Egyptian numerals
Numeral systems 

Hindu–Arabic numeral system 
East Asian 
Alphabetic 
Former 
Positional systems by base 
Nonstandard positional numeral systems 
List of numeral systems 
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 placevalued 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.
Contents
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 






 
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

Egyptian hieroglyphs could be written in both directions (and even vertically). This example is written lefttoright and topdown; on the original stone carving, it is righttoleft, and the signs are thus reversed.
Zero and negative numbers[edit]
nfr 
heart with trachea beautiful, pleasant, good 


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 ^{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:

There were special symbols for ^{1}⁄_{2} and for two nonunit fractions, ^{2}⁄_{3} (used frequently) and ^{3}⁄_{4} (used less frequently):



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

Addition and subtraction[edit]
For plus and minus signs, the hieroglyphs

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

while the numeral (30) was

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 signgroups containing more than one numeral, repeated as Roman numerals practiced. However, repetition of the same numeral for each placevalue was not allowed in the hieratic script. As the hieratic writing system developed over time, these signgroups 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]
 ^ "Egyptian numerals". Retrieved 25 September 2013.
 ^ "The Story of Numbers" by John McLeish
 ^ Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. Hoboken, NJ: John Wiley, 2011, p. 10
 ^ George Gheverghese Joseph (2011). The Crest of the Peacock: NonEuropean Roots of Mathematics (Third Edition). Princeton. p. 86. ISBN 9780691135267.
 ^ Cajori, Florian (1993) [1929]. A History of Mathematical Notations. Dover Publications. pp. pp. 229–230. ISBN 0486677664.
 ^ Callender, John B. (1975) Middle Egyptian, 1975
 ^ Loprieno, Antonio (1995) Ancient Egyptian: A Linguistic Introduction, Cambridge: Cambridge University Press, p. 71, 255
External links[edit]
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