HAMSIHAMSI

The History of Astronomical and Mathematical Sciences in India (HAMSI) Working Group is dedicated to fostering and disseminating research in the history of Indian exact sciences (including astronomy, mathematics, and related subjects), and integrating it into the history of mathematics in general.

Letter Numerals: the Kaṭapayādi system of numeration

Kaṭapayādi notation is a system of numeration for the verbal representation of numbers. The scheme designates each of the 33 Sanskrit consonants a corresponding number from 0 to 9.  The resulting letter-number associations are used to encode long strings of numbers into words.  The resulting text is thus one that has a double meaning: a verbal reading and as well as a numerical one. This system allows large numbers to be incorporated into versified text and to be more easily memorised.  Kaṭapayādi notation was most popular in the astral sciences in South India.

The earliest known usage of Kaṭapayādi dates back to the seventh century.  The details of this system  are expounded in Śaṅkaravarman's Sadratnamāla in a single verse:

nañāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ |
       mi
śre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ ||

[The letters,] na, ña and the [stand-alone] vowels a [etc.] represent “zero” The numeration [i.e. “one” through to “nine” are represented by the [consonant-groups] beginning with ka, ṭa, pa, and ya [going serially along the alphabet series.] In a mixed/conjunct consonant, only the last of the consonant counts. A consonant with a vowel-sound is not regarded [i.e. ignored] from the enumeration.

The correspondence between numerals and the letters of the Sanskrit alphabet is as follows:

1234567890
kakhagaghaṅacachajajhaña
ṭaṭhaḍaḍhaṇatathadadhana
paphababhama-----
yaralavaśaṣasaha--

As the above tables shows, the consonants ka, ṭa, pa, and ya all signify the digit "1".  The name, Kaṭapayādi, comes from these collection of consonants.

Remarks
  • The association of numbers to letters is in accordance with the table above; all numbers can be with multiple alphabets, e.g. the number 6 can be represented by either ca, ta or ṣa.
  • All stand alone vowels, e.g. ū or (as and when they arise in the verse) are regarded as 0.
  • The conjunct consonants are regarded with the consonant attached to a non-vowel (i.e. consonants without the vowel-stem) as valueless and only the last consonant is considered for its value, e.g. in case of the conjunct bhra which is formed as bh + ra, the value associated with this conjunct if only taken as ra, i.e. 2 (treating the vowel-less consonant, bh as valueless).
  • Fractional separations between numbers (e.g. decimal or sexagesimal) are generally not explicated in this system of numeration.
  • The numbers are to be considered with increasing place value from left to right in line, in keeping with the injunction, aṅkānām vāmato gati (a `reading' of the numbers from the left).
Examples

bhavati ("becomes") encodes the numerical string 4-4-6, indicating the number 644