Fracture: the history of fractions. The history of the appearance of ordinary fractions
One of the most difficult sections of mathematics to this daya day is considered a fraction. The history of fractions has more than one millennium. The ability to divide the whole into parts arose in the territory of ancient Egypt and Babylon. Over the years, the operations performed with fractions became more complicated, the form of their recording changed. Each state of the ancient world had its own characteristics in the "relationship" with this section of mathematics.
What is a fraction?
When there was a need to divide the whole intoparts without too much effort, then there were fractions. The history of fractions is inextricably linked with the solution of utilitarian problems. The term "fraction" has Arabic roots and comes from the word "break, divide". Since ancient times, in this sense, little has changed. The modern definition is as follows: a fraction is a part or a sum of parts of a unit. Accordingly, examples with fractions are the sequential execution of mathematical operations with fractions of numbers.
Today, there are two ways of recording them. Ordinary and decimal fractions arose at different times: the first are more ancient.
Came from the depths of centuries
For the first time, fractions were started in the territoryEgypt and Babylon. The approach of mathematicians of the two states had significant differences. However, the beginning was laid out the same way in both places. The first fraction was half or half. Then came a quarter, a third, and so on. According to archaeological excavations, the history of the occurrence of fractions is about 5 thousand years. For the first time, the numbers are found in Egyptian papyri and on Babylonian clay tablets.
Types of ordinary fractions today includeand the so-called Egyptian. They are the sum of several terms of the form 1 / n. The numerator is always one, and the denominator is a natural number. There are such fractions, no matter how hard to guess, in ancient Egypt. In the calculations, all the shares were tried to be written in the form of such sums (for example, 1/2 + 1/4 + 1/8). Separate designations only had fractions 2/3 and 3/4, the rest were divided into terms. There were special tables in which the fractions of the number were presented as a sum.
The oldest known mention of such asystem occurs in the Mathematical papyrus of Rind, dated to the beginning of the second millennium BC. It includes a table of fractions and mathematical problems with solutions and answers, represented in the form of fractions. The Egyptians were able to add, divide and multiply the number fractions. Fractions in the Nile valley were recorded with the help of hieroglyphics.
Representing the fraction of a number as a sum of termstype 1 / n, characteristic of ancient Egypt, was used by mathematicians not only of this country. Up to the Middle Ages, Egyptian pellets were used on the territory of Greece and other countries.
Development of mathematics in Babylon
Mathematics in the Babylonian kingdom looked otherwise. The history of the appearance of fractions is directly related to the peculiarities of the number system inherited from the predecessor of the ancient state, the Sumerian-Akkadian civilization. Calculating technology in Babylon was more convenient and perfect than in Egypt. Mathematics in this country solved a much larger range of tasks.
To judge the achievements of the Babylonians today,preserved clay tablets, filled with cuneiform. Due to the peculiarities of the material, they have reached us in large numbers. According to some scholars, mathematicians in Babylon before Pythagoras discovered a famous theorem, which undoubtedly testifies to the development of science in this ancient state.
Fracture: the story of fractions in Babylon
The number system in Babylon wassexagesimal. Each new rank differed from the previous one by 60. Such a system has been preserved in the modern world to designate the time and magnitude of angles. Fractions were also sexagesimal. Special characters 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 together with the state. Fractions written in a 60-tier system were used by ancient and Arab astronomers and mathematicians.
The history of ordinary fractions is not much enrichedin ancient Greece. The inhabitants of Hellas believed that mathematics should operate only in whole numbers. Therefore, expressions with fractions on the pages of the ancient Greek treatises were practically never met. However, a certain contribution to this section of mathematics was made by the Pythagoreans. They understood the fractions as ratios or proportions, and the unit was also considered indivisible. Pythagoras and his students built a general theory of fractions, learned to conduct all four arithmetic operations, and also comparison of 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. The 1/12 Assa was called an ounce. To denote fractions, there were 18 titles. Here are some of them:
The inconvenience of such a system was the impossibility of representing a number in the form of a fraction with a denominator of 10 or 100. Roman mathematicians overcame the difficulty by using interest.
Writing of ordinary fractions
In Antiquity, fractions already wrote to friends we knewway: one number above another. However, there was one significant difference. The numerator was located under the denominator. For the first time, fractals began to be written in ancient India. The modern way to us was used by the Arabs. But none of these peoples used a horizontal line to separate the numerator and denominator. For the first time it appears in the writings of Leonardo of Pisa, better known as Fibonacci, in 1202.
If the history of occurrence of ordinary fractionsbegan in Egypt, then the decimal appeared in China for the first time. In the Celestial Empire, they began to be used approximately from the III century BC. The history of decimal fractions began with the Chinese mathematician Liu Hui, who suggested using them in the extraction of square roots.
In the third century AD, decimals in Chinabegan to be used in the calculation of weight and volume. Gradually they began to penetrate deeper into mathematics. In Europe, however, decimal fractions began to be used much later.
Al-Kashi from Samarkand
Regardless of Chinese predecessorsThe decimals were discovered by the astronomer al-Kashi from the ancient city of Samarkand. He lived and worked in the XV century. His theory was expounded in the treatise "The key to arithmetic," which was published in 1427. Al-Kashi suggested using a new form of recording fractions. Both the whole and the fractional part were now written in one line. For their separation the Samarkand astronomer did not use a comma. He wrote the whole number and the fractional part in different colors, using black and red ink. Al-Qashi sometimes used a vertical line to separate.
Decimal fractions in Europe
A new kind of fractions began to appear in the worksEuropean mathematicians from the XIII century. It should be noted that with the labors of al-Qashi, as well as with the invention of the Chinese, they were not familiar. Decimal fractions appeared in the writings of Jordan Nemoraria. Then they were used already in the XVI century by François Viet. The French scientist wrote the "Mathematical Canon", which contained trigonometric tables. In them, Viet used decimals. To separate the whole and fractional part, the scientist used a vertical line, as well as a different font size.
However, these were only special cases of scientificuse. For the solution of everyday tasks, decimals in Europe began to be applied somewhat later. This happened thanks to the Dutch scientist Simon Stevin at the end of the 16th century. He published the mathematical work "The Tenth" in 1585. In it, the scientist expounded the theory of the use of decimal fractions in arithmetic, in the monetary system and for the definition of measures and weights.
Point, point, comma
Stevin also did not use the comma. He separated the two parts of the fraction by zero, circled.For the first time, the comma separated two parts of the decimal fraction only in 1592. In England, however, the point was used instead of it. In the United States, so far, decimals are written in this way.
One of the initiators of using both signspunctuation for the separation of the whole and fractional parts was the Scottish mathematician John Napier. He expressed his proposal in 1616-1617. The German scientist Johann Kepler also used the comma.
Fractions in Russia
On Russian soil, the first mathematiciandivision of the whole into parts, became the Novgorodian monk Kirik. In 1136 he wrote a work in which he outlined the method of "calculating the years". Kirik dealt with questions of chronology and calendar. In his work, he also cited the division of the hour into parts: the fifth, twenty-fifth and so on.
The division of the whole into parts was used in calculating the amount of tax in the XV-XVII centuries. The operations of addition, subtraction, division and multiplication with fractional parts were used.
The very word "fraction" appeared in Russia in the VIII century. It came from the verb "to split up, to divide into parts". For the name of fractions, our ancestors used special words. For example, 1/2 was designated as half or a half, 1/4 - four, 1/8 - one-half, 1/16 - half-hour, and so on.
A complete theory of fractions, which differs little frommodern, was outlined in the first textbook on arithmetic, written in 1701 by Leonti Filippovich Magnitsky. "Arithmetic" consisted of several parts. About the fractions in detail the author tells in the section "About the numbers of broken lines or with fractions". Magnitsky leads operations with "broken" numbers, different designations.
Today is still among the most difficultThe sections of mathematics are called fractions. The history of fractions was also not simple. Different peoples sometimes independently of each other, and sometimes borrowing the experience of predecessors, came to the need for the introduction, mastering and application of fractions of the number. Always the doctrine of fractions grew out of practical observations and due to pressing problems. It was necessary to divide the bread, mark out equal plots of land, calculate taxes, measure time, and so on. The features of the application of 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 division of algebra devoted to the parts of numbers has been formed, developed and successfully used today for a variety of needs, both practical and theoretical.