Egyptian numerals
Encyclopedia
The system of Ancient Egyptian numerals was used in Ancient Egypt until the early first millennium AD. 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, ciphered one to one onto the Egyptian alphabet.The Ancient Egyptian system used bases of ten. They also created the 365 day calendar. Most Egyptians still use this system today.
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 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
s could also be expressed, but only as sums of unit fraction
s, 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:
There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently):
If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
, the hieroglyphs
were used: if the feet pointed into the direction of writing, it signified addition
, otherwise subtraction
.
could also write out numerals 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 and the signs were used most of the time.
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
.
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:
Decimal system
Decimal system may refer to:* The decimal number system, used in mathematics for writing numbers and performing arithmetic.* The Dewey Decimal System, a subject classification system used in libraries....
, often rounded off to the higher power, written in hieroglyphs. The hieratic
Hieratic
Hieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
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. They also created the 365 day calendar. Most Egyptians still use this system today.
Digits and numbers
the following hieroglyphics were used to denote powersExponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
of ten:
| Value | 1 | 10 | 100 | 1,000 | 10,000 | 100,000 | 1 million Million One million or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione , from mille, "thousand", plus the augmentative suffix -one.In scientific notation, it is written as or just 106... , or many Many Many may refer to:* plural*A quantifier that can be used with count nouns - often preceded by "as" or "too" or "so" or "that"; amounting to a large but indefinite number; "many temptations"; "a good many"; "many directions"; more than a few, more than severalPlace names* Many, Moselle, a commune of... |
|---|---|---|---|---|---|---|---|
| Hieroglyph | or |
||||||
| Description | Single stroke | Heel bone | Coil of rope Rope A rope is a length of fibres, twisted or braided together to improve strength for pulling and connecting. It has tensile strength but is too flexible to provide compressive strength... |
Water lily Nymphaea lotus Nymphaea lotus, the Tiger Lotus, White lotus or Egyptian White Water-lily, is a flowering plant of the family Nymphaeaceae-Description:It grows in various parts of East Africa and Southeast Asia.... (also called Lotus) |
Finger | Tadpole Tadpole A tadpole or polliwog is the wholly aquatic larval stage in the life cycle of an amphibian, particularly that of a frog or toad.- Appellation :... or Frog Frog Frogs are amphibians in the order Anura , formerly referred to as Salientia . Most frogs are characterized by a short body, webbed digits , protruding eyes and the absence of a tail... |
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
Karnak
The Karnak Temple Complex—usually called Karnak—comprises a vast mix of decayed temples, chapels, pylons, and other buildings, notably the Great Temple of Amun and a massive structure begun by Pharaoh Ramses II . Sacred Lake is part of the site as well. It is located near Luxor, some...
shows the number 4622 as
Egyptian hieroglyphs could be written in both directions
Bi-directional text
Bi-directional text is text containing text in both text directionalities, both right-to-left and left-to-right . It generally involves text containing different types of alphabets, but may also refer to boustrophedon, which is changing text directionality in each row.Some writing systems of the...
(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
Fractions
Rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s could also be expressed, but only as sums of unit fraction
Unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n...
s, 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
Slash (punctuation)
The slash is a sign used as a punctuation mark and for various other purposes. It is now often called a forward slash , and many other alternative names.-History:...
, i.e., the numerator 1, and the positive denominator below. Thus, 1/3 was written as:
 |
There were special symbols for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently):
 |
 |
 |
If the denominator became too large, the "mouth" was just placed over the beginning of the "denominator":
 |
Addition and subtraction
For plus and minus signsPlus and minus signs
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...
, the hieroglyphs
were used: if the feet pointed into the direction of writing, it signified addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, otherwise subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
.
Written numbers
As with most modern day languages, the ancient Egyptian languageEgyptian language
Egyptian is the oldest known indigenous language of Egypt and a branch of the Afroasiatic language family. Written records of the Egyptian language have been dated from about 3400 BC, making it one of the oldest recorded languages known. Egyptian was spoken until the late 17th century AD in the...
could also write out numerals as words phonetically, just like one can write thirty instead of "30" in English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
. Thirty, for instance, was written as
while the number 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 papyrusPapyrus
Papyrus is a thick paper-like material produced from the pith of the papyrus plant, Cyperus papyrus, a wetland sedge that was once abundant in the Nile Delta of Egypt....
or ostraca
Ostracon
An ostracon is a piece of pottery , usually broken off from a vase or other earthenware vessel. In archaeology, ostraca may contain scratched-in words or other forms of writing which may give clues as to the time when the piece was in use...
, rather than being carved into hard stone (as were hieroglyphic texts), the vast majority of texts employing the Egyptian numeral system utilize the hieratic
Hieratic
Hieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
script. Instances of numerals written in hieratic can be found as far back as the Early Dynastic Period
Early Dynastic Period of Egypt
The Archaic or Early Dynastic Period of Egypt immediately follows the unification of Lower and Upper Egypt c. 3100 BC. It is generally taken to include the First and Second Dynasties, lasting from the Protodynastic Period of Egypt until about 2686 BC, or the beginning of the Old Kingdom...
. The Old Kingdom |Abusir Papyri
Abusir Papyri
The Abusir Papyri are the largest papyrus findings to date from the Old Kingdom in Ancient Egypt. The first papyri were discovered in 1893 at Abu Gorab near Abusir in northern Egypt. Their origins are dated to around the 24th century BC during the Fifth dynasty of Egypt, making them, even though...
are a particularly important corpus of texts that utilize hieratic numerals.
Boyer proved 50 years ago that hieratic
Hieratic
Hieratic refers to a cursive writing system that was used in the provenance of the pharaohs in Egypt and Nubia that developed alongside the hieroglyphic system, to which it is intimately related...
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
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
and the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
.
Egyptian words for numbers
The following table shows the reconstructed Middle Egyptian forms of the numerals (which are indicated by a preceding asterisk), followed by the transliteration of the hieroglyphsEgyptian hieroglyphs
Egyptian hieroglyphs were a formal writing system used by the ancient Egyptians that combined logographic and alphabetic elements. Egyptians used cursive hieroglyphs for religious literature on papyrus and wood...
used to write them, and finally the Coptic
Coptic language
Coptic or Coptic Egyptian is the current stage of the Egyptian language, a northern Afro-Asiatic language spoken in Egypt until at least the 17th century. Egyptian began to be written using the Greek alphabet in the 1st century...
numerals which descended from them and which give Egyptologists clues as to the vocalism
Vowel
In phonetics, a vowel is a sound in spoken language, such as English ah! or oh! , pronounced with an open vocal tract so that there is no build-up of air pressure at any point above the glottis. This contrasts with consonants, such as English sh! , where there is a constriction or closure at some...
of the original Egyptian numbers. The majuscule letter "A" in some reconstructed forms means that the quality of that vowel remains uncertain:
| Egyptian Transliteration Transliteration Transliteration is a subset of the science of hermeneutics. It is a form of translation, and is the practice of converting a text from one script into another... | English English language English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria... Translation Translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. Whereas interpreting undoubtedly antedates writing, translation began only after the appearance of written literature; there exist partial translations of the Sumerian Epic of... | Coptic Coptic language Coptic or Coptic Egyptian is the current stage of the Egyptian language, a northern Afro-Asiatic language spoken in Egypt until at least the 17th century. Egyptian began to be written using the Greek alphabet in the 1st century... (Sahidic dialect) |
|---|---|---|
| ‹ (masc. Grammatical gender Grammatical gender is defined linguistically as a system of classes of nouns which trigger specific types of inflections in associated words, such as adjectives, verbs and others. For a system of noun classes to be a gender system, every noun must belong to one of the classes and there should be... ) ‹ (fem. Grammatical gender Grammatical gender is defined linguistically as a system of classes of nouns which trigger specific types of inflections in associated words, such as adjectives, verbs and others. For a system of noun classes to be a gender system, every noun must belong to one of the classes and there should be... ) |
one | oua (masc.) ouei (fem.) |
| ‹ (masc.) ‹ (fem.) |
two | snau (masc.) snte (fem.) |
| ‹ (masc.) ‹ (fem.) |
three | šomnt (masc.) šomte (fem.) |
| ‹ (masc.) ‹ (fem.) |
four | ftoou (masc.) ftoe (fem.) |
| ‹ (masc.) ‹ (fem.) |
five | tiou (masc.) tie (fem.) |
| ‹ or (?) (masc.) ‹ or (?) (fem.) |
six | soou (masc.) soe (fem.) |
| ‹ (masc.) ‹ (fem.) |
seven | šašf(masc.) šašfe (fem.) |
| ‹ (masc.) ‹ (fem.) |
eight | šmoun (masc.) šmoune (fem.) |
| ‹ (masc.) ‹ (fem.) |
nine | psis (masc.) psite (fem.) |
| ‹ (masc.) ‹ (fem.) |
ten | mēt (masc.) mēte (fem.) |
| ‹ | twenty | jōt (masc.) jōti (fem.) |
| ‹ (masc.) ‹ (fem.) |
thirty | maab (masc.) maabe (fem.) |
| (?) ‹ (masc.) | forty | xme |
| ‹ | fifty | taeiou |
| ‹ or (?) | sixty | se |
| ‹ (masc.) | seventy | šfe |
| ‹ (masc.) | eighty | xmene |
| ‹ (masc.) | ninety | pstaiou |
| ‹ | one hundred | še |
| ‹ | two hundred | šēt |
| ‹ | one thousand | šo |
| ‹ |
ten thousand | tba |
| ‹ | one hundred thousand | |
| ‹ | one million | xax "many" |
External links
- http://egyptianmath.blogspot.com