Egyptian numerals

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The system of ancient Egyptian numerals was used in Ancient Egypt from around 3000 BC[1] until the early first millennium AD. It was a system of numeration based on multiples of ten, often rounded off to the higher power, written in hieroglyphs. The Egyptians had no concept of a place-valued system such as the decimal system.[2] The hieratic form of numerals stressed an exact finite series notation, ciphered one to one onto the Egyptian alphabet.

Digits and numbers[edit]

The following hieroglyphs were used to denote powers of ten:

Value 1 10 100 1,000 10,000 100,000 1 million, or
many
Hieroglyph
Z1
V20
V1
M12
D50
I8
C11
Description Single stroke Cattle hobble Coil of rope Water lily
(also called lotus)
Bent finger Tadpole Heh[3]

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

M12M12M12M12
V1 V1 V1
V1 V1 V1
V20V20Z1Z1

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.

Zero and negative numbers[edit]

nfr
 
heart with trachea
beautiful, pleasant, good
F35

By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr (𓄤), meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.[4]

Fractions[edit]

Rational numbers could also be expressed, but only as sums of unit fractions, i.e., sums of reciprocals of positive integers, except for ​23 and ​34. The hieroglyph indicating a fraction looked like a mouth, which meant "part":

D21

Fractions were written with this fractional solidus, i.e., the numerator 1, and the positive denominator below. Thus, ​13 was written as:

D21
Z1 Z1 Z1

There were special symbols for ​12 and for two non-unit fractions, ​23 (used frequently) and ​34 (used less frequently):

Aa13
 
D22
 
D23

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

D21
V1 V1 V1
V20 V20
V20 Z1

Addition and subtraction[edit]

For plus and minus signs, the hieroglyphs

D54andD55

were used: if the feet pointed into the direction of writing, it signified addition, otherwise subtraction.[5]

Written numbers[edit]

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. The word (thirty), for instance, was written as

Aa15
D36
D58

while the numeral (30) was

V20V20V20

This was, however, uncommon for most numbers other than one and two and the signs were used most of the time.

Hieratic numerals[edit]

As 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[when?] 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 human 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.

Egyptian words for numbers[edit]

The following table shows the reconstructed Middle Egyptian forms of the numerals (which are indicated by a preceding asterisk), the transliteration of the hieroglyphs used to write them, and finally the Coptic numerals which descended from them and which give Egyptologists clues as to the vocalism of the original Egyptian numbers. A breve (˘) in some reconstructed forms indicates a short vowel whose quality remains uncertain; the letter ‘e’ represents a vowel that was originally u or i (exact quality uncertain) but became e by Late Egyptian.

Egyptian transliteration Reconstructed vocalization English translation Coptic (Sahidic dialect)
per Callender 1975[6] per Loprieno 1995[7]
wꜥ(w) (masc.)
wꜥt (fem.)
*wíꜥyaw (masc.)
*wiꜥī́yat (fem.)
*wúꜥꜥuw (masc.) one ⲟⲩⲁ (oua) (masc.)
ⲟⲩⲉⲓ (ouei) (fem.)
snwj (masc.)
sntj (fem.)
*sínwaj (masc.)
*síntaj (fem.)
*sinúwwaj (masc.) two ⲥⲛⲁⲩ (snau) (masc.)
ⲥⲛ̄ⲧⲉ (snte) (fem.)
ḫmtw (masc.)
ḫmtt (fem.)
*ḫámtaw (masc.)
*ḫámtat (fem.)
*ḫámtaw (masc.) three ϣⲟⲙⲛ̄ⲧ (šomnt) (masc.)
ϣⲟⲙⲧⲉ (šomte) (fem.)
jfdw (masc.)
jfdt (fem.)
*j˘fdáw (masc.)
*j˘fdát (fem.)
*jifdáw (masc.) four ϥⲧⲟⲟⲩ (ftoou) (masc.)
ϥⲧⲟ (fto) or ϥⲧⲟⲉ (ftoe) (fem.)
djw (masc.)
djt (fem.)
*dī́jaw (masc.)
*dī́jat (fem.)
*dī́jaw (masc.) five ϯⲟⲩ (tiou) (masc.)
ϯ (ti) or ϯⲉ (tie) (fem.)
sjsw or jsw (?) (masc.)
sjst or jst (?) (fem.)
*j˘ssáw (masc.)
*j˘ssát (fem.)
*sáʾsaw (masc.) six ⲥⲟⲟⲩ (soou) (masc.)
ⲥⲟ (so) or ⲥⲟⲉ (soe) (fem.)
sfḫw (masc.)
sfḫt (fem.)
*sáfḫaw (masc.)
*sáfḫat (fem.)
*sáfḫaw (masc.) seven ϣⲁϣϥ̄ (šašf) (masc.)
ϣⲁϣϥⲉ (šašfe) (fem.)
ḫmnw (masc.)
ḫmnt (fem.)
*ḫ˘mā́naw (masc.)
*ḫ˘mā́nat (fem.)
*ḫamā́naw (masc.) eight ϣⲙⲟⲩⲛ (šmoun) (masc.)
ϣⲙⲟⲩⲛⲉ (šmoune) (fem.)
psḏw (masc.)
psḏt (fem.)
*p˘sī́ḏaw (masc.)
*p˘sī́ḏat (fem.)
*pisī́ḏaw (masc.) nine ⲯⲓⲥ (psis) (masc.)
ⲯⲓⲧⲉ (psite) (fem.)
mḏw (masc.)
mḏt (fem.)
*mū́ḏaw (masc.)
*mū́ḏat (fem.)
*mū́ḏaw (masc.) ten ⲙⲏⲧ (mēt) (masc.)
ⲙⲏⲧⲉ (mēte) (fem.)
mḏwtj, ḏwtj, or ḏbꜥty (?) (masc.)
mḏwtt, ḏwtt, or ḏbꜥtt (?) (fem.)
*ḏubā́ꜥataj (masc.) *(mu)ḏawā́taj (masc.) twenty ϫⲟⲩⲱⲧ (jouōt) (masc.)
ϫⲟⲩⲱⲧⲉ (jouōte) (fem.)
mꜥbꜣ (masc.)
mꜥbꜣt (fem.)
*máꜥb˘ꜣ (masc.) *máꜥb˘ꜣ (masc.) thirty ⲙⲁⲁⲃ (maab) (masc.)
ⲙⲁⲁⲃⲉ (maabe) (fem.)
ḥmw *ḥ˘mí (?) *ḥ˘méw forty ϩⲙⲉ (hme)
dyw *díjwu *díjjaw fifty ⲧⲁⲉⲓⲟⲩ (taeiou)
sjsjw, sjsw, or jswjw (?) *j˘ssáwju *saʾséw sixty ⲥⲉ (se)
sfḫjw, sfḫw, or sfḫwjw (?) *safḫáwju *safḫéw seventy ϣϥⲉ (šfe)
ḫmnjw, ḫmnw, or ḫmnwjw (?) *ḫamanáwju *ḫamnéw eighty ϩⲙⲉⲛⲉ (hmene)
psḏjw or psḏwjw (?) *p˘siḏáwju *pisḏíjjaw ninety ⲡⲥⲧⲁⲓⲟⲩ (pstaiou)
št *šúwat *ší(nju)t one hundred ϣⲉ (še)
štj *šū́taj *šinjū́taj two hundred ϣⲏⲧ (šēt)
ḫꜣ *ḫaꜣ *ḫaꜣ one thousand ϣⲟ (šo)
ḏbꜥ *ḏubáꜥ *ḏ˘báꜥ ten thousand ⲧⲃⲁ (tba)
ḥfn one hundred thousand
ḥḥ *ḥaḥ *ḥaḥ one million ϩⲁϩ (hah) "many"

See also[edit]

References[edit]

  • 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)

Notes[edit]

  1. ^ "Egyptian numerals". Retrieved 25 September 2013.
  2. ^ "The Story of Numbers" by John McLeish
  3. ^ Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. Hoboken, NJ: John Wiley, 2011, p. 10
  4. ^ George Gheverghese Joseph (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton. p. 86. ISBN 978-0-691-13526-7.
  5. ^ Cajori, Florian (1993) [1929]. A History of Mathematical Notations. Dover Publications. pp. pp. 229–230. ISBN 0-486-67766-4.
  6. ^ Callender, John B. (1975) Middle Egyptian, 1975
  7. ^ Loprieno, Antonio (1995) Ancient Egyptian: A Linguistic Introduction, Cambridge: Cambridge University Press, p. 71, 255

External links[edit]