One of the most difficult sections of mathematics to this day are fractions. 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 it became necessary to divide the 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 "break, divide." Since ancient times, little has changed in this sense. The modern definition is as follows: a fraction is a part or the sum of parts of a unit. Accordingly, examples with fractions represent the sequential execution of mathematical operations with fractions of numbers.
Today there are twothe way they are recorded. Ordinary and decimal fractions arose at different times: the former are more ancient.
Come from time immemorial
For the first time they began to operate with fractions on the territory of Egypt and Babylon. The approach of the mathematicians of the two states had significant differences. However, the beginning was the same 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 emergence of fractions has 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 are 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 it is to guess, in ancient Egypt. When calculating all the shares, they tried to write them down in the form of such sums (for example, 1/2 + 1/4 + 1/8). Only 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, dated to the beginning of the second millennium BC. It includes a table of fractions 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. Shots in the Nile Valleywere 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 states.
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 features of the number system inherited by the ancient state from its predecessor, the Sumerian-Akkadian civilization. The calculation technique in Babylon was more convenient and perfect than in Egypt. Mathematics in this country solved a much wider range of problems.
You can judge the achievements of the Babylonians today by the surviving clay tablets filled with cuneiform writing. Due to the characteristics of the material, they have come down to us in large numbers. According to some scientists, mathematicians in Babylon discovered a well-known theorem before Pythagoras, which undoubtedly indicates the development of science in this ancient state.
Fractions: the history of fractions in Babylon
The number system in Babylon was sexagesimal. Each new category differed from the previous one by 60. Such a system has been preserved in the modern world to indicate time and angles. Fractions were also sexagesimal. For recording, special icons were used. As in Egypt, the fraction examples contained separate symbols for 1/2, 1/3, and 2/3.
Babylonianthe system did not disappear with the state. Fractions written in the 60-decimal system were used by ancient and Arabic astronomers and mathematicians.
Ancient Greece
The history of ordinary fractions was not much enriched in ancient Greece. The inhabitants of Hellas believed that mathematics should operate only with whole numbers. Therefore, expressions with fractions on the pages of ancient Greek treatises practically did not occur. However, the Pythagoreans made a certain contribution to this branch of mathematics. They understood fractions as ratios or proportions, and they also considered the unit to be indivisible. Pythagoras and his students built a general theory of fractions, learned how to perform all four arithmetic operations, as well as how to compare fractions by reducing 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 assa was called an ounce. There were 18 names for fractions. Here are some of them:
- semis - half ass;
- sextante - the sixth of the ac;
- semiounce - half an ounce or 1/24 ace.
The inconvenience 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 locatedunder the denominator. For the first time, fractions began to be written in this way in ancient India. The Arabs began to use the modern way for us. But none of these 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 ordinary fractions began in Egypt, 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 proposed using them to extract square roots.
In the III century AD, decimal fractions in China began to be used 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 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 to use a new form of notation for fractions. Both integer and fractional parts were now written in 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. Al-Kashi sometimes also used a vertical bar to separate them.
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 Nemorarius. Then they were used already in the 16th century by Francois Viet. The French scientist wrote the "Mathematical Canon", which contained trigonometric tables. In them, Viet used decimal fractions. To separate the integer and fractional parts, the scientist used a vertical line, as well as a different font size.
However, these were only special cases of scientific use. To solve everyday problems, decimal fractions in Europe began to be used 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 outlined the theory of using decimal fractions in arithmetic, in the monetary system and to determine measures and weights.
Dot, dot, comma
Stevin didn't use a comma either. He separated the two parts of a fraction with a circled zero.
The first time a comma separated two parts of a decimal fraction was only in 1592. In England, however, the dot was used instead. In the United States, decimal fractions are still written in this way.
One of the initiators of the use of both punctuation marks to separate integer and fractional parts was the Scottish mathematician John Napier. He made his proposal in 1616-1617. comma usedand German scientist Johannes Kepler.
Fractions in Russia
On Russian soil, the first mathematician who outlined 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 "calculating years." Kirik de alt 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.
The division of the whole into parts was used when calculating the amount of tax in the XV-XVII centuries. Operations of addition, subtraction, division and multiplication with fractional parts were used.
The very word "fraction" appeared in Russia in the VIII century. It comes from the verb "to crush, divide into parts." Our ancestors used special words to name fractions. For example, 1/2 was designated as half or half, 1/4 - four, 1/8 - half an hour, 1/16 - half an hour 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 of broken lines or with fractions”. Magnitsky gives operations with "broken" numbers, their different designations.
Today, fractions are still among the most difficult sections 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 to introduce, master and use fractions of a number. The doctrine of fractions has always grown out of practical observations and thanks to vitalproblems. It was necessary to divide bread, mark equal plots of land, calculate taxes, measure time, and so on. Features of the use 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 section of algebra devoted to fractions of numbers has formed, developed and is successfully used today for a variety of needs, both practical and theoretical.
