Egyptian numerals

Numeral systems
Arabic Armenian Attic (Greek) Babylonian Chinese Egyptian Etruscan Greek Hebrew Indian Ionian (Greek) Japanese Khmer Mayan Roman Cyrillic Thai Binary (2) Octal (8) Decimal (10) Hexadecimal (16)
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Numeral systems

Arabic
Armenian
Attic (Greek)
Babylonian
Chinese
Egyptian
Etruscan
Greek
Hebrew
Indian
Ionian (Greek)
Japanese
Khmer
Mayan
Roman
Cyrillic
Thai

Binary (2)
Octal (8)
Decimal (10)
Hexadecimal (16)

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The system of Egyptian numerals was a numeral system used in ancient Egypt. It is a decimal system, written in both hieroglyphs and hieratic.

Contents

1 Digits and numbers

2 Fractions

3 Addition and subtraction

Digits and numbers

The following hieroglyphs were used to denote powers of ten:

Value	1	10	100	1,000	10,000	100,000	1 million, or infinity
Hieroglyph	<hiero>Z1</hiero>	<hiero>V20</hiero>	<hiero>V1</hiero>	<hiero>M12</hiero>	<hiero>D50</hiero>	<hiero>I8</hiero> or <hiero>I7</hiero>	<hiero>C11</hiero>
Description	Single stroke	Cattle hobble or yoke	Coil of rope	Water lily	Finger	Tadpole or Frog	Man with both hands raised

Multiples of these values were expressed by repeating the symbol as many times as needed. For instance, a stone carving from Karnak shows the number 4622 as

Egyptian hieroglyphs could be written in both directions (and even vertically). This example is written left-to-right and top-down; on the original stone carving, it is right-to-left, and the signs are thus inversed.

Fractions

Main article: Egyptian fraction

Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, except for 2/3 and 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part":

Fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Thus, 1/3 was written as:

$= \frac{1}{3}$

There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently):

$= \frac{1}{2}$

$= \frac{2}{3}$

$= \frac{3}{4}$

If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":

$= \frac{1}{331}$

Addition and subtraction

For plus and minus signs, the hieroglyphs

were used: if the feet pointed into the direction of writing, it signified addition, otherwise subtraction.

Written numbers

Besides this numeral system, the ancient Egyptian language could also write out numbers as words, just like one can write "thirty" instead of 30 in English. "Thirty", for instance, was written as

while the number 30 was

This was, however, uncommon for most numbers other than one and two.

Hieratic numerals

As most administrative and accounting texts were written on papyrus or ostraca, rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilise the hieratic script. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period. The Old Kingdom Abusir papyri are a particularly important corpus of texts that utilise hieratic numerals.

It is often thought that hieratic script uses a different numeral system, using individual signs for the numbers 1 to 9, multiples of 10 from 10 to 90, the hundreds from 100 to 900, and the thousands from 1000 to 9000. A large number like 9999 could thus be written with only four signs—combining the signs for 9000, 900, 90, and 9—as opposed to 36 hieroglyphs.

This difference is more apparent that real as these so-called "individual signs" are in fact merely scribal ligatures. In the oldest hieratic texts the individual numerals are clearly written, but during the Old Kingdom a series of standardised writings were developed for sign-groups containing more than one numeral. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing; this process continued into Demotic as well. However, it is incorrect to speak of these ligatured sign-groups as a different numeral system, just as it would be similarly incorrect to speak of a different spelling system when comparing equally ligatured sign-groups in literary hieratic texts with comparable hieroglyphic texts.

Two famous mathematical papyri using hieratic script are the Moscow and Rhind Mathematical Papyri.

References

Allen, James Paul. 2000. Middle Egyptian: An Introduction to the Language and Culture of Hieroglyphs. Cambridge: Cambridge University Press. Numerals discussed in §§9.1–9.6.
Gardiner, Alan Henderson. 1957. Egyptian Grammar; Being an Introduction to the Study of Hieroglyphs. 3rd ed. Oxford: Griffith Institute. For numerals, see §§259–266.
Goedicke, Hans. 1988. Old Hieratic Paleography. Baltimore: Halgo, inc.
Möller, Georg. 1927. Hieratische Paläographie: Die aegyptische Buchschrift in ihrer Entwicklung von der Fünften Dynastie bis zur römischen Kaiserzeit. 3 vols. 2nd ed. Leipzig: J. C. Hinrichs'schen Buchhandlungen. (Reprinted Osnabrück: Otto Zeller Verlag, 1965)