What is the Egyptian number system? History, description, examples
With the nonpositional Egyptian number system, which was used in us, the few remaining papyruses visually acquaint them. The examples of tasks and their solutions are so interesting in them that it remains only to regret that there are so few of them.
It can be seen from them that mathematics and the Egyptian number system were closely connected with economic needs and practical application. Every year after the spill of the Nile, it was necessary to restore the buildings, re-land the land plots, calculating the area and borders, keep a record of the harvest, the calendar.
What is positional and nonpositional numbered systems?
The answer lies in the title itself. If the position of a digit affects the result of calculations, we have a positional system of numbers, if not - non-positional.
If we write 12, that's twelve, and with the same numbers, 21 is twenty-one. According to the Egyptian number system: to write 12,you need to use the unit symbol twice and the tens symbol once, and 21 will look like one unit symbol and two tens symbols, that is, you just need to write three characters.
The non-positional ones are: the familiar Roman system, in which the numbers are denoted by Roman letters, the Slavic system, where each letter also denotes a certain number or number. The Roman system coped with its functions in Western Europe until the 16th century.
The number system we use in modern life is a positional decimal system.
Nonpositional systems were well suited to perform simple arithmetic operations, since complex calculations involved cumbersome recordings, which did not prevent the successful development of algebra and geometry in ancient Egypt.
How did the Egyptians think?
What is it - the Egyptian number system? To write any number, we used hieroglyphs, denoting certain numbers, the sum of which was equal to the desired value.
Special designations were available for the numbers 1, 10, 100, 1000, 10000, 100000, 1000000. When writing the required number, each designation was used up to 9 times. Record in the Egyptian number system was in ascending order: first, units, then dozens, hundreds, and so on.
And they wrote, as a rule, from right to left, but it was possible from left to right, the amount of this did not change. Vertical writing was also used, but then the countdown went from top to bottom.
Two ways of writing were used:
- Hieroglyphic, in which the adopted hieroglyphs were used.
- Hieratic, which was more schematic and convenient in practice.
The history of the Egyptian number system originated in ancient times, the first manuscripts with numbers refer to the second millennium BC. There was no money then, so the system was used for both incredible complexity and greatness of mathematical problems, and for solving everyday household problems.
After all, knowledge of mathematics was used in land surveying, and in building calendars and maps in astronomy, navigation, and in the construction of palaces, canals, and military fortifications.
The Egyptian non-positional number system was used until the 10th century AD.
It also had a mystical significance, the secret of which the priests took with them, but partially opened the world to Pythagoras. He has works in which he describes symbolic meanings that are given to digital hieroglyphs, written by him after his stay in Egypt. Therefore, their description belongs to the Egyptian number system.
Only a few papyri of those times survived, by which it can be understood that the level of mathematics was high. It is authentically known that the Greeks studied ancient Egyptian mathematics. One of the secret knowledge is the Egyptian non-positional number system.
Akhmes Papyrus dates back to 1650 BC, contains 84 mathematical tasks. It was found in Thebes, stored in the British Museum.
All tasks in the papyrus considered on specific examples of the Egyptian number system. They show examples of calculations with fractions, with integers, division and multiplication.
Calculations are given for finding the areas of geometric figures: a quadrilateral, a circle, a triangle.
Information from papyrus proves that the Egyptian mathematicians were able to extract the root, make arithmetic and geometric progression, equations with unknowns.
Interestingly, in the calculations, only aliquot fractions were used, in which the numerator was equal to one and was denoted by such a sign, and the denominator values were written below it, and all other fractions for calculations first needed to be expanded to aliquot fractions.But they were used and had the special designation of the fraction 2/3 and 3/4.
To bring the usual fractions in the state of aliquot on the Egyptian number system, it was necessary to work:
4/5 = 16/20 = 10/20 + 5/20 + 1/20 = 1/2+1/4 + 1/20
2/5 = 1/5 + 1/5, 2/7 = 1/4 + 1/28
3/7 = 12/28 = 24/56 = 14/56+7/56+3/56 = 1/4+1/8+1/18+1/56.
Fractions were formed in a modern way: by reducing to a common denominator, for many values there were numerous ready-made tables.
The Egyptians learned the desired result, not knowing the multiplication table, but using the knowledge that if one factor is doubled and the other factor is reduced, the result will not change:
Interestingly, this method of multiplication was known in Russia, and it was believed that it came from Ancient Egypt, and in Europe it was called Russian.
Thanks to the efforts of the scientist-egyptologist V. S. Golenischev, papyrus is stored in Moscow another 200 years older than the papyrus of the scribe Akhmes. The scientist bought it during his work in Thebes.
It was written in the hieratic way, in italics, it deals with 25 problems, given their description according to the Egyptian number system and solution. Its length is more than 5 m with a width of 7 cm. There are no comments to these problems, as in the previous papyrus, there are only mathematical calculations.
It shows that the Egyptians were able to calculate the areas of a triangle, a trapezium, a rectangle, a circle, as well as the volumes of a pyramid, a prism, a parallelepiped, a cylinder and a truncated pyramid with great accuracy, and many formulas completely coincide with modern ones.
In the Egyptian number system, the number pi was 3.16, which almost corresponded to the modern value 3.14, although at that time the value of 3 was used everywhere in the East.
All things are the essence of numbers.
It is believed that Pythagoras lived in Egypt for 22 years, deeply studying the geometry, philosophy, mysticism of numbers. Those discoveries that the Pythagorean school later made could well have been made in ancient Egypt.
Therefore, it is believed that the works of Pythagoras about the mysticism of the numbers that he wrote later are based on the secret knowledge he received from the Egyptian priests. They did not take foreigners to study, he came to them by a high patronage, after an interview with the main priest, who found him worthy to be privy to secrets.
Numbers were living entities, reflecting the properties of space, music, energy. Everything can be expressed through mathematics, describing visible phenomena with formulas, predict invisible, based on logic and mathematical laws.
The height, width of the base, the angle of inclination of the pyramid of Cheops in Egypt correspond to the mathematical rule for constructing the Pythagorean pyramid, which also confirms the interrelation of the discoveries made by him and knowledge obtained from the ancient Egyptian priests who used the Egyptian number system.
Working with numbers, ancient thinkers not only understood the essence of things, but could also influence them.
Studying the mathematics of ancient Egypt, which uses the Egyptian number system, one can only admire how much was discovered by people thousands of years before our era.