Mathematics in Ancient Egypt

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Mathematical Papyrus Rhind

Mathematics in Ancient Egypt refers to the history and application of daily calculation formulas.

overview

The earlier assumptions that ancient Egyptian mathematics developed very late are no longer tenable today. Almost simultaneously with the oldest writings in Mesopotamia and the Middle East , it was written around 3000 BC. Chr. In Egypt , the hieroglyphic writing out of necessity, with the emergence of the central government will be able to meet the requirements of the detention of processes in administration and business records by justice. This also gave rise to the characters for numbers and mathematics began to develop. Already in the late 4th millennium BC BC the Egyptians possessed mathematical knowledge and methods to cope with daily requirements, which concerned the quantitative and spatial relationships in the objective reality. So at the same time as the first evidence of the use of hieroglyphic writing, the first numerical signs can also be proven. After the unification of the empire, the discoveries required for Egyptian mathematics were made and the corresponding arithmetic procedures were developed on the basis of the requirements of the state administration up to around the 3rd dynasty . Only refinements were made later.

Without mathematical knowledge, the pyramid construction would be from approx. 2650 BC. Was not possible. The precisely calculated pyramids are a clear indication of the extensive mathematical knowledge in ancient Egypt. Like Roman numerals , Egyptian numbers were based on an additive system that did not have a separate character or position value description for the zero . In addition to addition and subtraction , fractions and solving equations with one variable were also known. The ancient Egyptians also knew methods for multiplication and division , as shown by arithmetic problems in the Rhind papyrus .

In contrast to finds from the same epoch from Mesopotamia , only a few mathematical calculations are documented from Egypt from the Old Kingdom . In a funerary inscription from the tomb of Metjen in Saqqara from the transition period from the third to the fourth dynasty, the calculation of the area of ​​a rectangle is handed down. Numbers found in temples and on stone monuments, however, give little insight into the types of calculations made. Reasons lie in the cumbersome and laborious writing of mathematical equations on an unsuitable surface. With the introduction of the papyri from the second half of the Middle Kingdom, the findings for mathematical proofs expanded.

swell

Today's knowledge of ancient Egyptian mathematics is mainly passed down through mathematical papyri. These are very similarly structured exercise books or textbooks that contain basic mathematical rules and practical exercise examples for students. The papyri were intended to prepare students for writing to cope with practical problems that awaited them in later daily working life.

The oldest and most famous of the mathematical papyri come from the Middle Kingdom and the Second Intermediate Period :

From the New Kingdom and later come:

numbers

The Egyptians used the decimal system , in contrast to the Babylonians , who calculated with the sexagesimal system (base 60). However, they did not have a position system , but wrote the number signs additively next to each other. For the number 1 they drew a vertical line and up to the number 9 the line was written nine times. At ten times, you took the next higher sign and used the same method again.

No more than seven different hieroglyphic characters were required for practical use . There was also a symbol for zero. In the hieroglyphic system, no digit is used. The symbol was necessary to indicate results (balance sheet) or a reference point (construction).

Hieroglyphic numerals
0 1 10 100 1,000 10,000 100,000 1,000,000
F35
.
Z1
V20
V1
M12
D50
I8
C11

A separate notation was used for the calculation of fractions , which was based on the addition of original fractions . So the break was z. B. as shown. The formation of Egyptian fractions is based on three simple basic rules:

  1. Find the largest stem fraction contained in the given fraction.
  2. Find the difference between these two fractions.
  3. Repeat steps 1 and 2 for the difference until the rest is a trunk fraction.
The representation of the stem fractions was done with the hieroglyph
r
that you put over the corresponding number. For some frequently used fractions like and there were special characters as an exception.
Special signs for breaks
Aa13
D21
Z1
D22
Z9
D23

The representation of numbers, also with compound fractions, enabled many possibilities of division and thus also the representation of small units of measurement and angle differences. The solutions found with the computing technology of the time are admirable.

Basic arithmetic

multiplication

Even if the process is alien to us today, the ancient Egyptians knew a method of multiplying in writing. They use the property that each multiplier can be represented as a sum of powers of 2 .

Ex. 13 * 12 = 156 the ancient Egyptians calculated as follows:

13 * 12       Unter den Multiplikator wird eine 1 geschrieben,
 1   12 /     der Multiplikand unverändert daneben. Dann werden
 2   24       beide Zahlen verdoppelt, bis der Multiplikator
 4   48 /     (in diesem Fall 13) zusammen addiert werden kann.
 8   96 /     (Hier: 8+4+1=13) Addiert man die rechten Zahlen derselben
+______       Zeilen so erhält man das Ergebnis (Hier: 12+48+96=156)
 13  156

division

The division works in a very similar way.

Example 143: 11 = 13 The ancient Egyptians made the following task out of it: Calculate with 11 until you find 143 (→ reverse multiplication)

143 : 11       Unter den Dividend wird die 1 geschrieben, der
  1   11 /     Divisor unverändert daneben.
  2   22       Diesmal muss jedoch sooft verdoppelt werden, bis auf der
  4   44 /     rechten Seite die Zahl des Dividenden zusammen addiert
  8   88 /     werden kann. (Hier: 11+44+88=143) Addiert man die linken
 +______       Zahlen der entsprechenden Zeilen, erhält man das Ergebnis.
 13  143       (Hier: 1+4+8=13)

algebra

In addition to simple arithmetic arithmetic tasks, those with "distributing" goods, food, beer, fodder etc. to a certain number of people or animals are the most common topics. In Task 5 of the Rhind Papyrus, for example, 8 loaves of bread are distributed among 10 people.

geometry

Due to the annually recurring flood of the Nile and the resulting blurring of the field boundaries by the deposited Nile mud, as well as the compulsion to re-divide the fields after the flood had passed, the ancient Egyptians were dependent on planimetric calculations of the areas of triangles and rectangles to avoid endless disputes about land ownership and land use and to develop trapezoids . Because of this practical relevance, geometry played a much greater role than arithmetic . The mathematical knowledge was based almost exclusively on empirical values. It was not just any abstract figures, but triangular or square fields that were calculated. The Egyptians were not concerned with mathematical proofs, but always with calculation rules, with “calculation recipes” with more or less good approximate values ​​as a result. The development of the geometry was closely linked to the needs of practice and oriented towards the requirements of field division and surveying, architecture and construction, as well as the measurement of space. Sesostris I designed the model of the nilometer .

With regard to the construction of grave pyramids they developed over time to calculate the base surface of the jacket, of the volume of a square truncated pyramid through . In the case of the pyramids, there was also speculation about the role of pi (Cheops pyramid) or the golden ratio in their construction or the use of Pythagorean triples . The approximate value (16/9) ² for the number of circles π (pi) was used when calculating the area of ​​the circle, as in the Rhind Papyrus ( problem 48 ). There is also a place in the older Moscow papyrus ( problem 10 ) where this value is used in the calculation rule for a curved surface, but its interpretation is uncertain.

Importance of Egyptian mathematics in ancient times

Egyptian mathematics and computational technology had a considerable influence on the development of a mathematical science in the Greek world. They were highly praised by Greek historians and considered a source of their own knowledge. Herodotus already reported in the 5th century BC Chr. , That the Greeks learned geometry from the Egyptians and the arithmetic of the Babylonern. Even Plato said in the 4th century. V. Chr. After a month-long stay in Heliopolis from the mathematical knowledge in contemporary Egypt with great respect.

See also

literature

  • Helmuth Gericke : Mathematics in Antiquity, Orient and Occident . 9th edition. Marixverlag, Wiesbaden 2005, ISBN 3-925037-64-0 .
  • Richard J. Gillings: Mathematics in the time of the pharaos , MIT Press 1972, Dover 1982
  • Annette Imhausen : Ägyptische Algorithmen: An investigation into the central Egyptian mathematical task texts . Harrassowitz, Wiesbaden 2003, ISBN 3-447-04644-9 .
  • Adel Kamel: A brilliant achievement - mathematics in ancient Egypt . In: Gabriele Höber-Kamel (Ed.): Kemet . Issue 4/2000. Kemet-Verlag, 2000, ISSN  0943-5972 , p. 31-37 .
  • Sybille Krämer: Symbolic machines. The idea of ​​formalization in a historical outline . Scientific Book Society, Darmstadt 1988, ISBN 3-534-03207-1 .
  • Johannes Lehmann: This is how the Egyptians and Babylonians calculated. 4000 years of mathematics in exercises . Urania, Leipzig / Jena / Berlin 1994, ISBN 3-332-00522-7 .
  • Marianne Michel: Les mathématiques de l'Égypte ancienne. Numération, métrologie, arithmétique, géométrie et autresproblemèmes (= Connaissance de l'Égypte ancienne. Volume 12). Editions Safran, Brussels 2014, ISBN 978-2-87457-040-7 .
  • Frank Müller-Römer : Mathematics Lessons in Ancient Egypt. In: Kemet. Volume 20, Issue 4, 2011, pp. 26-30, ISSN  0943-5972 .
  • André Pichot : The birth of science. Parkland-Verlag, Berlin 2000, ISBN 3-88059-978-5 .
  • Kurt Vogel: Pre-Greek Mathematics. Volume 1: Prehistory and Egypt. Schöningh, Paderborn 1958.
  • BL van der Waerden: Awakening Science. Egyptian, Babylonian and Greek mathematics . Birkhäuser, Basel 1966.
  • Armin Wirsching: The pyramids of Giza - mathematics built in stone . 2nd Edition. Books on Demand, Norderstedt 2009, ISBN 978-3-8370-2355-8 .

Notes and individual references

  1. Lexicon of Egyptology . Haarowitz, Wiesbaden 1982, ISBN 3-447-02262-0 , Volume IV, Sp. 118
  2. ^ Adel Kamel: Kemet , Heft 4/2000, p. 31.
  3. George Gheverghese Joseph: The Crest of the Peacock: Non-European Roots of Mathematics . Third edition. Princeton 2011, ISBN 978-0-691-13526-7 , p. 86.
  4. ^ Adel Kamel: Kemet , Heft 4/2000, p. 32.
  5. ^ Adel Kamel: Kemet , Heft 4/2000, p. 33.
  6. ^ Adel Kamel: Kemet , Heft 4/2000, pp. 33-34.
  7. ^ André Pichot: The birth of science . Parkland-Verlag, Cologne 2000, ISBN 3-88059-978-5 , p. 177.
  8. ^ Frank Müller-Römer : Mathematics lessons in ancient Egypt . In: Kemet , 20, 2011, Issue 4, pp. 26-30, ISSN  0943-5972