An alphabetic numeral system is a type of

numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...

. Developed in

classical antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...

, it flourished during the

early Middle Ages The Early Middle Ages (or early medieval period), sometimes controversially referred to as the Dark Ages, is typically regarded by historians as lasting from the late 5th or early 6th century to the 10th century. They marked the start of the Mi ...

. In alphabetic numeral systems,

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...

s are written using the characters of an

alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...

syllabary In the linguistic study of written languages, a syllabary is a set of written symbols that represent the syllables or (more frequently) moras which make up words. A symbol in a syllabary, called a syllabogram, typically represents an (option ...

, or another

writing system A writing system is a method of visually representing verbal communication, based on a script and a set of rules regulating its use. While both writing and speech are useful in conveying messages, writing differs in also being a reliable fo ...

. Unlike acrophonic numeral systems, where a numeral is represented by the first letter of the lexical name of the numeral, alphabetic numeral systems can arbitrarily assign letters to numerical values. Some systems, including the

Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...

, Georgian and

Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

systems, use an already established

alphabetical order Alphabetical order is a system whereby character strings are placed in order based on the position of the characters in the conventional ordering of an alphabet. It is one of the methods of collation. In mathematics, a lexicographical order is t ...

. Alphabetic numeral systems originated with

Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those ...

around 600 BC and became largely extinct by the 16th century. After the development of positional numeral systems like Hindu–Arabic numerals, the use of alphabetic numeral systems dwindled to predominantly ordered lists,

pagination Pagination, also known as paging, is the process of dividing a document into discrete pages, either electronic pages or printed pages. In reference to books produced without a computer, pagination can mean the consecutive page numbering to ind ...

, religious functions, and divinatory magic.

History

The first attested alphabetic numeral system is the Greek alphabetic system (named the Ionic or Milesian system due to its origin in west

Asia Minor Anatolia, tr, Anadolu Yarımadası), and the Anatolian plateau, also known as Asia Minor, is a large peninsula in Western Asia and the westernmost protrusion of the Asian continent. It constitutes the major part of modern-day Turkey. The re ...

). The system's structure follows the structure of the Egyptian demotic numerals; Greek letters replaced Egyptian signs. The first examples of the Greek system date back to the 6th century BC, written with the letters of the archaic Greek script used in

Ionia Ionia () was an ancient region on the western coast of Anatolia, to the south of present-day Izmir. It consisted of the northernmost territories of the Ionian League of Greek settlements. Never a unified state, it was named after the Ionia ...

. Other cultures in contact with Greece adopted this numerical notation, replacing the Greek letters with their own script; these included the Hebrews in the late 2nd century BC. The

Gothic alphabet The Gothic alphabet is an alphabet used for writing the Gothic language. Ulfilas (or Wulfila) developed it in the 4th century AD for the purpose of translating the Bible. The alphabet essentially uses uncial forms of the Greek alphabet, wi ...

adopted their own alphabetic numerals along with the Greek-influenced script. In

North Africa North Africa, or Northern Africa is a region encompassing the northern portion of the African continent. There is no singularly accepted scope for the region, and it is sometimes defined as stretching from the Atlantic shores of Mauritania in ...

, the Coptic system was developed in the 4th century AD, and the Ge'ez system in Ethiopia was developed around 350 AD. Both were developed from the Greek model. The Arabs developed their own alphabetic numeral system, the

abjad numerals The Abjad numerals, also called Hisab al-Jummal ( ar, حِسَاب ٱلْجُمَّل, ), are a decimal alphabetic numeral system/ alphanumeric code, in which the 28 letters of the Arabic alphabet are assigned numerical values. They have been ...

, in the 7th century AD, and used it for mathematical and astrological purposes even as late as the 13th century far after the introduction of the

Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

. After the adoption of Christianity,

Armenians Armenians ( hy, հայեր, ''hayer'' ) are an ethnic group native to the Armenian highlands of Western Asia. Armenians constitute the main population of Armenia and the ''de facto'' independent Artsakh. There is a wide-ranging diaspora ...

and

Georgians The Georgians, or Kartvelians (; ka, ქართველები, tr, ), are a nation and indigenous Caucasian ethnic group native to Georgia and the South Caucasus. Georgian diaspora communities are also present throughout Russia, Turkey, ...

developed their alphabetical numeral system in the 4th or early 5th century, while in the

Byzantine Empire The Byzantine Empire, also referred to as the Eastern Roman Empire or Byzantium, was the continuation of the Roman Empire primarily in its eastern provinces during Late Antiquity and the Middle Ages, when its capital city was Constantinopl ...

Cyrillic numerals Cyrillic numerals are a numeral system derived from the Cyrillic script, developed in the First Bulgarian Empire in the late 10th century. It was used in the First Bulgarian Empire and by South and East Slavic peoples. The system was used i ...

and

Glagolitic The Glagolitic script (, , ''glagolitsa'') is the oldest known Slavic alphabet. It is generally agreed to have been created in the 9th century by Saint Cyril, a monk from Thessalonica. He and his brother Saint Methodius were sent by the Byzan ...

were introduced in the 9th century. Alphabetic numeral systems were known and used as far north as England, Germany, and Russia, as far south as Ethiopia, as far east as Persia, and in North Africa from Morocco to Central Asia. By the 16th century AD, most alphabetic numeral systems had died out or were in little use, displaced by Arabic positional and Western numerals as the ordinary numerals of commerce and administration throughout Europe and the Middle East. The newest alphabetic numeral systems in use, all of them positional, are part of tactile writing systems for

visually impaired Visual impairment, also known as vision impairment, is a medical definition primarily measured based on an individual's better eye visual acuity; in the absence of treatment such as correctable eyewear, assistive devices, and medical treatment� ...

. Even though 1829 braille had a simple ciphered-positional system copied from Western numerals with a separate symbol for each digit, early experience with students forced its designer

Louis Braille Louis Braille (; ; 4 January 1809 – 6 January 1852) was a French educator and the inventor of a reading and writing system, named braille after him, intended for use by visually impaired people. His system is used worldwide and remains virtu ...

to simplify the system, bringing the number of available patterns (symbols) from 125 down to 63, so he had to repurpose a supplementary symbol to mark letters a–j as numerals. Besides this traditional system, another one was developed in France in the 20th century, and yet another one in the US.

Systems

An alphabetic numeral system employs the

letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet), a character representing one or more of the sounds used in speech; any of the symbols of an alphabet. * Letterform, the graphic form of a letter of the alpha ...

of a script in the specific order of the alphabet in order to express numerals. In Greek, letters are assigned to respective numbers in the following sets: 1 through 9, 10 through 90, 100 through 900, and so on. Decimal places are represented by a single symbol. As the alphabet ends, higher numbers are represented with various multiplicative methods. However, since writing systems have a differing number of letters, other systems of writing do not necessarily group numbers in this way. The

Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...

has 24 letters; three additional letters had to be incorporated in order to reach 900. Unlike the Greek, the

Hebrew alphabet The Hebrew alphabet ( he, אָלֶף־בֵּית עִבְרִי, ), known variously by scholars as the Ktav Ashuri, Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language and other Jewi ...

's 22 letters allowed for numerical expression up to 400. The Arabic abjad's 28 consonant signs could represent numbers up to 1000. Ancient Aramaic alphabets had enough letters to reach up to 9000. In mathematical and astronomical manuscripts, other methods were used to represent larger numbers.

Roman numerals Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ...

and Attic numerals, both of which were also alphabetic numeral systems, became more concise over time, but required their users to be familiar with many more signs.

Acrophonic Acrophony (; Greek: ἄκρος ''akros'' uppermost + φωνή ''phone'' sound) is the naming of letters of an alphabetic writing system so that a letter's name begins with the letter itself. For example, Greek letter names are acrophonic: the name ...

numerals do not belong to this group of systems because their letter-numerals do not follow the order of an alphabet. These various systems do not have a single unifying trait or feature. The most common structure is ciphered-additive with a decimal base, with or without the use of multiplicative-additive structuring for the higher numbers. Exceptions include the Armenian notation of Shirakatsi, which is multiplicative-additive and sometimes uses a base 1,000, and the Greek and Arabic astronomical notation systems.

Numeral signs

The tables below show the alphabetic numeral configurations of various writing systems. Greek alphabetic numerals – "Ionian" or "Milesian numerals" – (minuscule letters) :: Some numbers represented with Greek alphabetic numerals: :: = (3000 + 900 + 40 + 2) = 3942 :: = (600 + 60 + 6) = 666 :: Hebrew alphabetic numerals: :: The Hebrew writing system has only twenty-four consonant signs, so numbers can be expressed with single individual signs only up to 400. Higher hundreds – 500, 600, 700, 800, and 900 – can be written only with various cumulative-additive combinations of the lower hundreds (direction of writing is right to left): ::תק = (400+100) 500 ::תר = (400+200) 600 ::תש = (400+300) 700 ::תת = (400+400) 800 ::תררק = 400+200+200+100 = 900 Armenian numeral signs (minuscule letters): :: Unlike many alphabetic numeral systems, the Armenian system does not use multiplication by 1,000 or 10,000 in order to express higher values. Instead, higher values were written out in full using lexical numerals.

Higher numbers

As the alphabet ended, various multiplicative methods were used for the expression of higher numbers in the different systems. In the Greek alphabetic system, for multiples of 1,000, the ''hasta'' sign was placed to the left below a numeral-sign to indicate that it should be multiplied by 1,000. ::β = 2 ::͵β = 2,000 ::͵κ = 20,000 With a second level of multiplicative method – multiplication by 10,000 – the numeral set could be expanded. The most common method, used by Aristarchus, involved placing a numeral-phrase above a large M character (M = myriads = 10,000) to indicate multiplication by 10,000. This method could express numbers up to 100,000,000 (10⁸). could be represented as: :: According to

Pappus of Alexandria Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...

's report,

Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contributio ...

used another method. In it, the numerals above M = myriads = 10,000 represented the exponent of 10,000. The number to be multiplied by M was written after the M character.Greek number systems – MacTutor
/ref> This method could express 5,462,360,064,000,000 as: ::

Distinguishing numeral-phrases from text

Alphabetic numerals were distinguished from the words with special signs, most commonly a horizontal stroke above the numeral-phrase, but occasionally with dots placed to either side of it. The latter was manifested in the Greek alphabet with the ''hasta'' sign. 285 en gotique2

= 285 In Ethiopic numerals, known as Geʽez, the signs have marks both above and below them to indicate that their value is numerical. The Ethiopic numerals are the exception, where numeral signs are not letters of their script. This practice became universal from the 15th century onwards. Numeral signs of Ethiopic numerals with marks both above and below the letters: :: The direction of numerals follows the writing system's direction. Writing is from left to right in Greek, Coptic, Ethiopic, Ghotic, Armenian, Georgian, Glagolitic, and Cyrillic alphabetic numerals along with Shirakatsi's notation. Right-to-left writing is found in Hebrew and Syriac alphabetic numerals, Arabic abjad numerals, and Fez numerals.

Fractions

Unit fractions

Unit fractions were a method to express fractions. In Greek alphabetic notation, unit fractions were indicated with the denominator – alphabetic numeral sign – followed by small accents or strokes placed to the right of a numeral, known as a ''keraia'' (ʹ). Therefore, γʹ indicated one third, δʹ one fourth, and so on. These fractions were additive and were also known as

Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from ea ...

s. For example: . A mixed number could be written as such:

Astronomical fractions

In many astronomical texts, a distinct set of alphabetic numeral systems blend their ordinary alphabetical numerals with a base of 60, such as Babylonian sexagesimal systems. In the 2nd century BC, a hybrid of Babylonian notation and Greek alphabetic numerals emerged and was used to express fractions. Unlike the Babylonian system, the Greek base of 60 was not used for expressing integers. With this

sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

positional system – with a subbase of 10 – for expressing

fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

, fourteen of the alphabetic numerals were used (the units from 1 to 9 and the decades from 10 to 50) in order to write any number from 1 through 59. These could be a numerator of a fraction. The positional principle was used for the denominator of a fraction, which was written with an exponent of 60 (60, 3,600, 216,000, etc.). Sexagesimal fractions could be used to express any fractional value, with the successive positions representing 1/60, 1/60², 1/60³, and so on. The first major text in which this blended system appeared was

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

Almagest The ''Almagest'' is a 2nd-century Greek-language mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy ( ). One of the most influential scientific texts in history, it can ...

, written in the 2nd century AD. Astronomical fractions (with Greek alphabetic signs): :: :: This blended system did not use a

radix point A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...

, but the astronomical fractions had a special sign to indicate zero as a placeholder. Some late Babylonian texts used a similar placeholder. The Greeks adopted this technique using their own sign, whose form and character changed over time from early manuscripts (1st century AD) to an alphabetic notation. This sexagesimal notation was especially useful in astronomy and mathematics because of the division of the circle into 360 degrees (with subdivisions of 60 minutes per degree and 60 seconds per minute). In

Theon of Alexandria Theon of Alexandria (; grc, Θέων ὁ Ἀλεξανδρεύς; 335 – c. 405) was a Greek scholar and mathematician who lived in Alexandria, Egypt. He edited and arranged Euclid's '' Elements'' and wrote commentaries on w ...

's (4th century AD) commentary on the Almagest, the numeral-phrase expresses 1515 () degrees, 20 () minutes, and 15 () seconds.Thomas, Ivor. 1962. ''Selections Illustrating the History of Greek Mathematics'', vol. 1. Cambridge, MA: Harvard University Press. pp. 50–51. The degree's value is in the ordinary decimal alphabetic numerals, including the use of the multiplicative ''hasta'' for 1000, while the latter two positions are written in sexagesimal fractions. Arabs adopted astronomical fractions directly from the Greeks, and similarly Hebrew astronomers used sexagesimal fractions, but Greek numeral signs were replaced by their own alphabetic numeral signs to express both integers and fractions.

Alphabetic numeral systems

References

Sources

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