# Egyptian numerals

Part of a series on :
Numeral systems :
Place-value notation
Hindu–Arabic numerals
Western Arabic
Eastern Arabic
Bengali
Devanagari
Gujarati
Gurmukhi
Odia
Sinhala
Tamil
Malayalam
Telugu
Kannada
Dzongkha
Tibetan
Balinese
Burmese
Javanese
Khmer
Lao
Mongolian
Sundanese
Thai
East Asian systems
Contemporary
Chinese
Hokkien
Suzhou
Japanese
Korean
Vietnamese
Historic
Counting rods
Tangut
Other systems
History
Ancient
Babylonian
Post-classical
Cistercian
Mayan
Muisca
Pentadic
Quipu
Rumi
Contemporary
Cherokee
Kaktovik (Iñupiaq)
By radix/base
Common radices/bases
2
3
4
5
6
8
10
12
16
20
60
Non-standard radices/bases
Bijective (1)
Signed-digit (balanced ternary)
Mixed (factorial)
Negative
Complex (2i)
Non-integer (φ)
Asymmetric :
Sign-value notation
Non-alphabetic
Aegean
Attic
Aztec
Brahmi
Chuvash
Egyptian
Etruscan
Kharosthi
Prehistoric counting
Proto-cuneiform
Roman
Tally marks
Alphabetic
Abjad
Armenian
Alphasyllabic
Akṣarapallī
Āryabhaṭa
Kaṭapayādi
Coptic
Cyrillic
Geʽez
Georgian
Glagolitic
Greek
Hebrew :
List of numeral systems :
vte :
The system of ancient Egyptian numerals was used in Ancient Egypt from around 3000 BC 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. The hieratic form of numerals stressed an exact finite series notation, ciphered one-to-one onto the Egyptian alphabet.

## Digits and numbers

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 | Froglet | Heh |
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.

## Zero

| 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. However, the symbol nefer (nfr𓄤, "good", "complete", "beautiful") was apparently also used for two numeric purposes:
* 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.

## Fractions

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:
* $={\frac {1}{3}}$
Special symbols were used for 1⁄2 and for the non-unit fractions 2⁄3 and, less frequently, 3⁄4:
| $={\frac {1}{2}}$ | $={\frac {2}{3}}$ | $={\frac {3}{4}}$ |
If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
* $={\frac {1}{100}}$

## 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
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

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 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

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 | Reconstructed vocalization | English translation | Coptic (Sahidic dialect) |
| Egyptian transliteration | per Callender 1975 | per Loprieno 1995 | English translation | Coptic (Sahidic dialect) |
| 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" |