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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 positional notation 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.^{[citation needed]}

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
Gardiner's sign list ID	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 4,622 as:

Egyptian hieroglyphs could be written in both directions (and even vertically). In this example the symbols decrease in value from top to bottom and from left to right. On the original stone carving, it is right-to-left, and the signs are thus reversed.^{[citation needed]}

nfr	heart with trachea beautiful, pleasant, good

There was no symbol or concept of zero as a placeholder in Egyptian numeration and zero was not used in calculations.^[4] However, the symbol nefer (nfr𓄤, "good", "complete", "beautiful") was apparently also used for two numeric purposes:^[5]

• in a papyrus listing the court expenses, c. 1740 BC, it indicated a zero balance;
• in a drawing for Meidum Pyramid (and at other sites), nefer is used to indicate a ground level: height and depths are measured "above nefer" or "below nefer" respectively.

According to Carl Boyer, a deed from Edfu contained a "zero concept" replacing the magnitude in geometry.^[6]

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:

Special symbols were used for 1⁄2 and for the non-unit fractions 2⁄3 and, less frequently, 3⁄4:

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

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.^{[citation needed]}

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.^{[citation needed]}

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.^{[citation needed]}

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.^{[citation needed]}

Two famous mathematical papyri using hieratic script are the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus.^{[citation needed]}

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.^{[citation needed]}

Egyptian transliteration	Reconstructed vocalization		English translation	Coptic (Sahidic dialect)
	per Callender 1975[7]	per Loprieno 1995[8]
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"

• Egyptian language
• Egyptian mathematics

• 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.
• Hoffmann, Friedhelm (2024-03-11). "Aspects of Zero in Ancient Egypt". In Gobets, Peter; Lawrence Kuhn, Robert (eds.). The Origin and Significance of Zero: An Interdisciplinary Perspective. Brill. pp. 64–81. doi:10.1163/9789004691568_007. ISBN 978-90-04-69156-8.
• Joseph, G.G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press. ISBN 978-0-691-13526-7. Retrieved 2024-05-03.
• Möller, Georg (1927). Hieratische Paläographie: Die Ägyptische 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)

• Introduction to Hieroglyphs Numbers and Fractions at the Wayback Machine (archived September 29, 2007)
• Numbers and dates at the Wayback Machine (archived March 4, 2001)
• Egyptian Numbers at the Wayback Machine (archived January 12, 2004)
• Egyptian Math History