Historical remarks on the calculus of fractions

The oldest records used already fractions. About 4000 years ago, the Egyptian calculating book of Ahmes - the Papyrus Rhind - mentions this calculus in great detail. Egyptians only computed with simple fractions for which they placed a point on top of the number: =1/2, =1/3. etc. Other fractions had to be expressed in terms of fractions with numerators 1 and 2. Ahmed's book presents also decomposition of fractions of the form 2/(2n + 1) into simple fractions, for example, 2/5 = 1/3 + 1/15, 2/13 = 1/8 + 1/32 + 1/104, etc.; it shows how, after some practice, one perform the four kinds of operations with the aid of such sums.

Babylon was a second centre of the calculus. Babylonians, with whom astronomy reached a high level of development, preferred numbers which they could relate to astronomy. The sky's band within which the planets, the Sun and the Moon move, had a principal role and was subdivided into 12 images of animals. We have not been able to find out, whether this subdivision existed earlier. It occurred with the Babylonians from the start and is considered to be an ancient Babylonian tradition. Five planets were known to the Babylonians. Today, through telescopes, we know many more. 5 and 12 are numbers which recur frequently with the Babylonians. The year was subdivided into 12 months, the day into 12 double hours or Kaspu. The subdivision of the year by 12 then yielded 30 days per month. The additional days of the year were accounted for separately. Correspondingly, a double hour was subdivided into 30 double minutes. The week had 5 days, not 7 as today.

The subdivision of the circle into 360 degrees also occurred first in Babylon. The annual cycle was decomposed into 12 animal sector images, as well as into 24 sectors, the so-called Moon stations. This subdivision, transferred to the daily circuit, yielded the day's subdivision into 24 hours. The Babylonians developed a measuring system for time and space which they believed to have been discovered in the great book of the heavens.

The number 60, with many factors very suitable for mental arithmetic, was their basic number; it also turned up in their measures: 1 hour - 60 minutes, 1 minute = 60 seconds, etc. They grouped the numbers, not as we do today in tens, but in sixties. As with us the tens, hundreds, etc., the higher units received special names. For example, 1 Soss = 60 units, 1 Sar = 3600 units. The number 3721 thus became in terms of Sar and Sos 1.2.1. The sub-units thus were 1/60, 1/3600, etc. They always used fractions with 60, 3600, etc. in the denominator, in other words, they used the Sexagesimal System.

In comparison with the Egyptian system, the Babylonian calculus of fractions, effectively a forerunner of our decimal calculus of fractions, had the advantage that one deals with equally named fractions which not only can be compared with our simple fractions, but were readily added and subtracted. This advantage assisted the popularisation of the sexagesimal system at the end of the third century A.C. Their time and circle subdivision is still used today. Sexagesimal fractions were still used in Europe in the Middle Ages.

Like the numbers 5 and 12, also the number 7 (5 planets, Sun and Moon) had a certain meaning. You encounter it again in the subdivision of the week into 7 days. The spheres are the 7 orbits of the planets beside Sun and Moon; in their courses along the animal circuits, they generate harmonic sounds, which, however, are inaudible. In the theory of music, the scale again has 7 tones, to which were added later 5 half-tones: 5 + 7 = 12. Newton wrote about the 7 colours of the rainbow, although there are arbitrarily many of them, etc. Nevertheless, the number 7 has not contributed to the setting up of a number system, because it has no divisors apart from 7 and 1, while 60 has altogether 12 subdivisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

The Romans, a nation of administrators and lawyers, made 12 into the base number of their system. Their attitude towards mathematics was one of helplessness. However, since calculations were important in legal conflicts, one can perhaps explain thus the conspicuous phenomenon that they went their own way with the calculus of fractions. They created the so called 12 system. Their fractions had the denominator 12. Thus, 12/12 = 1 as, the subdivisions, the so called minutiae 11/12, 10/12, etc. had special names. A fraction such as 1/8 became 1½/12, while with many other fractions they used approximations. Addition and subtraction were as simple as in the Babylonian system, as the denominators were equal . The results of multiplications and divisions were mostly obtained from tables. This 12 or duodecimal system was widely used as late as the 12th Century. We find its traces, for example, in such concepts as the dozen = 12 items, the gross = 12 dozen items. In England, until decimation, the shilling had 12 pence.

Another stage of the development of the fraction calculus occurred in India. As a result of their position system, they were able to simplify calculations. They gave their arithmetic rules a form, which was already very similar to those of today. About 600 A.C., we can find a mode of writing almost like that in use today. They placed the numerator above the denominator: 2/3. In a mixed fraction, the integer was placed above the numerator, whence 2¾ took the form ²3₄, the numbers being placed above each other. The fraction bar is probably an Arabic contribution. It appears first in the Liber abaci (1202) of Leonardo of Pisa (the Italian mathematician Leonard Fibonacci (~ 1170 - ~ 1250)), who followed largely an Arabic manuscript.

The Greeks learned from the Egyptians and Babylonians the calculus of fractions and extended it. This Greek knowledge was taken over by the Arabs in combination with that of the Indians, so that we find in their treatises the Greek-Egyptian fraction rules, the Greek-Babylonian sexagesimal fractions and the pure Indian fraction calculus. Europe came into contact with these methods through the Arabs, but continued at first to use simultaneously the Roman method. While in the Middle Ages the scholars and better caculators had good command of the calculus of fractions, it was quite difficult to learn. Many people were afraid of it, especially since it had inherent difficulties. For example, the product of two fractions had to be smaller than each of its factors. This rule, which apparently was in contradiction to the products of integers, became a real obstacle to understanding and was often established with great subtleties in the Arithmetic books. The calculus of fractions had with people the reputation of being the most difficult part of it, a fact which still is with us in the phrase "to get lost in fractions".

Johann Müller of Königsberg in Franken (in Germany) (1436 - 1476) with the Latin name Regiomontanus undertook the unavoidable and necessary step to the decimal calculus which spelled the end of the use of other systems. However, a real decimal fraction system did then still not exist. Approximately one century passed until the French statesman François Vieta (1540 - 1603) and the Dutchman Simon Stevin (~1548 - ~ 1620) brought the manner of writing and calculations of ordinary decimal fractions to the level required for practical manipulations.

The advantages of decimal fractions, as compared with ordinary fractions, arises from the fact that, as in the sexagesimal and duodecimal systems, one has fractions with the same names. But its advantages are greater. One can compute with decimal fractions as with integers and must only take into account the decimal point. Also the manner of writing of these fractions is simpler than that of ordinary fractions.

Stevin, who recognized the extraordinary value of the decimal subdivision, demanded with far-sightedness from Governments the introduction of decimal coins, measuring and weight systems, in order to exploit fully its advantages. This demand was only met in 1799, that is, 200 years later, in France after the revolution. Almost all developed countries, with the exception of England, followed then. Germany introduced such systems in 1872, Russia and Turkey after the First World War. There was great resistance to this step which is reflected in the measures used still today for areas such as acre and rai.

Development tends to replace all measures, which do not fit into the decimal system, by decimal ones. For example, the scale of the thermometer into 100º was obvious, not the scales of Daniel Gabriel Fahrenheit (180) and René Antoin Réaumur (80). However, these trends also encountered hurdles. Thus in the decimal system 

For a given decimal number, the subdivisions after the decimal point are then clearly: 

Since the enormous advantages of decimal fractions, as compared with ordinary fractions, have been recognized, the importance of the ordinary fraction calculus has receded. It is now only being used with small numbers. You should restrict yourself to small denominators and execute all other calculations with decimal numbers. Results of common fractions are rarely "more exact" than those of decimal fractions; however, you must not be led astray, since the number of valid digits then becomes important.

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