Egyptian numerals

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The system of Egyptian numerals was a numeral system used in ancient Egypt. It was a decimal system, often rounded off to the higher power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, being ciphered one:one onto the beautiful Egyptian alphabet.The Egyptian system used bases of ten

1 Digits and numbers
2 Fractions
3 Addition and subtraction
4 Written numbers
5 Hieratic numerals
6 Egyptian words for numbers
7 See also
8 References
9 Notes
10 External links

[edit] Digits and numbers

The following hieroglyphs were used to denote powers of ten:

Value

100

1,000

10,000

100,000

1 million, or
infinity

Hieroglyph

Description

Single stroke

Cattle hobble
or yoke

Coil of rope

Water lily
(also called Lotus)

Finger

Tadpole
or Frog

Man with both
hands raised

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 inverted.

[edit] Fractions

Main article: Egyptian fraction

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:

$= \frac{1}{3}$

There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (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":

$= \frac{1}{331}$

[edit] Addition and subtraction

For plus and minus signs, the hieroglyphs

and

were used: if the feet pointed into the direction of writing, it signified addition, otherwise subtraction. ^[1]

[edit] Written numbers

Besides this numeral system, the ancient Egyptian language could also write out numbers as words, phonetically, just like one can write "thirty" instead of 30 in English. "Thirty", for instance, was written as

while the number 30 was

This was, however, uncommon for most numbers other than one and two.

[edit] Hieratic numerals

As most 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 utilise 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 utilise 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 Western 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 standardised 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.

[edit] Egyptian words for numbers

The following table shows the reconstructed Middle Egyptian forms of the numerals^[2] (which are indicated by a preceding asterisk), their transliterated forms in hieroglyphs (indicated between square brackets), and their later Coptic equivalents which give Egyptologists clues as to the vocalism of the original Egyptian numbers. The majiscule letter "A" in some reconstructed forms means that the quality of that vowel remains uncertain:

Egyptian Transliteration	English Translation	Coptic (Sahidic dialect)
wiʕyaw [wˁ.w] (masc.) wiʕīyat [wˁ.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.)
*ḏubāʕatay [ḏbˁ.ty]	twenty	jōt (masc.) jōti (fem.)
máʕbAʔ [mˁbȝ] (masc.) máʕbAʔat [mˁbȝ.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
*ḫaʔ [ḫȝ]	one thousand	šo
*ḏubaʕ [ḏbȝ]	ten thousand	tba
[hfn]	one hundred thousand
*ḥaḥ [ḥḥ]	one million	xax "many"

[edit] See also

[edit] 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)

[edit] Notes

^ Cajori, Florian [1929] (1993). A History of Mathematical Notations. Dover Publications, pp. 229-230. ISBN 0486677664.
^ John B. Callender, Middle Egyptian, 1975