Egyptian numerals
Numeral systems by culture 


Positional systems by base 
Decimal (10) 
Nonstandard positional numeral systems 
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The system of ancient Egyptian numerals was used in Ancient Egypt in around 3000 BC^{[1]} until the early first millennium AD. It was a system of numeration based on the scale of ten, often rounded off to the higher power, written in hieroglyphs, but they had no concept of a placevalued system such as the decimal system is.^{[2]} The hieratic form of numerals stressed an exact finite series notation, ciphered one to one onto the Egyptian alphabet. The Ancient Egyptian system used bases of ten.
Digits and numbers
The following hieroglyphics were used to denote powers of ten:
Value  1  10  100  1,000  10,000  100,000  1 million, or many 

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  Heel bone  Coil of rope  Water lily (also called Lotus) 
Bent Finger  Tadpole or Frog 
Man with both hands raised, perhaps 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
<hiero>M12M12M12M12!V1*V1*V1:V1*V1*V1!V20V20Z1Z1</hiero> 
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
nfr 
heart with trachea beautiful, pleasant, good 
<hiero>F35</hiero> 

By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, 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
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Rational numbers could also be expressed, but only as sums of unit fractions, i.e., sums of reciprocals of positive integers, except for Template:Frac and Template:Frac. The hieroglyph indicating a fraction looked like a mouth, which meant "part":
<hiero>D21</hiero> 
Fractions were written with this fractional solidus, i.e., the numerator 1, and the positive denominator below. Thus, Template:Frac was written as:
<hiero>D21:Z1*Z1*Z1</hiero> 
There were special symbols for Template:Frac and for two nonunit fractions, Template:Frac (used frequently) and Template:Frac (used less frequently):
<hiero>Aa13</hiero>  <hiero>D22</hiero>  <hiero>D23</hiero> 
If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
<hiero>D21:V1*V1*V1V20*V20:V20*Z1</hiero> 
Addition and subtraction
For plus and minus signs, the hieroglyphs <hiero>D54andD55</hiero> were used: if the feet pointed into the direction of writing, it signified addition, otherwise subtraction.^{[5]}
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. The word (thirty), for instance, was written as
<hiero>Aa15:D36D58</hiero> 
while the numeral (30) was
<hiero>V20V20V20</hiero> 
This was, however, uncommon for most numbers other than one and two and the signs were used most of the time.
Hieratic numerals
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 agoTemplate: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
The following table shows the reconstructed Middle Egyptian forms of the numerals^{[6]} (which are indicated by a preceding asterisk), followed by 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. The majuscule letter "A" in some reconstructed forms means that the quality of that vowel remains uncertain:
Egyptian Transliteration  English Translation  Coptic (Sahidic dialect) 

*wiʻyaw ‹ wꜥ.w (masc.) *wiʻīyat ‹ wꜥ.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.) 
*ḏubāʻatay ‹ ḏbꜥ.ty  twenty  jōt (masc.) jōti (fem.) 
*máʻbAʼ ‹ mꜥbꜣ (masc.) *máʻbAʼat ‹ mꜥbꜣ.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 
*ḫaʼ ‹ ḫꜣ  one thousand  šo 
*ḏubaʻ ‹ ḏbꜣ{{ safesubst:#invoke:Unsubst  date=__DATE__ $B=
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‹ hfn  one hundred thousand  
*ḥaḥ ‹ ḥḥ  one million  xax "many" 
See also
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)
Notes
 ↑ Template:Cite web
 ↑ "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
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ {{#invoke:citation/CS1citation CitationClass=book }}
 ↑ John B. Callender, Middle Egyptian, 1975
External links
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