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Fractions in Babylon, Egypt, Rome. Discovering Decimals PRESENTATION FOR USE AS A VISUAL AID IN EXTRACURRICULAR ACTIVITIES
Markelova G.V., mathematics teacher of the Gremyachinsky branch of the MBOU Secondary School. Keys

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On the origin of fractions
The need for fractional numbers arose as a result of practical human activity. The need to find the shares of a unit appeared among our ancestors when dividing the spoils after a hunt. The second significant reason for the appearance of fractional numbers should be considered the measurement of quantities using the selected unit of measurement. This is how fractions came into being.

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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.

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. Peoples went through many variants of writing fractions until they came to the modern notation.

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In the history of the development of fractional numbers, we encounter fractions of three types:
1) fractions or unit fractions in which the numerator is one, but the denominator can be any integer; 2) systematic fractions, in which the numerators can be any numbers, but the denominators can only be numbers of some particular type, for example, powers of ten or sixty;

3) general fractions in which the numerators and denominators can be any numbers. The invention of these three different types of fractions presented varying degrees of difficulty for mankind, so different types of fractions appeared in different eras.

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Fractions in Babylon

The Babylonians used only two numbers. A vertical line meant one unit, and an angle of two lying lines meant ten. They made these lines in the form of wedges, because the Babylonians wrote with a sharp stick on damp clay tablets, which were then dried and fired.

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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.

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Sexagesimal fractions
In ancient Babylon, a constant denominator of 60 was preferred. Sexagesimal fractions, inherited from Babylon, were used by Greek and Arab mathematicians and astronomers. 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. In this respect, sexagesimal fractions can be compared to our decimal fractions. Instead of the words “sixtieths”, “three thousand six hundredths” they said in short: “first small fractions”, “second small fractions”. This is where our words “minute” (Latin for “lesser”) and “second” (Latin for “second”) come from. So the Babylonian way of notating fractions has retained its meaning to this day.

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"Egyptian fractions"
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 so-called aliquot fractions (from the Latin aliquot - several) of the 1/n type - they are therefore sometimes also called “Egyptian”; these fractions had their own spelling: an elongated horizontal oval and under it the designation of the denominator. They wrote the remaining fractions as a sum of shares. The fraction 7/8 was written as fractions: ½+1/4+1/8.

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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 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.
1 troy ounce of gold - a measure of the weight of precious metals

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Discovering Decimals
For several millennia, humanity has been using fractional numbers, but they came up with the idea of ​​writing them in convenient decimals much later. Today we use decimals naturally and freely. 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.

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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.

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Using Decimals
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. much more often than ordinary fractions.
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.

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Using Decimals
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.

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List of sources
M.Ya.Vygodsky “Arithmetic and algebra in the Ancient World.” G.I. Glazer “History of mathematics at school.” I.Ya. Depman “History of Arithmetic”. Vilenkin N.Ya. “From the history of fractions” Friedman L.M. "We study mathematics." Fractions in Babylon, Egypt, Rome. Discovery of decimal fractions... prezentacii.com›History›Discovery of decimal fractions...mathematics "Fractions in Babylon, Egypt, Rome. Discovery of decimals... ppt4web.ru›…drobi…rime…desjatichnykh-drobejj.html Fractions in Babylon , Egypt, Rome. Discovery of decimal fractions"...powerpt.ru›…drobi-v…rime…desyatichnyh-drobey.html Egypt, Ancient Rome, Babylon. Discovery of decimal fractions."... uchportal.ru›Methodological developments›Discovery of decimal fractions. History of mathematics: ...Rome, Babylon. Discovery of decimal fractions... rusedu.ru›detail_23107.html 9Presentation: ...Ancient Rome, Babylon. Discovery of decimal fractions... prezentacii-powerpoint.ru›…drobi…vavilone…drobej/ Fractions in Babylon, Egypt, Rome. discovery of decimals... prezentacia.ucoz.ru›…drobi_v…desjatichnykh_drobej…

Fractions are still considered one of the most difficult areas of mathematics. The history of fractions goes back more than one thousand years. The ability to divide a whole into parts arose in the territory of ancient Egypt and Babylon. Over the years, operations performed with fractions have become more complex, and the form of their recording has changed. Each had its own characteristics in its “relationship” with this branch of mathematics.

What is a fraction?

When the need arose to divide a whole into parts without extra effort, then fractions appeared. The history of fractions is inextricably linked with the solution of utilitarian problems. The term “fraction” itself has Arabic roots and comes from a word meaning “to break, divide.” Little has changed in this sense since ancient times. The modern definition is as follows: a fraction is a part or sum of parts of a unit. Accordingly, examples with fractions represent the sequential execution of mathematical operations with fractions of numbers.

Today there are two ways to record them. arose at different times: the first are more ancient.

Came from time immemorial

For the first time they began to operate with fractions in Egypt and Babylon. The approach of the mathematicians of the two countries had significant differences. However, the beginning was made in the same way in both cases. The first fraction was half or 1/2. Then a quarter arose, a third, and so on. According to archaeological excavations, the history of the origin of fractions goes back about 5 thousand years. For the first time, fractions of a number are found in Egyptian papyri and on Babylonian clay tablets.

Ancient Egypt

Types of ordinary fractions today include the so-called Egyptian ones. They represent the sum of several terms of the form 1/n. The numerator is always one, and the denominator is a natural number. It’s hard to guess that such fractions appeared in ancient Egypt. When calculating, we tried to write down all shares in the form of such amounts (for example, 1/2 + 1/4 + 1/8). Only the fractions 2/3 and 3/4 had separate designations; the rest were divided into terms. There were special tables in which fractions of a number were presented as a sum.

The oldest known reference to such a system is found in the Rhind Mathematical Papyrus, dating from the beginning of the second millennium BC. It includes a fraction table and math problems with solutions and answers presented as sums of fractions. The Egyptians knew how to add, divide and multiply fractions of a number. Fractions in the Nile Valley were written using hieroglyphs.

The representation of a fraction of a number as a sum of terms of the form 1/n, characteristic of ancient Egypt, was used by mathematicians not only in this country. Until the Middle Ages, Egyptian fractions were used in Greece and other countries.

Development of mathematics in Babylon

Mathematics looked different in the Babylonian kingdom. The history of the emergence of fractions here is directly related to the peculiarities of the number system inherited by the ancient state from its predecessor, the Sumerian-Akkadian civilization. Calculation technology in Babylon was more convenient and more advanced than in Egypt. Mathematics in this country solved a much wider range of problems.

The achievements of the Babylonians today can be judged by the surviving clay tablets filled with cuneiform. Thanks to the peculiarities of the material, they have reached us in large quantities. According to some, a well-known theorem was discovered in Babylon before Pythagoras, which undoubtedly testifies to the development of science in this ancient state.

Fractions: The History of Fractions in Babylon

The number system in Babylon was sexagesimal. Each new digit differed from the previous one by 60. This system has been preserved in the modern world to indicate time and angles. Fractions were also sexagesimal. Special icons were used for recording. As in Egypt, examples with fractions contained separate symbols for 1/2, 1/3 and 2/3.

The Babylonian system did not disappear along with the state. Fractions written in the 60-digit system were used by ancient and Arab astronomers and mathematicians.

Ancient Greece

The history of ordinary fractions was little enriched in ancient Greece. The inhabitants of Hellas believed that mathematics should operate only with integers. Therefore, expressions with fractions were practically never found on the pages of ancient Greek treatises. However, the Pythagoreans made a certain contribution to this branch of mathematics. They understood fractions as ratios or proportions, and the unit was also considered indivisible. Pythagoras and his students built a general theory of fractions, learned to carry out all four arithmetic operations, as well as compare fractions by bringing them to a common denominator.

Holy Roman Empire

The Roman system of fractions was associated with a measure of weight called "ass". It was divided into 12 shares. 1/12 of an ace was called an ounce. There were 18 names for fractions. Here are some of them:

semis - half an assa;

sextante - the sixth part of the ass;

seven ounce - half an ounce or 1/24 ass.

The disadvantage of such a system was the impossibility of representing a number as a fraction with a denominator of 10 or 100. Roman mathematicians overcame the difficulty by using percentages.

Writing common fractions

In Antiquity, fractions were already written in a familiar way: one number over another. However, there was one significant difference. The numerator was located below the denominator. They first started writing fractions this way in ancient India. The modern method was used by the Arabs. But none of the named peoples used a horizontal line to separate the numerator and denominator. It first appears in the writings of Leonardo of Pisa, better known as Fibonacci, in 1202.

China

If the history of the emergence of ordinary fractions began in Egypt, then decimals first appeared in China. In the Celestial Empire they began to be used around the 3rd century BC. The history of decimal fractions began with the Chinese mathematician Liu Hui, who proposed their use in extracting square roots.

In the 3rd century AD, decimal fractions began to be used in China to calculate weight and volume. Gradually they began to penetrate deeper and deeper into mathematics. In Europe, however, decimals came into use much later.

Al-Kashi from Samarkand

Regardless of Chinese predecessors, decimal fractions were discovered by the astronomer al-Kashi from the ancient city of Samarkand. He lived and worked in the 15th century. The scientist outlined his theory in the treatise “The Key to Arithmetic,” which was published in 1427. Al-Kashi proposed using a new form of writing fractions. Both the integer and fractional parts were now written on one line. The Samarkand astronomer did not use a comma to separate them. He wrote the whole number and the fractional part in different colors using black and red ink. Sometimes al-Kashi also used a vertical line to separate.

Decimals in Europe

A new type of fractions began to appear in the works of European mathematicians in the 13th century. It should be noted that they were not familiar with the works of al-Kashi, as well as with the invention of the Chinese. Decimal fractions appeared in the writings of Jordan Nemorarius. Then they were used already in the 16th century by a French scientist who wrote the “Mathematical Canon,” which contained trigonometric tables. Vieth used decimal fractions in them. To separate the whole and fractional parts, the scientist used a vertical bar, as well as different font sizes.

However, these were only special cases of scientific use. Decimal fractions began to be used in Europe somewhat later to solve everyday problems. This happened thanks to the Dutch scientist Simon Stevin at the end of the 16th century. He published the mathematical work "Tenth" in 1585. In it, the scientist outlined the theory of using decimal fractions in arithmetic, in the monetary system and for determining weights and measures.

Dot, dot, comma

Stevin also did not use a comma. He separated the two parts of the fraction using a zero surrounded by a circle.

The first time a comma separated two parts of a decimal fraction was in 1592. In England, however, they began to use a dot instead. In the United States, decimals are still written this way.

One of the initiators of the use of both punctuation marks to separate the integer and fractional parts was the Scottish mathematician John Napier. He expressed his proposal in 1616-1617. The German scientist also used the comma

Fractions in Rus'

On Russian soil, the first mathematician to explain the division of the whole into parts was the Novgorod monk Kirik. In 1136, he wrote a work in which he outlined the method of “counting years.” Kirik dealt with issues of chronology and calendar. In his work, he also cited the division of the hour into parts: fifths, twenty-fifths, and so on.

Dividing the whole into parts was used when calculating the amount of tax in the 15th-17th centuries. The operations of addition, subtraction, division and multiplication with fractional parts were used.

The word “fraction” itself appeared in Rus' in the 8th century. It comes from the verb “to split, to divide into parts.” Our ancestors used special words to name fractions. For example, 1/2 was designated as half or a half, 1/4 as a quarter, 1/8 as a half, 1/16 as a half and so on.

The complete theory of fractions, not much different from the modern one, was presented in the first textbook on arithmetic, written in 1701 by Leonty Filippovich Magnitsky. "Arithmetic" consisted of several parts. The author talks about fractions in detail in the section “On numbers broken or with fractions.” Magnitsky gives operations with “broken” numbers and their different designations.

Today, fractions are still among the most difficult branches of mathematics. The history of fractions has not been simple either. Different peoples, sometimes independently of each other, and sometimes borrowing the experience of their predecessors, came to the need to introduce, master and use fractions of numbers. The study of fractions has always grown out of practical observations and thanks to pressing problems. It was necessary to divide bread, mark out equal plots of land, calculate taxes, measure time, and so on. The specifics of using fractions and mathematical operations with them depended on the number system in the state and on the general level of development of mathematics. One way or another, having overcome more than one thousand years, the section of algebra devoted to fractions of numbers has been formed, developed and is successfully used today for a variety of needs, both practical and theoretical.

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* * http://aida.ucoz.ru Horace From “The Science of Poetry” “Son of Albinus! Tell me: if we take five ounces and subtract one, what remains?” - “The third part of the ace.” "Wonderful! Well, you won’t waste your property! And if we add one to the previous five, what will be the total?” - “Half.” (Translation by M. Dmitriev.) http://aida.ucoz.ru

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* http://aida.ucoz.ru * The young Roman was right! Solving this problem we also got: 5/12-1/12=1/3; 5/12+1/12=1/2. http://aida.ucoz.ru

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* http://aida.ucoz.ru “Meticulously” Synonyms: precise, subtle, thorough, neat, conscientious, jewellery, punctual, pedantic, filigree, unforgettable. And this strange word “scrupulous” comes from the Roman name for 1/288 assa - “scrupulus”. http://aida.ucoz.ru

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* http://aida.ucoz.ru * The following names were also in use: “semis” - half an ace, “sextance” - a sixth of it, “semiounce” - half an ounce, that is, 1/24 of an ace, etc. .d. In total, 18 different names for fractions were used. To work with fractions, you had to remember the addition table and the multiplication table. 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 (2/3 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. http://aida.ucoz.ru

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After the victory, Gaius Julius Caesar decided to reward his vanguard and allocated them first 24 ounces, and then 36 ounces. How many aces did the detachment receive? Solution: 24 ounces is 2 asses, and 36 ounces is 3 aces, 3 +2 = 5 asses received by the squad. Answer: 5 aces. Misha Ivanov's problem

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The task of Angelina Glibina In ancient Rome, warriors who showed strength and courage in battle were honored. How many aces did it take to award 6 warriors if each was given 2 aces and 6 ounces? Solution: 6 multiplied by 2 aces, we get 12 aces - this was given for only 6 warriors, then we multiply 6 by 6, we get 36 ounces, and in one ace there are 12 ounces, we get 3 asses, we add 3 to 12, we get 15 asses . Answer: 15 aces.

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 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.

Slide 12 from the presentation "The History of Fractions".

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History of the origin of fractions

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.

There were only 18 different fractions in use:

SIMIS - half ace;

SEXSTANCE is the sixth part of it;

SECUNCTION – eighth;

TRIENS - third of the ass;

BES – two thirds;

OUNCE – twelfth of an ace;

SEVEN OUNCE – half an ounce.

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 base 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 Greek fractional notation, 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':

1/2 - half, half

1/3 – third

1/4 – even

1/6 – half a third

1/8 - half

1 / 12 – half a third

1/16 - half a half

1/24 – half and half a third (small third)

1 / 32 – half half half (small half)

1/5 – pyatina

1/7 - week

1/10 - tithe

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 those who do not know fractions do 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)

I.Ya.Depman “History of Arithmetic” (M. Enlightenment, 1959)