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History of the development of decimal fractions. Fractions: history of fractions. The history of the appearance of ordinary fractions. Operations with decimal fractions page 6

Ethics
06.12.2020

Andriyannikov Nikita

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

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SCIENTIFIC PRACTICAL CONFERENCE

Design and research work

Completed by: 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

A number expressed as a decimal sign
Both German and Russian will read it,
And the Yankees are the same.
DI. Mendeleev

Introduction.

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

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

Goals:

  1. Find out when and in which ancient sources decimal fractions were first mentioned.
  2. Trace how the notation of decimal fractions has changed over several centuries.
  3. Find out who was the first to enter a comma into a decimal fraction.

Tasks:

  1. Study and analyze the history of decimal fractions in various sources.
  2. Collect information using Internet resources and systematize the information received.
  3. Present the research results in the form of a presentation “The History of Decimals” using Power Point.

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

And outline ways to solve it...

NPOSH "Commonwealth"

Essay

"The history of decimal fractions"

Andriyannikov Nikita, 5B grade

2012

Mathematics is one of the oldest sciences, and its first steps are connected with the very first steps of the human mind. It arose in the work activities of people. Developing

Mathematics solved more and more precisely the complex problems that life itself posed to man. Trade, all production, and the economies of countries found themselves in a difficult situation in the 17th century. For sailors, accurate maps were needed, for merchants, quick and correct calculations without deception, for the construction of machines, ships, temples and dwellings - drawings verified to 1mm. Production developed, and the inability to quickly and accurately make calculations literally hindered the development of science and technology. Life presented scientists with the task of simplifying calculations, increasing their accuracy and speed. Decimal fractions satisfied these requirements.

Mathematicians came to decimal fractions in different times in Asia and Europe. The origin and development of decimal fractions in some Asian countries was closely related to metrology (the study of measures). Already in the 2nd century. BC. there was a decimal system of length measures.

(slide No. 2) Ancient China already used the decimal system of measures,
denoted fractions in words using measures of lengthchi, tsuni, lobes, ordinal, hairs, the finest, cobwebs.

(slide No. 3)

A fraction of the form 2.135436 looked like this: 2 chi, 1 cun, 3 lobes, 5 ordinal, 4 hairs, 3 finest, 6 cobwebs. Fractions were written this way for two centuries, and in the 5th century the Chinese scientist Tszyu-Chun-Zhi accepted not chi as a unit. Ah Zhang = 10 chi, then this fraction looked like this: 2 zhang, 1 chi, 3 cun, 5 lobes, 4 ordinal, 3 hairs, 6 finest, 0 cobwebs.

(slide 4)

Decimal fractions received a more complete and systematic interpretation in the works of the Central Asian scientist al-Kashi in the 20s of the 15th century.

The Central Asian city of Samarkand was in the 15th century. big cultural center. There, the famous observatory created by the prominent astronomer Ulugbek, the grandson of Tamerlane, worked in the 20s of the 15th century. a major scientist of that time -Jamshid Ghiyaseddin 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 successive denominators are 60 and its successive powers. By analogy, we introduced fractions in which the successive denominators are 10 and its successive powers.”

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

comma, but writes the whole part in blackink, and the fractional part in red or separates the whole part from the fractional partvertical line.

In 1579, decimal fractions were used in the “Mathematical Canon” of the French mathematician Francois Vieta (1540-1603), published in Paris. In this work, which is a collection of trigonometric tables, Viet decisively advocated the use of, as he put it, thousandths and thousands, hundredths and hundreds, tenths and tens, etc. instead of the sexagesimal system of integers and fractions. When writing decimal fractions, Vieth did not adhere to any one designation. Often he writes both the numerator and the denominator, sometimes he separates the digits of the integer part from the fractional part with a vertical bar, or the digits of the integer part are depicted in bold, or, finally, the digits of the fractional part are given in more small print and emphasizes. Fraction designation 2.135436 2 1579 F. Viet France

(slide No. 6) Al-Kashi's discovery of 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 No. 7) Flemish engineer and scientist Simon Stevin (1548-1620), about 150 years after al-Kashi, introduced the doctrine of decimal fractions in Europe.

He is considered the inventor of decimal fractions.Stevin, a native of Bruges, was first a merchant, then during Dutch revolution engineer in the troops of Moritz of Orange, who headed the republic. “To astrologers, farmers, volume measurers, barrel capacity checkers, stereometers in general, coin masters and all merchants - hello to Simon Stevin,” - this is how the inventor of decimal fractions addresses his readers in his book “Tenth” (1585). This small work (only 7 pages) contained an explanation of notation and rules for working with decimals. In the book, he tries to convince people to use decimals, saying that using them will "eliminatedifficulties, strife, mistakes, losses and other accidents, the usual companions of calculations." He wrote numbers fractional number in one line with the digits of an integer, while numbering them.

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

35 0 9 1 1 2 2 3

So, instead of a comma, there is a zero in a circle. In other circles or above the numbers, the decimal place is indicated: 1 – tenths, 2 – hundredths, etc. Stevin pointed to a large practical significance decimal fractions and persistently promoted them. He was the first scientist to demand the introduction of a decimal system of weights and measures.(slide No. 8)

The comma in the notation of fractions was first used in 1592, and in 1617. Scottish mathematician John Napier proposed separating decimals from a whole number with either a comma or a period.

Modern notation of decimal fractions i.e. separation of the whole part of the comma, proposed by Johannes Kepler (1571 - 1630). In countries where English is spoken (England, USA, Canada, etc.), a period is written instead of a comma. Fraction designation 2.135436 2.135436 2.135436 1571 - 1630 Kepler Germany In Russia, the first systematic information about decimal fractions is found in “Arithmetic” by Magnitsky (1703) C early XVII century, intensive penetration of decimal fractions into science and practice begins. The development of technology, industry and trade required increasingly cumbersome calculations, which were easier to perform with the help of decimal fractions. Decimal fractions became widely used in the 19th century after the introduction of the closely related metric system measures and weights. 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 instead of a comma they write a period, for example: 2.3 and read: two dot three.(slide No. 9)

In “Arithmetic, that is, the science of numbers” (1703), the first Russian teacher-mathematician Leonty Filippovich Magnitsky (1669-1739) devoted a separate chapter to decimal fractions. « M.V. Lomonosov called this book the gateway to his learning. The publication of Magnitsky's Book in 1703 was an important fact in the history of mathematical education in Russia. For half a century, the book was the “gateway of learning” for Russian youth striving for education. Magnitsky came from the people, 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 him to write 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.

During the project - research activities I found a lot of interesting and educational information on the history of mathematics. The work of finding the right material was useful and exciting. In the process of research, I found answers to all the questions that my manager and I posed before starting work: where and when were decimal fractions invented, who came up with the modern notation for these numbers. I did some research on how decimal notation has changed over the centuries and presented the results in a table.

Working on the project taught me to systematize the material I found, analyze the data and identify the necessary facts from large quantity information.

But the most important thing in working on the project is that in the process I learned how to work with the Power Point program, which gives me the opportunity in the future to 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. A journey into the history of mathematics or How people learned to count: A book for those who teach and learn. M.: Pedagogika-Press, 1995. 168 p.

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

From history The invention of decimal fractions is one of the greatest achievements of human culture. The rules for calculations with decimal fractions were described by the famous medieval scientist al-Kashi Jemshid Ibn Masud, who worked in Uzbekistan, near the city of Samarkand at the Ulegbek Observatory at the beginning of the 15th century. Al-Kashi wrote fractions on the same line with numbers in the decimal system, to separate the whole from the decimal, he used a vertical line or ink different color. His works were not known to European scientists for a long time, and only 150 years later were decimal fractions reinvented.




Test yourself Read the decimal fractions: A) 2.7; 11.4; 401.1; 0.8; 99.9; 909.9. B) 5.64; 21.87; 381, 77; 54.60; 0.55; 0.09; 2.02. B) 1.597; 12.882; 326.703; 0.321; 0.049; 0.001. Write decimal fractions. 7 whole 8 tenths 2 whole 25 hundredths 0 whole 92 hundredths 12 whole 3 hundredths 5 whole 187 thousandths 24 whole 24 thousandths


Historical reference The concept of an abstract decimal fraction first appeared in the 15th century. It was introduced by the eminent mathematician and astronomer Al-Cauchy ( full name Jemiad ibn – Masud al – Qoshi) in the work “The Key to Arithmetic” (1427). Al-Cauchy's discovery in Europe became known only 300 years later. Knowing nothing about Al-Cauchy’s discovery, decimal fractions were discovered for the second time, approximately 150 years after him, by the Flemish mathematician and engineer Simon Stevin in his work “Decimal” (1585). In Russia, the doctrine of decimal fractions was first taught by L.P. Magnitsky in his Arithmetic, the first Russian mathematics textbook. (1703) It was proposed in different ways to separate the whole part from the fractional part. Al-Koshi wrote the whole and fractional parts in one row, although he wrote them in different inks, or put a vertical line between them. S. Stevin, to separate the whole part from the fractional part, put a zero in the circle. The comma adopted in our time was proposed by the German astronomer J. Kepler (1571 - 1630).


Rule for comparing decimal fractions If the whole parts of decimal fractions are different, then the greater is the fraction that has more whole part. If the whole parts of decimal fractions are equal, then the fraction with more tenths is greater. If there are equal numbers of tenths, then the fraction that has more hundredths is larger, etc.


Test yourself Compare: 1.21 and 1.2 3.34 and 3.4 8.6 and 8.37 23.43 and 23.9 3.5601 and 4.48 85.113 and 85.13 148.05 and 14.805 6 .44806 and 6.601 and 35.6010


Rounding rule To round a number to a specified digit, you must: Separate all digits after this digit; Underline the first of those numbers that are separated, and determine which numbers include: 0; 1; 2; 3; 4 or 5; 6; 7; 8; 9 she is located; If the number 0 is underlined; 1; 2; 3; 4, then all numbers that are separated are replaced with zeros; if the number 5 is underlined; 6; 7; 8; 9, then 1 is added to the digit to which rounding is carried out, and all digits that are separated are replaced with zeros.


Rule of addition (subtraction) To add (subtract) decimal fractions, you need to: Equalize the number of decimal places in these fractions; Write them below each other so that the comma is written under the comma; Perform addition (subtraction) without paying attention to the comma; Place a comma under the comma in the given fractions in your answer.




From history The rules for calculations with decimal fractions were described by the famous scientist al-Kashi Jemshid Ibn Masud at the beginning of the 15th century. He wrote fractions in the same way as is customary now, but did not use a comma: he wrote the fractional part in red ink or separated it with a vertical line. But in Europe they did not find out about this, and only 150 years later the scientist Simon Stephen wrote down decimal fractions in a rather complex way: in place of the decimal point, a zero in a circle. A comma or period to separate a whole part has been used since the 17th century. In Russia, L. F. Magnitsky outlined decimal fractions in 1703 in the first mathematics textbook “Arithmetic, that is, the science of numerals.”


The rule for multiplying a decimal fraction by a place unit To multiply a decimal fraction by a place unit, it is enough to move the decimal point in the fraction to the right as many places as there are zeros in the place unit. If in a decimal fraction the number of digits to the right of the decimal point is less than the number of zeros in the decimal unit, then the required number of zeros can be added to the right of the fractional part of the decimal fraction. 213.84 * 10 = 2,138.4; 97.2 * 100 = 97.20 * 100 = 9,720; 74.3379 * = .9.


The rule for dividing a decimal fraction by a place unit To divide a decimal fraction by a place unit, it is enough to move the decimal point in the fraction to the left as many places as there are zeros in the place unit. If in a decimal fraction the number of digits to the left of the decimal point (digits of the whole part of the fraction) is less than the number of zeros in the digit unit, then to the left before the highest significant figure You can add as many zeros to an entire part of a fraction as there are missing ones. 213.84: 10 = 21.384; 9.72: 100 = 0.0972; 74.03: = 0.07403.


Rule for multiplying decimals To multiply a decimal by natural number, you need to: 1) multiply it by this number, ignoring the comma; 2) in the resulting product, separate as many digits on the right with a comma as there are in the decimal fraction separated by a comma. When multiplying decimal fractions, be indifferent to their commas. You should, I can tell you in advance, multiply them like natural numbers. And in the resulting product, On the right, a comma in each case, Separate as many characters as possible, three, five, six... How many are there in the factors.


Rule for dividing decimal fractions To divide a decimal fraction by a natural number, you must: 1) divide the fraction by this number, ignoring the comma; 2) put a comma in the quotient when the division of the whole part ends. If the whole part less than divisor, then the quotient starts from zero integers. To divide a number by a decimal fraction, you must: in the dividend and in the divisor, move the decimal point to the right by as many digits as there are after the decimal point in the divisor; After that, divide by a natural number.




History of origin. Fractions appeared in ancient times. When dividing up spoils, when measuring quantities, and in other similar cases, people encountered the need to introduce fractions. But there was no single recording of fractions, as well as whole numbers. Fractions appeared in ancient times. When dividing up spoils, when measuring quantities, and in other similar cases, people encountered the need to introduce fractions. But there was no single recording of fractions, as well as whole numbers.


Fractions in Egypt. Fractions in Egypt. The ancient Egyptians already knew how to divide 2 objects into three people; for this number -2/3- they had a special symbol. By the way, this was the only fraction used by Egyptian scribes that did not have a unit in the numerator - all other fractions certainly had a unit in the numerator. The ancient Egyptians already knew how to divide 2 objects into three, for this number -2/3- they have there was a special badge. By the way, this was the only fraction used by Egyptian scribes that did not have a unit in the numerator - all other fractions certainly had a unit in the numerator. The Egyptians also knew how to multiply and divide fractions. But to multiply, you had to multiply fractions by fractions, and then, perhaps, use the table again. The situation is even more complicated. The Egyptians also knew how to multiply and divide fractions. But to multiply, you had to multiply fractions by fractions, and then, perhaps, use the table again. The situation with division was even more complicated.


Fractions in Greece. The Greeks, like the Egyptians, originally had fractions with only a numerator, equal to one, and wrote them down in words, and later with symbols, for example, a fraction was written like this: ٧ א The Greeks, like the Egyptians, initially had fractions only with a numerator equal to one, and they wrote them down in words, and later with symbols, for example, a fraction was written like this: ٧ א Heron of Alexandria (1st century BC) used fractions general view and wrote them down without a fractional line, putting the numerator and denominator side by side, and writing the numerator with one stroke, and writing the denominator twice and marking it with two strokes, for example, writing it like this: ßεε. Heron of Alexandria (1st century BC) used general fractions and wrote them without a fractional line, putting the numerator and denominator side by side, and writing the numerator with one stroke, and writing the denominator twice and marking it with two strokes, for example, writing it like this: ßεε . The Greeks had a sign that replaced the word “it turns out”; this sign was called “gygnestai”. The Greeks had a sign that replaced the word “it turns out”; this sign was called “gygnestai”. Diophantus (3rd century AD) wrote fractions almost the same way as we do, only the denominator was written above the line, and the numerator, the word particle and then the denominator below the line. Diophantus (3rd century AD) wrote fractions almost the same way as we do, only the denominator was written above the line, and the numerator, the word particle and then the denominator below the line.


Decimal fractions in antiquity Some elements of the decimal fraction are found in the works of many European scientists in the XII, XIII, XIV centuries. The complete theory of decimal fractions was given by the Uzbek scientist Jamshid Ghiyaseddin al-Kashi in the book “The Key to Arithmetic,” published in 1424. But this work did not reach European scientists in a timely manner. Only 150 years after the publication of this book (1585), the Flemish scientist Simon Stevin, in his book “On Decimal,” described the rules for operating with decimal fractions. He is considered the inventor of decimal fractions. Stevin wrote decimal fractions like this: 0.3752= or 5.693= Other authors used the notation 3.7= 3 7 or 3/7, or the whole part was written in ink of one color, the fractional part in ink of a different color.


Modern decimals Modern notation, i.e. separating the whole part of a comma, suggested Kepler (gg.). Modern recording, i.e. separating the whole part of a comma, suggested Kepler (gg.). In countries where they speak English (England, USA, Canada, etc.), even now they write a dot instead of a comma, for example, 2,3 they write 2.3 and read: two dot three. In countries where they speak English (England, USA, Canada, etc.), even now they write a dot instead of a comma, for example, 2,3 they write 2.3 and read: two dot three.



Slide 2

Slide 3

Introduction

Slide 4

For several millennia, humanity has been using fractional numbers, but they came up with the idea of ​​writing them in convenient decimals much later.

Slide 5

In Ancient China they already used the decimal system of measures, denoting fractions in words using measures of length CHI: tsuni, fractions, ordinal, hairs, finest, cobwebs.

Slide 6

Fraction 2.135436 looked like this:

2 chi, 1 cun, 3 lobes, 5 ordinal, 4 hairs, 3 finest, 6 cobwebs. 2 zhang, 1 chi, 3 cun, 5 lobes, 4 ordinal, 3 hairs, 6 finest, 0 cobwebs. In the 5th century, the Chinese scientist Tszyu-Chun-Zhi accepted not “CHI” as a unit, but 1ZHANG=10 CHI. Drobvida 2.135436 looked like this:

Slide 7

The Arab mathematician al-Uklisidi tried to write down the decimal fraction using numbers and certain signs in the 10th century in the “Book of Sections on Indian Arithmetic.” Some elements of the decimal fraction are found in the works of many European scientists in the 12th - 14th centuries.

Slide 8

The complete theory of decimal fractions was given by the Uzbek scientist Jamshid Ghiyaseddin al-Kashiv in the book “The Key to Arithmetic,” published in 1424, in which he showed the recording of fractions in one line with numbers in the decimal system and gave rules for operating with them. The scientist used several ways to write fractions: he used either a vertical line or black and red ink. But this work did not reach European scientists in a timely manner!

Slide 9

From the history of decimals

Hartmann Beyer (1563-1625) "Decimal Logistics"

Slide 10

From the history

Al-Kashi Jemshid Ibn Masud For example: the number 2.75 looked like this: 275 or 2 / 75 Simon Stevin: For example: the number 24.56 looked like this: 2456 012

Slide 11

In his book “The Tenth,” he not only sets out the theory of decimal fractions, but also tries to convince people to use them, saying that when they are used, “difficulties, strife, errors, losses and other accidents, the usual companions of calculations, are eliminated.” He is considered the inventor of decimal fractions. Only at the end of the 16th century the idea of ​​writing fractional numbers in decimals came to a certain Simon Stevin from Flanders. In his book “The Tenth” (1585), he sets out the theory of decimal fractions and proposes writing the digits of a fractional number on the same line with the digits of a whole number, while numbering them. For example, the number was written like this: 0.3752 = or 5.13=

Slide 12

From the history of decimals

Here's how they would write the number 3.1415: Girard Albert (1595, Saint-Mihiel - 1632, The Hague), Dutch mathematician, student of Simon Stevin. 3 1 4 1 5 0 1 2 3 4 0 I II III IV 3. 1 4 1 5 3 1415 S. Stevin J. H. Beyer A. Girard

Slide 13

1617 - Scottish mathematician John Napier proposed separating decimals from a whole number with either a comma or a period. 1592 - a comma is used for the first time in writing fractions. 1571 - Johannes Kepler proposed the modern notation of decimal fractions, i.e. separating the whole part by a comma. Before him, there were other options: 3.7 was written as 3(0)7 or 3\ 7 or integer and fractional parts in different inks. 1703 - In Russia, the doctrine of decimal fractions was presented by L.F. Magnitsky in the textbook “Arithmetic, that is, the science of numerals.” In countries where they speak English (England, USA, Canada, etc.), they still write a period instead of a comma, for example: 2.3





















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For many centuries, in the languages ​​of peoples, a broken number was called a fraction. The need for fractions arose at an early stage of human development. So, apparently, dividing a dozen fruits among a large number of participants in the hunt forced people to turn to fractions. The first fraction was half. In order to get half from one, you need to divide the unit, or “break” it into two. This is where the name broken numbers comes from. Now they are called fractions. There are three types of fractions: units (aliquots) or fractions (for example, 1/2, 1/3, 1/4, etc.). Systematic, i.e. fractions in which the denominator is expressed by a power of a number (for example, a power of 10 or 60, etc.). General type, in which the numerator and denominator can be any number. There are “false” fractions - irregular and “ real” – correct.

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Writing fractions in Egypt The Egyptians tried to write all fractions as sums of fractions, that is, fractions of the form 1/n. For example, instead of 8/15 they wrote 1/3 + 1/5. The only exception was the fraction 2/3. In the Ahmes papyrus there is a task: “Divide 7 loaves among 8 people.” If you cut each loaf 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 fractions: 1/2 + 1/4 + 1/8. This means that each person should be given half a loaf, a quarter of a loaf, and an eighth of loaf; Therefore, we cut four loaves in half, two loaves into 4 parts and one loaf into 8 shares, after which we give each one a part.

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Adding such fractions was inconvenient. After all, both terms can contain equal parts, and then upon addition a fraction of the form 2/n will appear. But the Egyptians did not allow such fractions. Therefore, the Ahmes papyrus begins with a table in which all fractions of this type from 2/5 to 2/99 are written as sums of shares. This table was also used to divide numbers. The Egyptians also knew how to multiply and divide fractions. But to multiply, you had to multiply fractions by fractions, and then, perhaps, use the table again. The situation with division was even more complicated.

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The Babylonians took a completely different path. They only worked with sexagesimal fractions. Since the denominators of such fractions are the numbers 60, 602, 603, etc., then fractions such as 1/7, 1/11, 1/13 could not be accurately expressed through sexagesimal ones: they were expressed approximately through them. We still use such fractions to denote time and angles. For example, the time is 3h.17m.28s. can be written like this: 3.17 "28" hours (read 3 whole, 17 sixties 28 three thousand six hundredths of an hour). Instead of the words “sixtieths”, “three thousand six hundredths” they said in short: “first small fractions”, “second small fractions”. From this came the words minute (in Latin - lesser) and second (from Latin - second). The Babylonian way of notating fractions has retained its significance to this day. Since the Babylonians' number system was positional, they worked with sexagesimal fractions using the same tables as for natural numbers.

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Interesting system fractions were in Ancient Rome. It was based on dividing a unit of weight into 12 parts, which was called ass. The twelfth part of an ace was called an ounce. And the path, time and other quantities were compared with a visual thing - weight. For example, a Roman might say that he walked seven ounces of a path or read five ounces of a book. In this case, of course, it was not a question of weighing the path or the book. This meant that 7/12 of the journey had been completed or 5/12 of the book had been read. And for fractions obtained by reducing fractions with a denominator of 12 or splitting twelfths into smaller ones, there were special names.

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The Roman system of fractions and measures was duodecimal. Even now they sometimes say: “He studied this issue thoroughly.” This means that the issue has been studied to the end, that not even the slightest ambiguity remains. And the strange word “scrupulous” comes from the Roman name for 1/288 assa - “scrupulus”. The following names were also in use: “semis” - half an ace, “sextanes” - a sixth of it, “semiounce” - half an ounce, that is, 1/24 of an ace, etc. In total, 18 different names for fractions were used. To work with fractions, it was necessary to remember both the addition table and the multiplication table for these fractions. Therefore, the Roman merchants knew for sure that when adding triens (1/3 assa) and sextans, the result is semis, and when multiplying imp (2/3 assa) by sescunce (3/2 ounce, that is, 1/8 assa), an ounce is obtained. To facilitate the work, special tables were compiled, some of which have come down to us.

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Greece The Greeks associated the study of relationships and fractions with music. In addition to arithmetic and geometry, Greek mathematics included music. The Greeks called music that part of arithmetic that deals with relationships and proportions. The Greeks also created the scientific theory of music. They knew: the longer the stretched string, the “lower” the sound it makes; that a short string produces a high sound. However, musical instrument not one, but several strings, and in order for all the strings to sound “in agreement” when played, pleasing to the ear, the length of their sounding parts must be in a certain ratio. For example, for the pitches of the sounds produced by two strings to differ by an octave, their lengths must be in a ratio of 1:2. Likewise, a fifth has a ratio of 2:3, a fourth a ratio of 3:4, etc.

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From the history of fraction notation Modern system notation of fractions with a numerator and denominator was created in India. Only there they wrote the denominator at the top and the numerator at the bottom and did not write a fractional line. The Arabs began to write fractions exactly as they do now. In Ancient China, they used a decimal system of measures and denoted fractions in words using chi length measures: tsuni, fractions, ordinal, hairs, finest, cobwebs. A fraction of the form 2.135436 looked like this: 2 chi, 1 cun, 3 lobes, 5 ordinal, 4 hairs, 3 finest, 6 cobwebs. Fractions were written this way for two centuries, and in the 5th century the Chinese scientist Tszyu-Chun-Zhi accepted not chi as a unit, but zhang = 10 chi, then this fraction looked like this: 2 zhang, 1 chi, 3 tsun, 5 shares, 4 ordinal, 3 hairs, 6 finest, 0 cobwebs.