The history of fractions or who needs them. Presentation on the topic "history of the emergence of ordinary fractions"

History of the origin of fractions

Chuiko A.V.

5, secondary school st. Shokai

Hand. Riplinger L.A.

Introduction

The need for fractional numbers arose in humans at a very early stage of development. Already the division of the spoils, consisting 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 since ancient times have had a need to measure length, area, volume, time and other quantities. The result of measurements cannot always be expressed in a natural number; parts of the measure used must also be taken into account. Historically, fractions originated from the process of measurement.

The need for more accurate measurements led to the fact that the initial units of measure began to be split 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 quantities were measured by this smaller unit.

Fractions in Ancient Rome

The Romans used the basic unit of mass measurement, and also the monetary unit was “ass”. The ass was divided into 12 equal parts - ounces. All fractions with a denominator of 12 were added from them, that is, 1/12, 2/12, 3/12... Over time, ounces began to be used to measure any quantity.

This is how the Romans arose duodecimal fractions, that is, fractions whose denominator has always been a number 12 . Instead of 1/12, the Romans said “one ounce”, 5/12 – “five ounces”, etc. Three ounces was called a quarter, four ounces a third, six ounces a half.

Fractions in Ancient Egypt

For many centuries, the Egyptians called fractions “broken numbers,” and the first fraction they were introduced to was 1/2. It was followed by 1/4, 1/8, 1/16, ..., then 1/3, 1/6, ..., i.e. the simplest fractions called unit or basic fractions. Their numerator is always one. Only much later did the Greeks, then the Indians and other peoples, begin to use fractions of a general form, called ordinary, in which the numerator and denominator can be any natural numbers.

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

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

One of the first known references to Egyptian fractions is the mathematical Rhind papyrus. Three older texts that mention Egyptian fractions are the Egyptian Mathematical Leather Scroll, the Moscow Mathematical Papyrus, and the Akhmim Wooden Tablet. The Rhind Papyrus includes a table of Egyptian fractions for rational numbers of the form 2/ n, as well as 84 mathematical problems, their solutions and answers, written in the form of Egyptian fractions.

The Egyptians put the hieroglyph ( er, "[one] of" or re, mouth) above the number to indicate a unit fraction in ordinary notation, but in sacred texts a line was used. Eg:

They also had special symbols for the fractions 1/2, 2/3 and 3/4, which could also be used to write other fractions (greater than 1/2).

They wrote the remaining fractions as a sum of shares. They wrote the fraction in the form
, but the “+” sign was not indicated. And the amount
written in the form . Consequently, this notation for mixed numbers (without the “+” sign) has been preserved since then.

Babylonian sexagesimal fractions

The inhabitants of ancient Babylon about three thousand years BC created a system of measures similar to our metric one, only it was based not on the number 10, but on the number 60, in which the smaller unit of measurement was part of the higher unit. This system was completely followed by the Babylonians for measuring time and angles, and we inherited from them the division of hours and degrees into 60 minutes, and minutes into 60 seconds.

Researchers explain in different ways the appearance of the sexagesimal number system among the Babylonians. 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 calculations.

Sixtieths were common in the life of the Babylonians. That's why they used sexagesimal fractions whose denominator is always the number 60 or its powers: 60 2, 60 3, etc. In this respect, sexagesimal fractions can be compared to our decimal fractions.

Babylonian mathematics influenced Greek mathematics. Traces of the Babylonian sexagesimal number system have lingered in modern science in the measurement of time and angles. The division of hours into 60 minutes, minutes into 60 seconds, circles into 360 degrees, degrees into 60 minutes, minutes into 60 seconds has been preserved to this day.

The Babylonians made valuable contributions to the development of astronomy. Scientists of all nations used sexagesimal fractions in astronomy until the 17th century, calling them astronomical in fractions. In contrast, the general fractions that we use were called ordinary.

Numbering and fractions in Ancient Greece

Since the Greeks worked with fractions only sporadically, they used different notations. Heron and Diophantus, the most famous arithmeticists among ancient Greek mathematicians, wrote fractions in alphabetical form, with the numerator placed below the denominator. But in principle, preference was given to either fractions with a unit numerator or sexagesimal fractions.

The shortcomings of the Greek notation for fractional numbers, including the use of sexagesimal fractions in the decimal number system, were not due to flaws in the fundamental principles. The shortcomings of the Greek number system can rather be attributed to their insistence on rigor, which markedly increased the difficulties associated with analyzing the relationship of incommensurable quantities. The Greeks understood the word "number" as a set of units, so what we now consider as a single rational number - a fraction - the Greeks understood as the ratio of two whole numbers. This explains why fractions were rarely found in Greek arithmetic.

Fractions in Rus'

In Russian handwritten arithmetic of the 17th century, fractions were called fractions, later “broken numbers.” In old manuals we find the following names of fractions in Rus':

Slavic numbering was used in Russia until the 16th century, then the decimal positional number system gradually began to penetrate into the country. It finally supplanted the Slavic numbering under Peter I.

Fractions in other states of antiquity

In the Chinese “Mathematics in Nine Sections,” reductions of fractions and all operations with fractions already take place.

In the Indian mathematician Brahmagupta we find a fairly developed system of fractions. He comes across different fractions: both basic and derivatives with any numerator. The numerator and denominator are written in the same way as we do now, but without a horizontal line, but are simply placed one above the other.

The Arabs were the first to separate the numerator from the denominator with a line.

Leonardo of Pisa already writes fractions, placing in the case of a mixed number, the whole number on the right, but reads it in the same way as is customary among us. Jordan Nemorarius (XIII century) divides fractions by dividing the numerator by the numerator and the denominator by the denominator, likening division to multiplication. To do this, you have to supplement the terms of the first fraction with factors:

In the 15th – 16th centuries, the study of fractions takes on a form that is already familiar to us and is formalized into approximately the same sections that are found in our textbooks.

It should be noted that the section of arithmetic about fractions has long been one of the most difficult. It’s not for nothing that the Germans still have a saying: “Getting into fractions,” which meant getting into a hopeless situation. It was believed that anyone who does not know fractions does not know arithmetic.

Decimals

Decimal fractions appeared in the works of Arab mathematicians in the Middle Ages and independently of them 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 work “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. Dividing them into 60 parts is not as convenient as dividing them 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 fractions, if introduced instead of sexagesimal ones, 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 scientists of the Middle Ages. In Western Europe 16th century. Along with the widespread decimal system for representing integers, 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 recording 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 he compiled. In 1585 he published the book Tithes, in which he explained decimal fractions.

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

The development of industry and trade, science and technology 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 of weights and measures. For example, in our country, in agriculture and industry, decimal fractions and their special form - percentages - are used much more often than ordinary fractions.

Literature:

M.Ya.Vygodsky “Arithmetic and algebra in the Ancient World” (M. Nauka, 1967)

G.I. Glazer “History of mathematics in school” (M. Prosveshcheniye, 1964)

Abstract of the dissertation

... stories ordinary fractions. 1.1 Emergence fractions. 3 1.2 Fractions in Ancient Egypt. 4 1.3 Fractions in Ancient Babylon. 7 1.4 Fractions in Ancient Rome. 8 1.5 Fractions in Ancient Greece. 9 1.6 Fractions ... origin, – at which the numerator fractions was writing...

1. Topic: “history of ordinary fractions and practical application of knowledge about them”

   Lesson

   Teacher's word stories: Good afternoon! The topic of today's lesson is " Story ordinary fractions and practical... with Babylonian numbering, gives information about sexagesimal fractions. Origin The sexagesimal number system among the Babylonians is associated...

2. History of the Middle Ages, volumes 1 and 2, edited

   Abstract of the dissertation

   Processed jointly by its members, gradually fragmented for small individual families who received... in France. M, 1953. Thierry O. Experience storiesorigin and successes of the third estate // Tvri O. Elect...

History of fractions

Fractions appeared in ancient times. When dividing up spoils, when measuring quantities, and in other similar cases, people encountered the need to introduce fractions.

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 so-called basic fractions): 1/2; 1/3; 1/28; ... . If the Egyptian needed to use other fractions, he represented them as a sum of base fractions. For example, instead of 8/15 they wrote 1/3+1/5. Sometimes it was convenient.

In the Ahmes papyrus there is a task:

And in Egyptian this problem was solved like this: The fraction 7/8 was written in the form of 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, four loaves were cut in half, two loaves into 4 parts and one loaf into 8 shares, after which everyone was given a part of it.

But 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 kind from 2/5 to 2/99 are written as a sum of shares.

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 difficult.

In ancient Babylon they preferred the opposite - a constant denominator equal to 60. Sexagesimal fractions, inherited from Babylon, were used by Greek and Arab mathematicians and astronomers. But it was inconvenient to work on natural numbers written in the decimal system and fractions written in the sexagesimal system. But working with ordinary fractions was already quite difficult. Therefore, the Dutch mathematician Simon Stevin proposed switching to decimal fractions

An interesting system of fractions was 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.

Even now they sometimes say: “He studied this issue carefully.” This means that the issue has been studied to the end, that not even the smallest ambiguity remains. And the strange word “scrupulously” comes from the Roman name for 1/288 assa - “scrupulus”. The following names were also in use: “semis” - half an ass, “sextans” - a sixth of it, “semiounce” - half an ounce, i.e. 1/24 asses, etc. In total, 18 different names for fractions were used. To work with fractions, you had to remember the addition table and the multiplication table for these fractions.

Therefore, the Roman merchants firmly knew that when adding triens (1/3 assa) and sextans, the result is semis, and when multiplying imp (2/3 assa) by sescunce (2/3 ounce, i.e. 1/8 assa), the result is an ounce . To facilitate the work, special tables were compiled, some of which have come down to us.

The modern system of writing 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.

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Recording 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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Babylon 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 had a positional number system, they worked with sexagesimal fractions using the same tables as for natural numbers.

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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 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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From the history of ordinary fractions Work of 6th grade student Daniil Kakurin Supervisor: Rozhko I.A.

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We have such a fraction, The whole story will go on about it, It consists of numbers, And between them, like a bridge, The fraction line lies, Above the line is the numerator, Know, Below the line is the denominator, Such a fraction must certainly be called ordinary.

Object of research: History of the emergence of ordinary fractions Subject of research: Ordinary fractions Hypothesis: If there were no fractions, could mathematics develop? Research methods: - working with literature - searching for information on the World Wide Web - working with fractions in a game form Purpose of the work: - expanding knowledge about the origin of fractions - studying the sequence of improving the recording of ordinary fractions Tasks: do an analysis: - why are fractions written this way? - who came up with such notations? - is there any further development?

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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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The first European scientist who began to use and disseminate the modern notation of fractions was the Italian merchant and traveler Fibonacci (Leonardo of Pisa). In 1202 he introduced the word fraction.

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Fractions in Ancient Egypt.

The first fraction was half. It was followed by 1/4,1/8,1/16,..., then 1/3,1/6, etc., i.e. the simplest fractions, parts of a whole, called units. The ancient Egyptians expressed any fraction as the sum of only the base fractions. The Egyptians wrote on papyri, that is, on scrolls made from the stems of large tropical plants that bore the same name. The most important in content is the Ahmes papyrus, named after one of the ancient Egyptian scribes. By whose hand it was written. Its length is 544 cm and width is 33 cm.

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It is kept in London, in the British Museum. It was acquired in the last century by the Englishman Rind and is therefore sometimes called the Rind papyrus. This ancient mathematical document is entitled: “Ways by which one can come to understand all the dark things, all the secrets contained in things.”

The papyrus is a collection of solutions to 84 problems of an applied nature; these problems relate to operations with fractions, determining the area of ​​a rectangle, there are also arithmetic problems on proportional division, determining the relationship between the amount of grain and the resulting bread or beer, etc. However, for solving these problems no general rules are given, not to mention already about attempts at some theoretical generalizations.

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In the Papyrus of Ahmes there is such a task - to divide the seven loaves equally between eight people.

A modern schoolchild would most likely solve the problem this way: you need to cut each loaf into 8 equal parts and give each person one piece from each loaf. And here is how this problem was solved on papyrus: Each person should be given half, a quarter and an eighth of bread. Now it is clear that you need to cut 4 loaves in half, 2 loaves into 4 pieces, and only one loaf into 8 pieces. And if our schoolchild would have to make 49 cuts, then Ahmes would have to make only 17, i.e. the Egyptian method is almost 3 times more economical.

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To decompose non-unit fractions into the sum of unit ones, there were ready-made tables, which Egyptian scribes used for the necessary calculations.

This table helped to carry out complex arithmetic calculations in accordance with accepted canons. Apparently, the scribes memorized it, just as schoolchildren now memorize the multiplication table. 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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Babylon.

In ancient Babylon, a high level of culture was achieved in the third millennium BC. The Sumerians and Akkadians who inhabited Ancient Babylon wrote not on papyrus, which did not grow in their country, but on clay. By pressing a wedge-shaped stick on soft clay tiles, signs that looked like wedges were applied. That is why such writing is called cuneiform.

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The vertical wedge was designated 1; 60; 602; 603,...The horizontal wedge meant 10. To write 62 we did this: the gap

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Fractions in Ancient Rome.

An interesting system of fractions was in ancient Rome. It was based on dividing a unit of weight into 12 parts, which was called ass. The twelfth share 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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Ancient Greece.

Fractions were not found in Greek works on mathematics. Greek scientists believed that mathematics should deal only with integers. They left fractions to be tinkered with by merchants, artisans, as well as land surveyors, astronomers and mechanics. But the old proverb says: “Drive nature through the door, it will fly in through the window.” Therefore, fractions penetrated into the strictly scientific works of the Greeks, so to speak, “from the back door.” In Greece, along with unit, “Egyptian” fractions, common, ordinary fractions were also used. Among the different notations, the following was used: the denominator is on top, the numerator of the fraction is below it.

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Even 2-3 centuries before Euclid and Archimedes, the Greeks were fluent in arithmetic operations with fractions. In the VI century. BC. lived the famous scientist Pythagoras.

It is said that when asked how many students attended his school, Pythagoras replied: “Half study mathematics, a quarter study music, the seventh are silent, and besides this, there are three women.”

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In Rus', fractions were called fractions, later “broken numbers.” For example, these fractions were called generic or basic. Half, half –1 2 Quarter – 1 4 Half – 1 8 Half and half – 1 16 Pyatina – 1 5 Third – 1 3 Half a third –1 6

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From the history of fraction notation.

The modern system of writing 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. In the 15th century, in Uzbekistan, the mathematician and astronomer Jamshid Giyaseddin al-Kashi wrote down the fraction in one line with numbers in the decimal system and gave rules for operating with them. He used several ways to write fractions: either he used a vertical line, or black and red ink.

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Old problems with fractions.

In the work of the famous Roman poet of the 1st century BC. e. Horace describes a conversation between teachers and students in one of the Roman schools of this era: Teacher. Let Albin's son tell me how much is left if one ounce is taken away from five ounces? Student. One third. Teacher. Right. You will be able to take care of your property. Solution: 4 oz 4 oz 4 oz Answer: 1/3

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Problem from the Papyrus of Ahmes (Egypt, 1850 BC)

“A shepherd comes with 70 bulls. They ask him: “How many of your large herd are you bringing?” The shepherd replies: “I bring two-thirds of a third of the cattle. Count them!” Solution: 1) 70:2·3=105 heads - this is 1/3 of livestock 2) 105·3=315 heads of livestock Answer: 315 head of livestock

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Thank you for your attention!

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Literature

1.History of arithmetic. Depman, 1965 2.History of mathematics from Descartes to the mid-19th century. Willeitner, 1960 3. Encyclopedia for children Avanta + mathematics. 4.Children's encyclopedia. M., 1965

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