Egyptian numerals

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.
The following hieroglyphs were used to denote powers of ten:
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.
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:
According to Carl Boyer, a deed from Edfu contained a "zero concept" replacing the magnitude in geometry.
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 the signs were used most of the time.
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.
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.