# Egyptian numerals

 If characters are not displayed correctly in the following, this is due to the font, see Representation of hieroglyphic numbers in Unicode and Help with display problems .
Numeral in the 3rd millennium BC in the Karnak temple complex in Luxor
Nile flood heights in Egyptian numerals on the White Chapel of Sesostris I.

The Egyptian numeric script (also called Egyptian numerals or numerals ) has been a feature since the beginning of the 3rd millennium BC. Chr. Attested hieroglyphic numerals , with which positive rational numbers ( whole and fractional ) were written additively . In their further development to the hieratic number font, hieratic italic characters took the place of these number hieroglyphs from the middle of the 3rd millennium, with a simplification of the principle of additive character repetition.

## Hieroglyphic numerals

### Natural numbers

The Egyptians used a decimal number system in which there was a separate symbol for each power of ten from 1 to 1,000,000. Any natural number (positive whole number) was written with the largest possible powers of ten, ordered according to size, which were stated so often until the total number was obtained. The pronunciation of the numbers can only be partially reconstructed today, since inscriptions usually only contain the number signs, so it is not known with certainty how tens and ones were pronounced in combination ( twenty-five ). The important numbers are: wa 1; senu 2; chemet 3; fedu 4; diu 5; seresu 6; sefech 7; chemenu 8; pesedj 9; medj 10; djebaty 20 (unsure); maba 30; hem 40; diyu 50; ser 60; sefech 70; chemen 80; pesdjeyu 90; shet 100.

1 10 100 1,000 10,000 100,000 1,000,000
Simple line Beef rack Rope loop Water lily finger Tadpole
or
frog
Heh (ancient Egyptian god of infinity)

Example for the number 305, with three hundreds and five units:

In the 2nd century BC An inscription was placed on the Temple of Horus in Edfu , in which the areas of temple lands were calculated. According to today's, but not certain, interpretation, square and triangular parcels were roughly calculated from the lengths of the sides according to a general formula for quadrilaterals; in triangular parcels, the fourth side was set to zero and the hieroglyph was used as a symbol
("Nothing"). So maybe you already knew the number zero .

### Fractions

In order to be able to carry out the division completely, the Egyptians used common fractions of natural numbers, which they represented by adding up ancestral fractions , ie fractions with the numerator 1, as well as the fraction 2/3. The fractions were originally in smaller units of measure.

General fractions were written by writing the denominator under the symbol of the mouth, which also meant the grain measure Ro (320 Ro = 1 Heqat ) and was abbreviated hieratic with a point, demotic with an oblique line, but the denominator 2 for 2 / 3 was used and 1/2 was the half symbol. To simplify the calculation of fractions, the Egyptians created tables of stem fraction breakdowns of general fractions and used auxiliary numbers that corresponded to the counters of today's fraction calculation. Based on the Egyptian form, stem fractions are now represented in Latin transcription by the overlined denominator and 2/3 by a double overlined 3.

General fractions
2/3 1/2 1/3 1/4 ... 1/9 1/10 1/11 1/12 ...
${\ displaystyle {\ overline {\ overline {3}}}}$ ${\ displaystyle {\ overline {2}}}$ ${\ displaystyle {\ overline {3}}}$ ${\ displaystyle {\ overline {4}}}$ ... ${\ displaystyle {\ overline {9}}}$ ${\ displaystyle {\ overline {10}}}$ ${\ displaystyle {\ overline {11}}}$ ${\ displaystyle {\ overline {12}}}$ ...

...

...

So was z. B. 5/12 written as follows:

${\ displaystyle = {\ frac {1} {3}} + {\ frac {1} {12}} = {\ frac {5} {12}}}$

If the denominator had too many digits, the mouth was only placed above the front digits of the denominator:

${\ displaystyle = {\ frac {1} {331}}}$

## Hieratic and demotic numerals

The hieroglyphs were too cumbersome to write for everyday use, however, so the hieratic script appeared alongside them as their simplified form as early as the middle of the 3rd millennium . Repetitions of numerals were each reduced to a single character. As a result, with four characters for the powers of ten 1, 10, 100 and 1,000 as well as 32 (4 times 8) characters for their multiplication, a system of a total of 36 number characters was available for writing the numbers 1 to 9,999. By eliminating the repetition of characters, a four-digit number could be written as a sequence of a maximum of four hieratic numerals instead of a maximum of 36 hieroglyphic characters. From the middle of the 7th century BC A further simplification to the demotic script took place. The hieratic and demotic numerals remained in use until they were replaced by the Greek numbers in the Hellenistic period .

## literature

• Georges Ifrah: Universal History of Numbers. (Translation from the French by Alexander von Plasen, editor Peter Wanner) Special edition of the 2nd edition, Parkland, Cologne 1998, ISBN 3-880-59-956-4 , p. 230 ff., P. 265 ff.
• Kurt Vogel : Pre-Greek Mathematics. Volume I: Prehistory and Egypt (= mathematical study books. No. 1). Schroedel, Hanover; Schöningh, Paderborn 1958.

## Individual evidence

1. ^ Alan Gardiner: Egyptian Grammer: being an introduction to the study of hieroglyphs. 3rd, revised edition, Griffith institute / Ashmolean museum, Oxford 1979, ISBN 978-0-900416-35-4 , pp. 191-192.
2. Helmuth Gericke: Mathematics in antiquity and the Orient. Springer, Berlin a. a. 1984, ISBN 978-0-387-11647-1 , pp. 58-60.
3. K. Vogel: Pre-Greek Mathematics. Vol. I, 1958, p. 44 f.
4. K. Vogel: Pre-Greek Mathematics. Vol. I, 1958, p. 37 ff.
5. K. Vogel: Pre-Greek Mathematics. Vol. I, 1958, p. 34 f.
6. K. Vogel: Pre-Greek Mathematics. Vol. I, 1958, pp. 35 ff.