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The system of ancientEgyptian numerals was used inAncient 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 inhieroglyphs.
The Egyptians had no concept of apositional notation such as thedecimal system.[2] Thehieratic 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 denotepowers of ten:
![1,000,000 𓁨 Heh[3]](/mt/?noimg=&dark=on&url=https%3a%2f%2fen.wikipedia.org%2fwiki%2fFile%3aHiero-numeral_1_000_000.jpg)
Multiples of these values were expressed by repeating the symbol as many times as needed. For instance, a stone carving fromKarnak shows the number 4,622 as:
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Egyptian hieroglyphs could be written inboth 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 |
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There was no symbol or concept ofzero as a placeholder in Egyptian numeration and zero was not used in calculations.[4] However, the symbolnefer (nfr𓄤, "good", "complete", "beautiful") was apparently also used for two numeric purposes:[5]
According toCarl Boyer, a deed fromEdfu contained a "zero concept" replacing the magnitude in geometry.[6]
Rational numbers could also be expressed, but only as sums ofunit fractions,i.e., sums ofreciprocals of positive integers, except for2⁄3 and3⁄4. The hieroglyph indicating a fraction looked like a mouth, which meant "part":
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Fractions were written with this fractionalsolidus,i.e., thenumerator 1, and the positivedenominator below. Thus,1⁄3 was written as:
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Special symbols were used for1⁄2 and for the non-unit fractions2⁄3 and, less frequently,3⁄4:
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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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As with most modern day languages, the ancientEgyptian language could also write out numerals as words phonetically, just like one can writethirty instead of "30" inEnglish. The word (thirty), for instance, was written as
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while the numeral (30) was
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This was, however, uncommon for most numbers other than one and the signs were used most of the time.[7]
As administrative and accounting texts were written onpapyrus orostraca, rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilize thehieratic script. Instances of numerals written in hieratic can be found as far back as theEarly Dynastic Period. The Old KingdomAbusir Papyri are a particularly important corpus of texts that utilize hieratic numerals.[citation needed]
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Boyer proved 50 years ago[when?] thathieratic 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 asRoman 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 intoDemotic, as well.[citation needed]
Two famous mathematical papyri using hieratic script are theMoscow Mathematical Papyrus and theRhind Mathematical Papyrus.[citation needed]
The following table shows the reconstructed Middle Egyptian forms of the numerals (which are indicated by a preceding asterisk), thetransliteration of thehieroglyphs used to write them, and finally theCoptic numerals which descended from them and which giveEgyptologists clues as to thevocalism 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]
| Egyptiantransliteration | Reconstructed vocalization | Englishtranslation | Coptic (Sahidic dialect) | |
|---|---|---|---|---|
| per Callender 1975[8] | per Loprieno 1995[9] | |||
| 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" |
