Ka-ṭa-pa-yā-di (Devanagari: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്‍പ്പേര്|പരല്‍പ്പേര്) of numerical notation is an ancient Indian 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.

## Rules and practices

See these documents for descriptions of katapayadi:

``````नज्ञावचश्च शून्यानि संख्या: कटपयादय:|
मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:||
``````

Transiliteration:

``````nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ
miśre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ
``````

Translation: na (न), nya (ञ) and a (अ)-s i.e. vowels represent zero. The (nine) integers are represented by consonant group beginning with ka, ṭa, pa, ya. In a conjunct consonant, the last of the consonants alone will count. A consonant without vowel is to be ignored.

Explanation:

The assignment of letters to the numerals are as per the following arrangement.

1234567890
ka क കkha ख ഖga ग ഗgha घ ഘnga ङ ങca च ചcha छ ഛja ज ജjha झ ഝnya ञ ഞ
ṭa ट ടṭha ठ ഠḍa ड ഡḍha ढ ഢṇa ण ണta त തtha थda द ദdha ध ധna न ന
pa प പpha फ ഫba ब ബbha भ ഭma म മ-----
ya य യra र രla ल ലva व വśha श ശsha ष ഷsa स സha ह ഹ--
• Consonants have numerals assigned as per the above table. For example, ba (ब) is always three 3 whereas 5 can be represented by either nga (ङ) or ṇa (ण) or ma (म) or śha (श).
• All stand-alone vowels like a (अ) and (ऋ) are assigned to zero 0.
• In case of a conjuct, consonants attached to a non-vowel will not be valueless. For example, kya (क्या) is formed by k (क्) + ya (य) + a (अ). The only consonant standing with a vowel is ya (य). So the corresponding numeral for kya (क्या) will be 1.
• There is no way of representing decimal separator in the system.
• Indians used the Hindu-Arabic numeral system for numbering, traditionally written in increasing place values from left to right. This is as per the rule aṅkānām vāmato gati (अङ्कानाम् वामतो गति) which means numbers go from left to right.

### Variations

• The Dravidian consonant, ḷ (Malayalam: ള, Devanagari: ळ), of the Dravidian languages of south India is employed in works using the Kaṭapayādi System, like the Madhava’s sine table.
• Apparently, some practicers doesn’t map the stand-alone vowels to zero. But considers as valueless.