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 |
miśre 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:
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
---|---|---|---|---|---|---|---|---|---|
ka | kha | ga | gha | ṅa | ca | cha | ja | jha | ña |
ṭa | ṭha | ḍa | ḍha | ṇa | ta | tha | da | dha | na |
pa | pha | ba | bha | ma | - | - | - | - | - |
ya | ra | la | va | śa | ṣa | sa | ha | - | - |
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 ṛ (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
bhavati ("becomes") encodes the numerical string 4-4-6, indicating the number 644