Egyptian numerals

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 place-valued 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

or
Description Single stroke Heel bone Coil of rope Water lily
(also called Lotus)
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

Egyptian hieroglyphs could be written in both directions (and even vertically). This example is written left-to-right and top-down; on the original stone carving, it is right-to-left, and the signs are thus reversed

Zero and negative numbers

nfr

heart with trachea
beautiful, pleasant, good

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

Main article: Egyptian fraction

Irrational numbers could also be expressed, but only as sums of unit fractions, i.e., sums of reciprocals of positive integers, except for 23 and 34. 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, 13 was written as:

$= \frac{1}{3}$

There were special symbols for 12 and for two non-unit fractions, 23 (used frequently) and 34 (used less frequently):

$= \frac{1}{2}$
$= \frac{2}{3}$
$= \frac{3}{4}$

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

$= \frac{1}{331}$

For plus and minus signs, the hieroglyphs

 and

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

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[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 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[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)
*wiʻyawwꜥ.w (masc.)
*wiʻīyatwꜥ.t (fem.)
one oua (masc.)
ouei (fem.)
*sínwaysn.wy (masc.)
*síntaysn.ty (fem.)
two snau (masc.)
snte (fem.)
*ḫámtawḫmt.w (masc.)
*ḫámtatḫmt.t (fem.)
three šomnt (masc.)
šomte (fem.)
*yAfdáwỉfd.w (masc.)
*yAfdátỉfd.t (fem.)
four ftoou (masc.)
ftoe (fem.)
*dīyawdỉ.w (masc.)
*dīyatdỉ.t (fem.)
five tiou (masc.)
tie (fem.)
*yAssáwsỉs.w or ỉs.w (?) (masc.)
*yAssátsỉs.t or ỉs.t (?) (fem.)
six soou (masc.)
soe (fem.)
*sáfḫawsfḫ.w (masc.)
*sáfḫatsfḫt (fem.)
seven šašf(masc.)
šašfe (fem.)
*ḫAmānawḫmnw (masc.)
*ḫAmānatḫmnt (fem.)
eight šmoun (masc.)
šmoune (fem.)
*pAsīḏawpsḏw (masc.)
*pAsīḏatpsḏt (fem.)
nine psis (masc.)
psite (fem.)
*mūḏawmḏw (masc.)
*mūḏatmḏt (fem.)
ten mēt (masc.)
mēte (fem.)
*ḏubāʻatayḏbꜥ.ty twenty jōt (masc.)
jōti (fem.)
*máʻbAʼmꜥbꜣ (masc.)
*máʻbAʼatmꜥbꜣ.t (fem.)
thirty maab (masc.)
maabe (fem.)
*ḥAmí (?) ‹ ḥm.w (masc.) forty xme
*díywudy.w fifty taeiou
*yAssáwyusỉsy.w or ỉswy.w (?) sixty se
*safḫáwyusfḫy.w (masc.) seventy šfe
*ḫamanáwyuḫmny.w (masc.) eighty xmene
*pAsiḏawyupsḏy.w (masc.) ninety pstaiou
*šáwatš.t one hundred še
*šūtayš.ty two hundred šēt
*ḫaʼḫꜣ one thousand šo
*ḏubaʻḏbꜣ[dubious ]
[these do not match]
ten thousand tba
hfn one hundred thousand
*ḥaḥḥḥ one million xax "many"

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

1. ^ "Egyptian numerals". Retrieved 25 September 2013.
2. ^ "The Story of Numbers" by John McLeish
3. ^ Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. Hoboken, NJ: John Wiley, 2011, p. 10
4. ^ George Gheverghese Joseph (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton. p. 86. ISBN 978-0-691-13526-7.
5. ^ Cajori, Florian (1993) [1929]. A History of Mathematical Notations. Dover Publications. pp. pp. 229–230. ISBN 0-486-67766-4.
6. ^ John B. Callender, Middle Egyptian, 1975