Alphasyllabic numeral system

Last updated July 26, 2023

Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, but syllables of a script, and numerals are represented with these syllable-signs.^[1] On the basic principle of these systems, numeric values of the syllables are defined by the consonants and vowels which constitute them, so that consonants and vowels are - or are not in some systems in case of vowels - ordered to numeric values. While there are many hundreds of possible syllables in a script, and since in alphasyllabic numeral systems several syllables receive the same numeric value, so the mapping is not injective.

Alphasyllabaries

The basic principle of the Indian alphasyllabaries is a set of 33 consonant-signs, which are combined with a set of about 20 diacritic marks that indicate vowels of the brahmi scripts, these produce a set of signs for syllables; unmarked consonant-signs denote the syllable with the inherent vowel ’a’.

Indian alphasyllabic numeration

Starting around 500 AD, Indian astronomers and astrologers began to use this new principle for numeration with assigning numeral values to the phonetic signs of various Indian alphasyllabic scripts – the brahmi scripts.^[2] Earlier 20th-century scholars supposed that the Indian grammarian Pāṇini used alphasyllabic numerals already in the 7th century BC.^[3] Since there is no direct evidence for any alphasyllabic numeration in India until about 510 AD, recently this theory is not supported.^[4]

These systems, known collectively as varnasankhya systems, were considered to be distinct from other Indian systems – i.e. brahmi or kharosthi numerals - that had abstract numeral-signs.^[5] Alike the alphabetic systems of Europe and the Middle East, these systems used phonetic signs of a script for numeration, but they were more flexible than those. Three significant systems of them: Āryabhaṭa numeration, katapayadi system, and the aksharapalli numerals.

Alphasyllabic numeration are very important for understanding Indian astronomy, astrology, and numerology, since Indian astronomical texts were written in Sanskrit verse, which had strict metrical form. These systems had the advantage of being able to give any word a numerical value, and to find many words corresponding to one given number. This made possible the construction of various mnemonics to aid scholars and students, and would have served a prosodic function.

Structure

Aryabhata numeration table (varga consonants, and avarga consonants are ordered to 1-25, and 30 -100; vowels to the powers of 100) Aryabhata-Code-Tabelle.png — Āryabhaṭa numeration table (varga consonants, and avarga consonants are ordered to 1-25, and 30 -100; vowels to the powers of 100)

Structure of the Indian alphasyllabic numeration systems differs basically from one another. Though in each of the systems consonants and vowels are ordered to numeric values, thereby each syllable has a numeric value, but on the base of each system's own rules. In various systems the V, CV, CCV syllables receive different values, and the methods, how the numbers are represented by these syllables, are quite different.

Āryabhaṭa numeration system operates on the additive principle, so that the number's value, which is represented in it, is computed as the sum of each syllable's numeric value. In his mapping, the consonants are ordered from 1 to 25, then by tens from 30 to 100. Each successive vowel is ordered to the different exponent of 100. In Āryabhaṭa numeration’s the diacritic signs, which mark vowels, multiply the value of the syllable’s consonant by the given power of 100. Direction of his script is right to left, which reflects the order of the Sanskrit lexical numerals.^[6]
In katapayadi system, syllables have the numeric values only from 0 to 9. To each V, CV and CCV syllable is given a value between 0 and 9. In this way each number between 0 and 9 are ordered to several syllables. Unlike Aryabhata's system, changing the vowel in the syllable doesn’t change the syllable’s numerical value. The number’s value, which is represented in this way, is given as positional number with one syllable on each position. Direction of this script is right to left.^[7]

Numerals of the katapayadi system
1	2	3	4	5	6	7	8	9	0
ka क క ക	kha ख ఖ ഖ	ga ग గ ഗ	gha घ ఘ ഘ	nga ङ జ్ఞ ങ	ca च చ ച	cha छ ఛ ഛ	ja ज జ ജ	jha झ ఝ ഝ	nya ञ ఞ ഞ
ṭa ट ట ട	ṭha ठ ఠ ഠ	ḍa ड డ ഡ	ḍha ढ ఢ ഢ	ṇa ण ణ ണ	ta त త ത	tha थ థ ഥ	da द ద ദ	dha ध ధ ധ	na न న ന
pa प ప പ	pha फ ఫ ഫ	ba ब బ ബ	bha भ భ ഭ	ma म మ മ	-	-	-	-	-
ya य య യ	ra र ర ര	la ल ల ല	va व వ വ	śha श శ ശ	sha ष ష ഷ	sa स స സ	ha ह హ ഹ	-	-

In aksharapalli system, syllables were assigned the numerical values 1–9, 10–90, but never as high as 1000. According to S. Chrisomalis there was never a single regular system for correlating signs with numeral values in this system. It was used widely for paginating books, aksharapalli numerals were written in the margins from top to bottom.^[8]

Systems

Related Research Articles

An abugida, sometimes known as alphasyllabary, neosyllabary or pseudo-alphabet, is a segmental writing system in which consonant–vowel sequences are written as units; each unit is based on a consonant letter, and vowel notation is secondary. This contrasts with a full alphabet, in which vowels have status equal to consonants, and with an abjad, in which vowel marking is absent, partial, or optional – in less formal contexts, all three types of script may be termed "alphabets". The terms also contrast them with a syllabary, in which the symbols cannot be split into separate consonants and vowels.

The Hindu–Arabic numeral system is a decimal place-value numeral system that uses a zero glyph as in "205".

Āryabhaṭa numeration is an alphasyllabic numeral system based on Sanskrit phonemes. It was introduced in the early 6th century in India by Āryabhaṭa, in the first chapter titled Gītika Padam of his Aryabhatiya. It attributes a numerical value to each syllable of the form consonant+vowel possible in Sanskrit phonology, from ka = 1 up to hau = 10¹⁸.

<span class="mw-page-title-main">Katapayadi system</span>

Kaṭapayādi system of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered.

Aksharapalli is a certain type of alphasyllabic numeration scheme extensively used in the pagination of manuscripts produced in India in pre-modern times. The name Aksharapalli can be translated as the letter system. In this system the letters or the syllables of the script in which the manuscript is written are used to denote the numbers. In contrast to the Aksharapalli system, the ordinary decimal system is called the Ankapalli system.

An alphabetic numeral system is a type of numeral system. Developed in classical antiquity, it flourished during the early Middle Ages. In alphabetic numeral systems, numbers are written using the characters of an alphabet, syllabary, or another writing system. 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, Georgian and Hebrew systems, use an already established alphabetical order. Alphabetic numeral systems originated with Greek numerals 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, religious functions, and divinatory magic.

A is a vowel of Indic abugidas. In modern Indic scripts, A is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . Bare consonants without a modifying vowel sign have the "A" vowel inherently, and thus there is no modifier sign for "A" in Indic scripts.

Ā is a vowel of Indic abugidas. In modern Indic scripts, Aa is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, "Ā" comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent short "A" vowel.

I is a vowel of Indic abugidas. In modern Indic scripts, I is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, "I" comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ī is a vowel of Indic abugidas. In modern Indic scripts, Ii is derived from the early "Ashoka" Brahmi letter . As an Indic vowel, Ī comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

U is a vowel of Indic abugidas. In modern Indic scripts, U is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, U comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ū is a vowel of Indic abugidas. In modern Indic scripts, Ū is derived from the early "Ashoka" Brahmi letter . As an Indic vowel, Ū comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ṛ is a vowel symbol, or vocalic consonant, of Indic abugidas. In modern Indic scripts, Ṛ is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, Ṛ comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ṝ is a vowel-like letter of Indic abugidas, often referred to as a "vocalic R̄". In modern Indic scripts, Ṝ is derived from the early "Ashoka" Brahmi letter . As an ostensible Indic vowel, Ṝ comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ḷ or Vocalic L is a vowel symbol of Indic abugidas. In modern Indic scripts, Ḷ is derived from the Brahmi letter . As an Indic vowel, Ḷ comes in two normally distinct forms: 1) as an independent letter, and 2) as a sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ḹ is a vowel of Indic abugidas. In modern Indic scripts, Ḹ is derived from the early "Ashoka" Brahmi letter . As an Indic vowel, Ḹ comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

E is a vowel of Indic abugidas. In modern Indic scripts, E is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, E comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Ai is a vowel of Indic abugidas. In modern Indic scripts, Ai is derived from the early "Ashoka" Brahmi letter . As an Indic vowel, Ai comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

O is a vowel of Indic abugidas. In modern Indic scripts, O is derived from the early "Ashoka" Brahmi letter after having gone through the Gupta letter . As an Indic vowel, O comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

Au is a vowel of Indic abugidas. In modern Indic scripts, Au is derived from the middle "Kushana" Brahmi letter , and the Gupta letter . As an Indic vowel, Au comes in two normally distinct forms: 1) as an independent letter, and 2) as a vowel sign for modifying a base consonant. Bare consonants without a modifying vowel sign have the inherent "A" vowel.

References

↑ Stephen Chrisomalis (2010). Numerical Notation: A Comparative History. Cambridge University Press. p. 205. ISBN 9780521878180 . Retrieved 2019-07-05.
↑ S. Chrisomalis 2010: p. 206.
↑ Datta and Singh 1962 [1935]
↑ S. Chrisomalis 2010: p. 206.
↑ Ifrah 1998: p. 483.
↑ S. Chrisomalis 2010: p. 208.
↑ S. Chrisomalis 2010: p. 209.
↑ S. Chrisomalis 2010: p. 212.

Sources

Stephen Chrisomalis (2010). Numerical Notation: A Comparati-ve History. Cambridge University Press. ISBN 9780521878180 . Retrieved 2019-07-05.
Datta, Bibhutibhusan; Singh, Avadhesh Narayan (1962) [1935]. History of Hindu Mathematics. Bombay: Asia Publishing House.
Georges Ifrah: The Universal History of Numbers. From Prehistory to the Invention of the Computer. John Wiley & Sons, New York, 2000, ISBN 0-471-39340-1.