions in ancient China

In ancient China, they already used the decimal system of measures, denoted the fraction with words, using measures of the length of chi: cuni, shares, ordinal, hairs, the thinnest, cobwebs. A fraction of the form 2.135436 looked like this: 2 chi, 1 cun, 3 shares, 5 ordinal, 4 hairs, 3 finest, 6 cobwebs. This is how fractions were written for two centuries, and in the 5th century, the Chinese scientist Zu-Chun-Zhi took not chi as a unit, but zhang = 10 chi, then this fraction looked like this: 2 zhang, 1 chi, 3 cun, 5 shares, 4 ordinal, 3 hairs, 6 thinnest, 0 cobwebs.

Fractions in Ancient Rome

An interesting system of fractions was in Ancient Rome. It was based on a division into 12 parts of a unit of weight, which was called ass. The twelfth of an ace was called an ounce. And the way, time and other quantities were compared with a visual thing - weight. For example, a Roman could say that he walked seven ounces of the road or read five ounces of a book. At the same time, of course, it was not about weighing the path or the book. It meant that 7/12 of the way was covered or 5/12 of the book was read. And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names.

Even now, it is sometimes said: "He scrupulously studied this issue." This means that the issue has been studied to the end, that not even the slightest ambiguity has remained. And the strange word "scrupulously" comes from the Roman name 1/288 assa - "scrupulus". There were also such names in use: "semis" - half of the ass, "sextans" - its sixth share, "seven ounce" - half an ounce, i.e. 1/24 ass, etc. In total, 18 different names of fractions were used. To work with fractions, it was necessary to remember the addition table and the multiplication table for these fractions. Therefore, Roman merchants firmly knew that when adding a triens (1/3 ass) and sextans, a semis is obtained, and when a demon (2/3 ass) is multiplied by a sescution (2/3 ounces, i.e. 1/8 ass), an ounce is obtained . To facilitate the work, special tables were compiled, some of which have come down to us.

Fractions in Rus'

In Russian, the word "fraction" appeared only in the VIII century. The word "fraction" comes from the word "crush, break, break into pieces." Among other peoples, the name of the fraction is also associated with the verbs "break", "break", "shatter". In the first textbooks, fractions were called "broken numbers". The following names of fractions in Rus' were found in old manuals:

- half, half, - third,

- four, - half a third,

- half an hour, - half a third,

- half a half, - half a half a third (small third),

- half an hour and a half (small quarter), - five,

- week, - tithe.

Ancient mathematicians did not consider 100/11 a fraction. The remainder of the division of 1 pound is proposed to be exchanged for eggs, which could be bought 91 pieces. If 91:11 then you get 8 eggs and 3 eggs in the balance. The author recommends giving them to the one who shared, or changing them for salt to salt the eggs.

Decimals.

For several millennia, humanity has been using fractional numbers, but it thought of writing them in convenient decimal places much later.

Why did people switch from ordinary fractions to decimals? Yes, because the actions with them are simpler, especially addition and subtraction.

Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently in ancient China. But even earlier, in ancient Babylon, fractions of the same type were used, only sexagesimal.

Later, the scientist Hartmann Beyer (1563-1625) published the essay “Decimal Logistics”, where he wrote: “... I noticed that technicians and artisans, when they measure any length, very rarely and only in exceptional cases express it in integers of the same name; usually they have to either take small measures, or resort to fractions. In the same way, astronomers measure quantities not only in degrees, but also in fractions of a degree, i.e. minutes, seconds, etc. Their division into 60 parts is not as convenient as the division into 10, 100 parts, etc., because in the latter case it is much easier to add, subtract, and generally perform arithmetic operations; It seems to me that decimal parts, if introduced instead of sexagesimal, would be useful not only for astronomy, but also for all kinds of calculations.

Today we use decimals naturally and freely. However, what seems natural to us served as a real stumbling block for the scientists of the Middle Ages. Western Europe in the 16th century along with the widespread decimal representation of whole numbers, sexagesimal fractions were used everywhere in calculations, dating back to the ancient tradition of the Babylonians. It took the bright mind of the Dutch mathematician Simon Stevin to bring the record of both integer and fractional numbers into a single system. Apparently, the impetus for the creation of decimal fractions was the tables of compound interest compiled by him. In 1585, he published the book "Tithing", in which he explained decimal fractions.

From the beginning of the 17th century, intensive penetration of decimal fractions into science and practice begins. In England, a dot was introduced as a sign separating the integer part from the fractional part. The comma, like the dot, was proposed as a separator in 1617 by the mathematician Napier.

The development of industry and commerce, science and technology required more and more cumbersome calculations, which were easier to perform with the help of decimal fractions. Decimal fractions were widely used in the 19th century after the introduction of the metric system of measures and weights, closely related to them. For example, in our country, in agriculture and industry, decimal fractions and their particular form - percentages - are used much more often than ordinary fractions.

Fractions in music.

The Pythagoreans, who studied music a lot and deified the number, believed that the Earth was spherical and located in the center of the Universe: there is no reason for it to be displaced or stretched in one direction. The Sun, Moon and 5 planets (Mercury, Venus, Mars, Jupiter and Saturn) move around the Earth. The distances from them to our planet are such that they seem to make up a seven-stringed harp, and when they move, beautiful music arises - the music of the spheres. Usually people do not hear it because of the vanity of life, and only after death some of them will be able to enjoy it. And Pythagoras heard it during his lifetime.

His students, the Pythagoreans, who studied music a lot and deified the number, investigated how much the tone of the string rises if it is pressed in the middle, or a quarter of the distance of one of the ends, or a third. It was found that the simultaneous sounding of two strings is pleasing to the ear if their lengths are related as 1:2, or 2:3, or 3:4, which corresponds to musical intervals of octave, fifth and fourth. Harmony turned out to be closely related to fractions, which confirmed the main idea of the Pythagoreans: "the number rules the world" ...

So fractions played a decisive role in music. And now, in the generally accepted notation, a long note - a whole one - is divided into halves (twice as short), quarters, eighths, sixteenths and thirty-seconds.

I study at a music school and I know that 6/8 is three quarters and that there are eight sixteenths in one half. When I learn a new piece, I count out loud every note in a measure (“one and, two and ...”) without even suspecting that I am counting ordinary fractions. Thus, the rhythmic pattern of any piece of music created by European culture, no matter how complex it may be, is determined by binary fractions.

In the process of cognition of reality, mathematics plays an ever-increasing role. Today there is no such field of knowledge where mathematical concepts and methods would not be used to one degree or another. Problems, the solution of which was previously considered impossible, are successfully solved through the use of mathematics, thereby expanding the possibilities of scientific knowledge. Mathematics has always been an integral and essential part of human culture, it is the key to understanding the world around us, the basis of scientific and technological progress and an important component of personality development.

The first fraction that people met was half. Although the names of all the following fractions are associated with the names of their denominators (three - "third", four - "quarter", etc.), this is not the case for half - its name in all languages has nothing to do with the word "two". The next fraction was a third. These and some other fractions are found in the oldest mathematical texts that have come down to us, compiled 5000 years ago - ancient Egyptian papyri and Babylonian cuneiform tablets.
Both the Egyptians and the Babylonians had special notations for the fractions 1/3 and 2/3 that did not match the notations for other fractions.

The Egyptians tried to write down all fractions as sums of shares, i.e. fractions of the form 1/n.

The only exception was the fraction 2/3. For example, instead of 8/15 they wrote 1/3 + 1/5. Sometimes it was convenient. In a papyrus written by the Egyptian scribe Ahmes, there is a task: to divide seven loaves among eight people. If you cut each bread into 8 pieces, you will have to make 49 cuts. And in Egyptian this problem was solved like this. The fraction 7/8 was written as shares: 1/2 + 1/4 + 1/8. Now it is clear that you need to cut 4 loaves in half, 2 loaves into 4 parts and only one loaf into 8 parts (17 cuts in total).

But adding fractions written as fractions was inconvenient. After all, the same parts can enter into both terms, and then, when added, a fraction of the form 2/n will appear. And the Egyptians did not allow such fractions. Therefore, the papyrus of Ahmes begins with a table in which all fractions of the form 2/n, from 2/5 to 2/99, are written as sums of shares. With the help of this table, the division of integers was also performed. For example, how to divide 5 by 21:

The Egyptians also knew how to multiply and divide fractions. But when multiplying, you had to multiply fractions by fractions, and then, perhaps, use the table again. Division was even more difficult.

The Babylonians went the other way. The fact is that the number system in Babylon was sexagesimal - each unit of the next category was 60 times more than the previous one. For example, the entry 14 "42" 38 meant the number 14 602 + 42 60 + 38, i.e. in our entry the number 52 × 958 (only the Babylonians used not our numbers, but other designations made up of wedges). Therefore the Babylonians had fractions not decimal, but sexagesimal. In fact, we still use such fractions in terms of time and angles. For example, the time 3 hours 17 minutes 28 seconds can also be written like this: 3.17 "28" h ( 3 integers are read, 17 sixties 28 three thousand six hundredth hours). Latin - lesser) and second (in Latin - second).So the Babylonian way of designating fractions has retained its meaning to this day.

Not all fractions can be represented as final sexagesimals, just as not all fractions are written as final decimals. For example, fractions like 1/7, 1/11, 1/13 cannot be written in sexagesimal form. But they can be replaced with sexagesimal fractions with any degree of accuracy. This is what the Babylonians did.

Therefore, in 1585, the Dutch mathematician and engineer Simon Stevin suggested switching to decimal fractions. At first they were written very difficult, but gradually they switched to modern recording. Even a century and a half before Stevin, decimal fractions were introduced by the astronomer al-Kashi, who worked at the Samarkand observatory Ulugbek, but his work remained unknown to European mathematicians.

Now in computers, as you, of course, know, binary fractions are used. They look like 0.101101. It is curious that binary fractions were used, in fact, in Ancient Rus', where there were such fractions as half, four, half-four, half-half-four, etc. .

An interesting system of fractions was in Ancient Rome. It was based on dividing the unit of measurement of the weight of the ass into 12 parts. The twelfth of an ace was called an ounce. And the path, time, etc. compared with a visual thing - weight. For example, a Roman could say that he walked seven ounces of the road or read 5 ounces of a book.

In this case, of course, it was not about weighing the path or the book. It simply said that 7/12 of the way was covered or 5/12 of the book was read. And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names.

Even now, it is sometimes said that “he scrupulously studied this issue”. This means that the issue has been studied to the end, that not a single smallest ambiguity remains. And the strange word “scrupulously” comes from the Roman name 1/288 assa - skripulus. There were also such names in use: semis - half of the ass, sextans - its sixth share, seven ounce - half an ounce, i.e. 1/24 ass, etc. A total of 18 different names were used. To work with fractions, it was necessary for these fractions to remember both the addition table and the multiplication table. Therefore, Roman merchants knew for sure that when adding a triens (1/3 ass) and sextans, a semis is obtained, and when a bes (two thirds of an ass) is multiplied by a sescution (3/2 ounce, i.e. 1/8 ass), an ounce is obtained. At the same time, they well understood that they did not multiply the weights themselves (multiplying weight by weight does not make sense), but the fractions expressing these weights. To facilitate the work, special tables were compiled, some of which have come down to us.

So the role that the number 60 played in Ancient Babylon, and the number 2 in Ancient Rus', the number 12 played in Ancient Rome - the Roman system of fractions and measures was duodecimal (although they wrote the numbers according to the decimal system, only in a different way than this we do). Due to the fact that numbers like 1/10n are not expressed in the form of final duodecimal fractions, the Romans did not know how to represent the result of division by 10, 100, etc. shot. For example, one Roman mathematician, dividing 1001 asses by 100, first obtained 10 asses, then split the ass into ounces, etc., but, of course, did not get rid of the remainder.

There were no fractions in Greek writings on mathematics. Greek scientists believed that mathematics should deal only with whole numbers.
With fractions, they left merchants, artisans, as well as surveyors, astronomers and mechanics to mess around. But the old proverb says: "Drive nature through the door, it will fly in through the window." Therefore, even in the strictly scientific writings of the Greeks, fractions penetrated, so to speak, “from the rear”. In addition to arithmetic and geometry, Greek mathematics included ... music. The Greeks called music that part of our arithmetic, which deals with relationships and proportions. Why such a strange name? The fact is that the Greeks also created a scientific theory of music. They knew that the longer the stretched string, the lower, “thicker” the sound it makes. They knew that a short string made a high pitched sound. But every musical instrument has not one, but several strings. In order for all the strings to sound "according" when played, pleasant to the ear, the length of their sounding parts must be in a certain ratio. For example, in order for the pitches of sounds emitted by two strings to differ by an octave, their lengths must be related as 1:2. Similarly, a quint corresponds to a ratio of 2:3, a quart to a ratio of 3:4, and so on. Therefore, the doctrine of relationships, of fractions, was associated with the Greeks with music.

The modern system of writing fractions with a numerator and a denominator was created in India. Only there they wrote the denominator from above, and the numerator from below and did not write a fractional line. And the Arabs began to write fractions exactly as they do now.

Literature

1. Vilenkin N. Ya. From the history of fractions. / Kvant, No. 5/1987.

2. Ancient Egyptian problem. / “To the World of Informatics” No. 66 (“Computer Science” No. 1/2006).

3. Number systems. / “To the world of informatics” No. 90, 93 (“Computer science” No. 9, 17/2007).

4. Abacus in Russia. / “To the world of informatics” No. 69, 71 (“Computer science” No. 4, 6/2006).

Koksunova Ilyana

The educational and research work of a 8th grade student examines the history of the origin of fractions. The paper examines the history of modern recording of fractions, the origin of the names of some fractions.

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Ministry of Education and Science

REPUBLIC OF KALMYKIA

TSELINNY AREA

MOKU "KHAR - BULUK SECONDARY SCHOOL"

Educational research work:

"HISTORY OF THE ORIGIN OF SHOT"

Author : Koksunova Ilyana

8th grade student.

Supervisor : Muchkaeva Elena Chudeevna, mathematic teacher.

Introduction Goals and objectives of the study Place and duration of the study Research methods The emergence of fractions
From specific fractions to basic
Hexadecimal fractions
Fraction system of ancient Rome
Writing fractions among the Greeks
Modern notation for common fractions
Decimal Conclusion Literature Application

But there is no tone of arithmetic,

Already in the whole defendant,

And in shares nothing

Answer possible.

There about you rejoice,

Be able to in parts.

L.F. Magnitsky

I. Introduction.

Fractions arise when a natural number is divided into equal parts: in two, three parts, ten parts, etc. But it is not enough to know what a fraction is. You need to be able to compare them, perform actions on fractions, solve all sorts of problems with fractions.

Since ancient times, people had to not only count objects (which required natural numbers), but also measure length, time, area, and make payments for purchased or sold goods. It was not always possible to express the result of measurement or the cost of goods in natural numbers. also had to take into account parts, measure shares . This is how fractions were born. In practical life, fractions are absolutely necessary. As ideas about natural numbers arose, ideas also arose about the fractions of units, or rather, about the fractions of a whole specific object. Thus, the emergence of the idea of the number 2 led to the idea of half, half of half, and so on. The appearance of a natural number n led to the idea of a fraction of the formnow called aliquot , or generic , or main .

II. Goals and objectives of the study.

Target : 1. Study the history of fractions.

2. Study the history of the writing system and the names of fractions in different countries.

To achieve this goal, I set myself the following tasks:

Collect material about the history of fractions.
To study the history of the classification of fractional numbers.
Reveal the names of fractions are applicable at the present time..

III. Place and duration of the study: 1 year.

Khar-Buluk village

Research methods:

The method of working with scientific - popular literature, documents.
comparison method.

History of common fractions.

People often have to divide the whole into parts. The most famous share is, of course, half. Words with the prefix "floor" can be heard, perhaps, every day: half an hour, half a kilogram, half a roll.

But there are other uses as well. For example, a quarter, tenth, hundredth. When are shares formed? Then, when one object (a loaf of bread, a sheet of paper) or a unit of measurement (an hour, a kilogram) is divided into equal parts. A fraction is each of the equal parts of a unit. The name of the share depends on how many equal parts the unit is divided into. They divided the name of the share “half” into two parts, into three - “third”, into four - “quarter”. And if it’s five, six, seven parts, then they use the words “fifth, sixth, seventh,” etc. Quarters are called fourths in a different way, thirds - thirds, and halves - second parts.

To record any share, use a horizontal dash. It is called a fractional line. A unit is placed above it, and under the line the number of equal parts into which the unit is divided is written. For example, the second, twenty-first, one hundred and fifth parts are written:, . They read: “one second”, “one twenty-one”, “one hundred and fifth”. If the number of equal parts into which the unit is divided is indicated by the letter n , then this letter is written under the fractional bar:. They read: "one enna".

Why are shares needed? The answer is very simple: when measuring quantities, it is often impossible to get by with only whole units. Imagine, for example, that we were allowed to use only whole meters to measure length. How then would we be able to measure a person's height? Or sports results in jumping? In such cases, centimeters are used.

And in technology, smaller fractions of a meter are often needed - thousandths. They, as you know, are called millimeters. And larger fractions of a meter are useful, for example, tenths. How are fractions made from fractions? Take, for example, the number "two ninths". It is not a natural number, but not a fraction of one. This is the sum of two equal shares. For numbers that are either fractions or sums of fractions, use the common name - fractional numbers . Fractional numbers are called and simply fractions.

A FRACTION IS OR A SHARE OR THE SUM OF SEVERAL EQUAL PARTS. So the number "two ninths" is a fraction. It is written in numbers:. Fraction is equal to the sum of two equal ninths: = .

To write a fraction, a fractional bar and two natural numbers are used. Under the fractional line write denominator fractions. It shows what fractions make up a fraction. Above the line is written numerator fractions. It shows how many parts a fraction is.

In the most ancient written sources that have come down to us - Babylonian clay tablets and Egyptian papyri, there are not only natural numbers, but also fractions.

Fractions were needed to express the result of measuring length, mass, area in cases where the unit of measurement did not fit into the measured value an integer number of times.

Then a new, smaller unit of measurement was introduced. The names of these new units of measurement became the first names of fractions. For example, fractionstill called "half"; among the Romans, the word “ounce” was at first the name of the twelfth of a unit of mass, but then the ounce began to denote one twelfth of any value (they said: “Seven ounces of the way”, that is, seven twelfths of the way).

In Russian, the word fraction "appeared in the 8th century, it came from the verb" crush "- to break, break into pieces. In the first textbooks of mathematics (in the 17th century), fractions were called “broken numbers”. Among other peoples, the naming of fractions is also associated with the verbs "break", "break", "shatter".

The modern designation of fractions originates in ancient India; the Arabs also began to use it, and from them in the 12th-14th centuries it was borrowed by Europeans. In the beginning, fractions were not written with a slash; for example numbers there were 2 recorded so . the feature of the fraction began to be used constantly only about 300 years ago. The first European scientist whobegan to use and distribute the modern notation of fractions, was an Italian merchant and traveler, the son of a city clerk Fibonacci ( Leonardo of Pisa). In 1202, he introduced the word "fraction". Titles " numerator "and" denominator "introduced in the XIII century by Maxim Planud - Greek monk, scientist - mathematician.

The emergence of fractions.

The appearance of aliquot fractions is very characteristic of the initial development of the concept of number in any ancient civilization. This is the first appearance of fractions as a result of the process of crushing the whole into parts; this can explain the appearance of aliquot fractions of the formfor small n (for example, n= 2, 3, 4, 6, 8,10), because dividing a unit by a larger number in the practice of that time was hardly encountered.

Another (main) source of fractions is the process of measurement, which appeared along with the account. Any measurement is always based on some value (length, volume, weight, etc.). The choice of one or another unit, which serves as the basis for a system of measures, is determined by the specific historical situation.

Measures in their development have passed approximately the same stages as numbers. At the first stages of the development of human society, measurements were made by eye. With the further development of society, some natural measures appeared: foot length , palm width etc. .

The existence of such ancient measures is evidenced by the names of the measures of length, which have survived to this day. These measures are ft (foot length), inch (width of the thumb at its base), yard, cubit (distance from the end of the fingers to the elbow), palm (palm width).

Of all the measures of length, the most firmly entered into the life of the Russian people arshin . (It should be noted that the length of the measures varied depending on the terrain and conditions of use). This is evidenced by a large number of sayings and turns of folk speech: “measure on your own arshin”, “as if swallowed an arshin”, etc. The need for a more accurate measurement led to the fact that the original units of measures began to be divided into two, three, etc. parts. As a result of fragmentation, smaller units of measure received individual names, and quantities began to be measured already in these smaller units.

This is how the first concrete fractions arose as parts of certain definite measures. Only much later the names of these specific fractions began to serve to designate the same parts of quantities, and then for abstract fractions.

From specific fractions to basic.

There is every reason to believe that initially only binary fractions existed. Later they were joinedand its binary divisions. So, dividing the arshin into 16 inches meets the requirement that, , , shares would be expressed in whole numbers of vershoks. Such a binary system of division of the basic unit is clearly expressed in the old Russian system for measuring fields and some other quantities. So, in the XV century. plows began to be used as a unit for measuring field areas (plow = 800 quarters; quarter =tithes), as well as half a sokha, half - half a sokha (a quarter of a plow), half a quarter of a plow, etc.

In connection with the division of various units of measurement into parts, fractions of the form were widespread in Rus': half =, quarter = , half hour = , floor - half hour = , floor - floor - half an hour or a small quarter =, third = , half third = , half-half third = , half-half-half-third, or a small third = etc.

Hexadecimal fractions.

In ancient Babylon, there were sexagesimal fractions, that is, they were written, for example, in the form of 4; 52; 03. This meant: 4+ + .

The Babylonians only worked with sexagesimal fractions. Because the denominators of such fractions are the numbers 60, 60 2 , 60 3 etc., then fractions such as, it was impossible to express exactly through sexagesimals: they expressed approximately through them. Because The Babylonians had a positional number system, they acted with sexagesimal fractions using the same tables as for natural numbers.

The system of fractions of ancient Rome.

An interesting system of fractions was in ancient Rome - duodecimal. It was based on dividing into 12 parts of a unit of weight, which was called ass . A copper coin, and subsequently a unit of weight - ass The Romans divided into twelve equal parts - ounces . The twelfth of an ace was called an ounce. And the way, time and other quantities were compared with a visual thing - weight. For example, a Roman could say that he walked seven ounces of the road or read five ounces of a book. It meant that passedpath or readbooks. And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names.

Even now it is sometimes said: "He scrupulously studied this issue." This means that the issue has been studied to the end, that not a single smallest ambiguity remains. And the strange word "scrupulously" comes from the Roman name assa - "scrupulus". There were also such names:"semis" - half ass, "sextans" - one sixth of it"seven ounce" - half ounces, i.e. ace, etc. Total applied18 different names for fractions. To work with fractions, it was necessary for these fractions to remember both the addition table and the multiplication table. Therefore, the Roman merchants knew for sure that when adding triens (assa) and sextans are semis, and multiplying the demon (assa) by sescution (ounces, that is ace) is an ounce. To facilitate the work, special tables were compiled, some of which have come down to us.

Due to the fact that there are no fractions with denominators 10 or 100 in the duodecimal system, the Romans found it difficult to divide by 10, 100, etc. When dividing 1001 asses by 100, one Roman mathematician first received 10 asses, then split the ass into ounces, etc. e. But he did not get rid of the remainder. In order not to deal with such calculations, the Romans began to use percentages. They took interest from the debtor (that is, money in excess of what was given in debt). At the same time, they said: not “the interest will be 16 hundredths of the amount of the debt”, but “for every 100 sesterces of debt you will pay 16 sesterces of interest”. And the same thing is said, and there was no need to use fractions! Since the words "one hundred" sounded in Latin "about a centum", then the hundredth part was called percent. And although now fractions, and especially decimal fractions, are known to everyone, percentages are still used in financial calculations and in planning, that is, in various areas of human activity. Previously, they also used ppm - so called thousandths (in Latin "about a mille" - per thousand). Unlike percentages, which are denoted by the sign%, ppm denote ‰.

The writing of fractions among the Greeks.

There were no fractions in Greek writings on mathematics. Greek scientists believed that mathematics should deal only with whole numbers. They provided merchants, artisans, as well as astronomers, surveyors, mechanics and other “black people” with fractions. But the old proverb says: "Drive nature through the door - it will fly in through the window." Therefore, even in the strictly scientific writings of the Greeks, fractions penetrated from the “reverse”. In addition to arithmetic and geometry, Greek science included music. The Greeks called music the doctrine of harmony. This teaching was based on that part of our arithmetic, which deals with ratios and proportions. The Greeks knew that the longer the stretched string, the lower the sound it makes, and the short string makes a high sound. But every musical instrument has not one, but several strings. In order for all the strings to sound "according" when played, pleasant to the ear, the lengths of their sounding parts must be in a certain ratio. Therefore, the doctrine of ratios and fractions was used in Greek music theory.

Due to the fact that Greek scientists did not recognize fractional numbers, they had difficulty measuring quantities. The Greek mathematician could not say that the length of one segment is three times the length of the other. After all, these lengths could turn out to be fractional numbers, or even not be expressed at all by the numbers known to the Greeks, and therefore it was impossible to apply the multiplication operation to them. The Greek scientists had to come up with a way to get by in science without expressing lengths, areas and volumes in numbers (merchants and artisans calmly did this, not paying attention to the philosophies of scientists). For this, it was necessary to create a doctrine of the ratios of magnitudes, the equality of such ratios, and so on. The equality of two relations was then called the Latin word "proportion" (the Greeks used the Greek word "analogy" for this).

Modern notation for common fractions.

It should be noted that the department of arithmetic about fractions has long been one of the most confusing. So, those who did not know fractions were not recognized as versed in arithmetic. It was hard to master the fractions. Even the most educated people of the Middle Ages found operations with fractions very difficult. This happened because there were no general methods of actions with fractions and there were no recording of fractions, they were added, multiplied and divided according to various “recipes”.

The modern system of writing fractions with a numerator and a denominator was created in India. Indians widely used "ordinary" fractions. Our notation of ordinary fractions with the numerator and denominator was adopted in India as early as the 8th century BC. however, without a fractional bar. Only there they wrote the denominator from above, and the numerator from below. And the Arabs began to write fractions exactly as they do now.

Decimal.

The beginning of a new stage in the history of fractions was decimal fractions. The introduction of decimal fractions, along with the decimal number system, is one of the most important moments in the history of arithmetic, and therefore of all mathematics in general. Already in the III century. among the peoples of China, who used the decimal system of measures, decimal fractions began to appear, which acted as named numbers - units of the decimal system of measures.

Some hints of decimal fractions were found among the peoples of India, and then among the peoples of the Middle East. Al - Uklidisi (X century) was the first mathematician of the countries of Islam who used decimal fractions and understood their importance. At an-Nasawi (d. ca. 1030) there are hints of decimal fractions when extracting a square root (when extracting a square root, if it is not completely extracted, they assigned as many zeros to the root expression as it was necessary to get extra characters in the root). In Europe, a similar method of extracting square roots was first used by a Spanish monk.John of Seville(XII century). Decimal fractions were used in his treatise by a Baghdad scholar al-Baghdadi (1002 - 1071).

At the end of the 16th century, decimal fractions appeared. When calculating with decimal fractions, very numbers with a very large number of digits were obtained. Such a number of characters was not necessary for practice. Therefore, it was necessary to round off the answers received, to conduct approximate calculations. The Russian mathematician and shipbuilder Academician Alexei Nikolaevich Krylov (1863 - 1945) did a lot for the development of approximate calculations. Now, to facilitate calculations, machines have been built that count surprisingly fast. In one second, these machines can perform millions of arithmetic operations (additions, subtractions, multiplications and divisions) on multi-digit numbers.

In science and industry, in agriculture, in calculations, decimal fractions are used much more often than ordinary ones.

This is due to the simplicity of the rules for calculating decimal fractions, their similarity to the rules for operations with natural numbers. The rules for calculating with decimal fractions were described by the famous scientist of the Middle Agesal-Kashi Jemshd Ibn Mas'ud, who lived in the city of Samarkand in the observatory of Ulugbek at the beginning of the 15th century.

Al-Kashi wrote down decimal fractions in the same way as is customary now, but did not use a comma: he wrote down the fractional part in red ink or separated it with a vertical line.

But at that time they did not learn about this in Europe, and only after 150 years decimal fractions were reinvented by a Flemish engineer scientistSimon Stevin. Stevin wrote decimals rather hard.

For example, the number 24.56 looked like this: 2405162 or 2456 - instead of a comma zero in a circle (or 0 above the whole part), the numbers 1, 2, 3, ... marked the position of the remaining characters.

A comma or a period to separate the whole part from the fractional began to be used from the XVII.

In Russia, the doctrine of decimal fractions outlinedLeonty Filippovich Magnitskyin 1703 in the first textbook of mathematics "Arithmetic, that is, the science of numerals."

Our numbering is decimal. This name comes from the rule: the unit of each digit is 10 times the unit of the previous least significant digit.

The unit digit is the smallest in the notation of natural numbers. The unit of the previous least significant digit must be 10 times less than the unit of each digit.

So people agreed to the right of the category of units to place the category tenths shares. And to indicate where units end and tenths begin, put before tenths comma .

For example, the entry 34.2 stands for the number. Number 5 can be written: 5.9.

The digits to the right of the decimal point can be continued further. What will the unit of the second such order of the series mean? For the rule to hold, it must be 10 less than. So this is: 10, i.e. .

1st place after the decimal point - tenths,

2nd digit after the decimal point - hundredths,

3rd place after the decimal point - thousandths.

A fraction written with numbers and a comma is called a decimal fraction, a fraction written with a fractional bar is called an ordinary fraction.

Like natural numbers, any decimal fraction can be represented as a sum of bit terms.

dozens

units

tenths

hundredths

thousandths

ten-thousandths

hundred-thousandths

millions

ten millionth

hundred millionth

billionth

Let's try to write an ordinary fractiondecimal fraction. To do this, you need to divide the numerator by the denominator. Having calculated several quotient digits, we will see the pattern with which these digits appear. It can be seen that only 6 will be obtained. But you can continue like this without end. Therefore, the resulting fraction is calledinfinite decimal. It is completely impossible to write it down. So somewhere you have to break off the record and put an ellipsis. It is only necessary to understand the regularity with which the numbers go one after another. For fractionwe found such a pattern above. You can write: =0,6666...

Infinite decimals are also numbers. They can be added and subtracted, multiplied and divided, compared with each other. They are compared in the same way as final (i.e. regular) decimals. For example, 10.63186318... > 10.631846318...,since in the discharge of hundred thousandths the first number has the number 6, and the second - 4.

Let's discard all the digits in an infinite decimal fraction, starting with a certain digit. We will get the final decimal fraction. For example, from the fraction 0.666666... you can get final fractions 0.6; 0.66; 0.666; 0.6666 Each of them is said to be -shortcoming approximationgiven infinite decimal. From these approximations, one can build an infinite chain of inequalities: 0.6

Now we will again discard all the digits in the infinite decimal fraction, starting from a certain digit, but we will increase the last digit by one. Then we again get the final decimal fraction. It will be greater than the given infinite decimal fraction. They call herover-approximation. For example, for the number 0.666666,.. fractions 0.7; 0.67; 0.667; ... are oversized approximations. Each of these fractions is greater than the number 0.666666...; and the more digits a fraction contains, the closer it is to that number.

The more digits are taken in the approximation of a given number, the closer the resulting final decimal fraction is to the given number..

Remembering that =0.6666... we can get many approximate equalities.

It is easy to see that when translating some ordinary fractions, infinite decimal fractions are obtained, where one or a group of digits begins to repeat from a certain place. Such a repeating group of numbers is called period infinite decimal fraction, and the fraction itself is called periodic. The final decimal fraction can also be considered periodic - its period consists of zero.

Every rational number can be written as a periodic decimal. And vice versa, if a number is written as a periodic decimal fraction, then it is rational. But, besides rational numbers, there are other numbers. This is what Pythagoras discovered. He proved an amazing thing:It turns out that the length of the diagonal of a unit square cannot be written as a rational number!And an infinite decimal fraction is possible. In the same way, it is impossible to write the numbers as a periodic fractionπ, e.

Conclusion.

In science and industry, in agriculture, decimal fractions are used much more often than ordinary ones. This is due to the simplicity of the rules for calculating decimal fractions, their similarity to the rules for operations with natural numbers.

While doing this, I found out that

1. The history of fractions has an ancient origin.

2. The rules for calculating decimal fractions were described by the famous medieval scientist al-Kashi Jamshid Ibn Masud, who worked in the city of Samarkand in the observatory of Ulugbek at the beginning of the 15th century.

3. In Europe, decimal fractions were reinvented by the Flemish scientist and engineer Simon Stevin in the late 16th and early 17th centuries.

4. In Russia, the doctrine of decimal fractions was expounded by Leonty Filippovich Magnitsky in 1703 in the first mathematics textbook "Arithmetic, that is, the science of numerals."

I also learned the history of the writing system and the names of fractions in different countries and their application in modern mathematics.

The work on the history of the origin and recording of numbers is very interesting and multifaceted, and you can search and find a lot of interesting information, both about the origin of numbers and about their application in practice.

Literature.

Vilenkin N.Ya. and others. Mathematics 6th grade. M.: Enlightenment, 1993.
Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. M.: Education, 1989 .
Rybnikov K.A. History of mathematics. Moscow: Nauka, 1994.
Stroik D.Ya. Brief essay on the history of mathematics. Moscow: Nauka, Fizmatlit, 1990.
Shevrin L.N. and others. Mathematics: Textbook - an interlocutor for 5 - 6 cells. M.: Education, 1989.
Yushkevich A.P. Mathematics in its history. M.: Nauka, 1996.

Yard - the main measure of length in England, this measure was established by decree of King Henry I. The length of a yard is currently approximately 0.9144 m