Egyptian numerals
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The system of Egyptian numerals was a numeral system used in ancient Egypt. It was a decimal system, often rounded off to the higher power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, being ciphered one:one onto the Egyptian alphabet.The Egyptian system used bases of ten.
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[edit] 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 |
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| Hieroglyph |
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or
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| Description | Single stroke | Cattle hobble or yoke |
Coil of rope | Water lily (also called Lotus) |
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
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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 reversed.
[edit] Fractions
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":
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Fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Thus, 1/3 was written as:
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There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently):
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If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
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[edit] Addition and subtraction
For plus and minus signs, the hieroglyphs
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were used: if the feet pointed into the direction of writing, it signified addition, otherwise subtraction. [1]
[edit] Written numbers
As with most modern-day languages, the ancient Egyptian language could also write out numerals as words phonetically, just like one can write thirty instead of 30 in English. Thirty, for instance, was written as
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while the number 30 was
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This was, however, uncommon for most numbers other than one and two, and also the signs were used a lot in their time.
[edit] 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 utilize 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 utilize hieratic numerals.
Boyer proved 50 years ago that hieratic script used 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. Boyer saw the new hieratic numerals as ciphered, mapping one number onto one Egyptian letter for the first time in Western history. Greeks adopted the new system, mapping their counting numbers onto two of their alphabets, the Doric and Ionian.
In the oldest hieratic texts the individual numerals were clearly written in a ciphered relationship to the Egyptian alphabet. But during the Old Kingdom a series of standardized writings had developed for sign-groups containing more than one numeral, repeated as Roman numerals practiced. However, repetition of the same numeral for each place-value was not allowed in the hieratic script. As the hieratic writing system developed over time, these sign-groups were further simplified for quick writing; this process continued into Demotic as well.
Two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus.
[edit] Egyptian words for numbers
The following table shows the reconstructed Middle Egyptian forms of the numerals[2] (which are indicated by a preceding asterisk), their transliterated forms in hieroglyphs (indicated between square brackets), and their later Coptic equivalents which give Egyptologists clues as to the vocalism of the original Egyptian numbers. The majuscule letter "A" in some reconstructed forms means that the quality of that vowel remains uncertain:
| Egyptian Transliteration | English Translation | Coptic (Sahidic dialect) |
|---|---|---|
| *wiʕyaw [wˁ.w] (masc.) *wiʕīyat [wˁ.t] (fem.) |
one | oua (masc.) ouei (fem.) |
| *sínway [sn.wy] (masc.) *síntay [sn.ty] (fem.) |
two | snau (masc.) snte (fem.) |
| *ḫámtaw [ḫmt.w] (masc.) *ḫámtat [ḫmt.t] (fem.) |
three | šomnt (masc.) šomte (fem.) |
| *yAfdáw [ỉfd.w] ([masc.) *yAfdát [ỉfd.t] (fem.) |
four | ftoou (masc.) ftoe (fem.) |
| *dīyaw [dỉ.w] (masc.) *dīyat [dỉ.t] (fem.) |
five | tiou (masc.) tie (fem.) |
| *yAssáw [sỉs.w or ỉs.w (?)] (masc.) *yAssát [sỉs.t or ỉs.t (?)] (fem.) |
six | soou (masc.) soe (fem.) |
| *sáfḫaw [sfḫ.w] (masc.) *sáfḫat [sfḫt] (fem.) |
seven | šašf(masc.) šašfe (fem.) |
| *ḫAmānaw [ḫmnw] (masc.) *ḫAmānat [ḫmnt] (fem.) |
eight | šmoun (masc.) šmoune (fem.) |
| *pAsīḏaw [psḏw] (masc.) *pAsīḏat [psḏt] (fem.) |
nine | psis (masc.) psite (fem.) |
| *mūḏaw [mḏw] (masc.) *mūḏat [mḏt] (fem.) |
ten | mēt (masc.) mēte (fem.) |
| *ḏubāʕatay [ḏbˁ.ty] | twenty | jōt (masc.) jōti (fem.) |
| *máʕbAʔ [mˁbȝ] (masc.) *máʕbAʔat [mˁbȝ.t] (fem.) |
thirty | maab (masc.) maabe (fem.) |
| *ḥAmí (?) [ḥm.w] (masc.) | forty | xme |
| *díywu [dy.w] | fifty | taeiou |
| *yAssáwyu [sỉsy.w or ỉswy.w (?)] | sixty | se |
| *safḫáwyu [sfḫy.w] (masc.) | seventy | šfe |
| *ḫamanáwyu [ḫmny.w] (masc.) | eighty | xmene |
| *pAsiḏawyu [psḏy.w] (masc.) | ninety | pstaiou |
| *šáwat [š.t] | one hundred | še |
| *šūtay [š.ty] | two hundred | šēt |
| *ḫaʔ [ḫȝ] | one thousand | šo |
| *ḏubaʕ [ḏbȝ] | ten thousand | tba |
| [hfn] | one hundred thousand | |
| *ḥaḥ [ḥḥ] | one million | xax "many" |
[edit] See also
[edit] 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)
[edit] Notes
- ^ Cajori, Florian [1929] (1993). A History of Mathematical Notations. Dover Publications, pp. 229-230. ISBN 0486677664.
- ^ John B. Callender, Middle Egyptian, 1975
[edit] External links
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