To this day, fractions are considered to be the most difficult sections of mathematics. The history of fractions has more than one millennium. The ability to divide the whole into parts arose on the territory of ancient Egypt and Babylon. Over the years, operations with fractions have become more complicated, and the form of their recording has changed. Each state of the ancient world had its own peculiarities in the "relationship" with this section of mathematics.
What is a fraction?
When it became necessary to divide the whole into parts without unnecessary efforts, then fractions appeared. The history of fractions is inextricably linked with the solution of utilitarian problems. The term "fraction" has Arabic roots and comes from a word meaning "break, divide." Since ancient times, little has changed in this sense. The modern definition is as follows: a fraction is a part or a 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. Ordinary and decimal fractions arose at different times: the former are more ancient.
Came from the depths of time
For the first time, fractions began to be operated on the territory of Egypt and Babylon. The approach of mathematicians of the two states had significant differences. However, the beginning was laid equally there and there. The first fraction was half or 1/2. Then came a quarter, a third, and so on. According to archaeological excavations, the history of the appearance of fractions has about 5 thousand years. For the first time, fractions of the number are found in Egyptian papyrus and on Babylonian clay tablets.
Ancient Egypt
Types of ordinary fractions today include the so-called Egyptian. They represent the sum of several terms of the form 1 / n. The numerator is always one, and the denominator is a natural number. Such fractions appeared, no matter how hard to guess, in ancient Egypt. In the calculations, they tried to record all the shares in the form of such amounts (for example, 1/2 + 1/4 + 1/8). Only 2/3 and 3/4 fractions had separate designations, the rest were divided into terms. There were special tables in which the shares of the number were presented as a sum.
The oldest known mention of such a system is found in the Rinda Mathematical Papyrus dating from the beginning of the second millennium BC. It includes a table of fractions and mathematical problems with solutions and answers presented in the form of sums of fractions. The Egyptians knew how to add, divide and multiply fractions of a number. Fractions in the Nile Valley were recorded using hieroglyphs.
Representation of the fraction of the number as the 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.
The development of mathematics in Babylon
The mathematics in the Babylonian kingdom looked different. The history of the appearance of fractions here is directly related to the features of the number system, which the ancient state inherited from its predecessor, the Sumero-Akkadian civilization. The calculating technique in Babylon was more convenient and perfect than in Egypt. Mathematics in this country solved a much larger range of problems.
One can judge the achievements of the Babylonians today by the preserved clay tablets filled with cuneiform writing. Due to the characteristics of the material, they came to us in large quantities. According to some scholars, mathematicians in Babylon before Pythagoras discovered the famous theorem, which undoubtedly testifies to the development of science in this ancient state.
Fractions: The Story of Fractions in Babylon
The number system in Babylon was six-decimal. Each new category differed from the previous one by 60. Such a system has been preserved in the modern world to designate time and angle values. Fractions were also sixty decimal. For recording, special icons were used. As in Egypt, frac examples contained separate characters for 1/2, 1/3, and 2/3.
The Babylonian system did not disappear with the state. Fractions written in a 60-tier system were used by ancient and Arab astronomers and mathematicians.
Ancient Greece
The history of common fractions is not much enriched in ancient Greece. The inhabitants of Hellas believed that mathematics should only operate with integers. Therefore, expressions with fractions on the pages of ancient Greek treatises practically did not occur. However, a certain contribution to this section of mathematics was made by the Pythagoreans. They understood fractions as relations 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 comparing fractions by bringing them to a common denominator.
Holy Roman Empire
The Roman fraction system was associated with a weight measure called “ass.” She divided into 12 shares. 1/12 Assa was called an ounce. To designate fractions, there were 18 names. Here are some of them:
The inconvenience of such a system was the impossibility of representing the 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 manner familiar to us: one number over another. However, there was one significant difference. The numerator was located under the denominator. For the first time, fractions began to be written in ancient India. The modern way we began to use the Arabs. But none of these peoples used the 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 appearance of ordinary fractions began in Egypt, then decimals first appeared in China. In the Celestial Empire, they began to be used from about the 3rd century BC. The history of decimals began with the Chinese mathematician Liu Hui, who suggested using them to extract square roots.
In the 3rd century AD, decimal fractions in China began to be used in calculating weight and volume. Gradually they began to penetrate deeper into mathematics. In Europe, however, decimals began to be used much later.
Al-Kashi from Samarkand
Regardless of Chinese predecessors, decimal fractions were discovered by 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 saw the light in 1427. Al-Kashi suggested using a new form for recording fractions. Both the integer and fractional parts are now written in one line. The Samarkand astronomer did not use a comma to separate them. He wrote an integer and a 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 kind of fractions began to appear in the works of European mathematicians from 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 Nemorarii. Then they were used in the 16th century by François Viet. The French scientist wrote The Mathematical Canon, which contained trigonometric tables. In them, Viet used decimal fractions. To separate the whole and fractional parts, the scientist used a vertical line, as well as different font sizes.
However, these were only special cases of scientific use. To solve everyday problems, decimal fractions in Europe began to be applied a little later. This happened thanks to the Dutch scientist Simon Stevin at the end of the XVI century. He published the mathematical work "Tenth" in 1585. In it, the scientist outlined the theory of the use of decimal fractions in arithmetic, in the monetary system and for determining measures and weights.
Dot, dot, comma
Stevin also did not use a comma. He separated the two parts of the fraction with the help of zero circled in a circle.
For the first time, a comma separated two parts of a decimal only in 1592. In England, however, a dot was used instead. In the United States, decimal fractions are still written this way.
One of the initiators of using both punctuation marks to separate the whole and fractional parts was the Scottish mathematician John Napier. He expressed his proposal in 1616-1617. The German scientist Johannes Kepler also used a comma.
Fractions in Russia
On Russian soil, the first mathematician who stated 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 "reckoning years." Kirik dealt with issues of chronology and calendar. In his work, he cited including the division of hours into parts: fifths, twenty-fifths, 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 "crush, divide into parts." Our ancestors used special words to name fractions. For example, 1/2 was designated as half or half a half, 1/4 - even, 1/8 - half, 1/16 - half and so on.
A 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 the numbers of broken lines or with fractions." Magnitsky gives operations with "broken" numbers, their different designations.
Today, fractions are still called among the most complex branches of mathematics. The history of fractions was also not simple. Different peoples, sometimes independently of each other, and sometimes borrowing the experience of their predecessors, came to the need for the introduction, development and application of fractions of a number. The doctrine of fractions has always grown out of practical observations and thanks to pressing problems. It was necessary to divide the bread, mark out equal tracts of land, calculate taxes, measure time and so on. 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, overcoming more than one thousand years, a section of algebra devoted to the fractions of numbers has formed, developed and is being successfully used today for a wide variety of needs, both practical and theoretical.