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 ṛ (ऋ) are assigned to zero. In case of a conjunct, consonants attached to a non-vowel will be valueless. For example, kya (क्य) is formed by, k (क्) + y (य्) + a (अ). The only consonant standing with a vowel is ya (य). So the corresponding numeral for kya (क्य) will be 1. There is no way of representing the 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 "अङ्कानां वामतो गतिः" which means numbers go from right to left. Variations The consonant, ḷ (Malayālam: ള, Devanāgarī: ळ, Kannada: ಳ) is employed in works using the Kaṭapayādi system, like Mādhava's sine table. Late medieval practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless. Usage Mathematics and astronomy Mādhava's sine table constructed by 14th century Kerala mathematician-astronomer Mādhava of Saṅgama·grāma employs the Kaṭapayādi system to enlist the trigonometric sines of angles. Karaṇa·paddhati, written in the 15th century, has the following śloka for the value of pi (π) അനൂനനൂന്നാനനനുന്നനിത്യൈ- സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്‍- വ്യാസസ്തദര്‍ദ്ധം ത്രിഭമൗര്‍വിക സ്യാത്‌ Transliteration anūnanūnnānananunnanityai ssmāhatāścakra kalāvibhaktoḥ caṇḍāṃśucandrādhamakuṃbhipālair vyāsastadarddhaṃ tribhamaurvika syāt It gives the circumference of a circle of diameter, anūnanūnnānananunnanityai (10,000,000,000) as caṇḍāṃśucandrādhamakuṃbhipālair (31415926536). Śaṅkara·varman's Sad·ratna·mālā uses the Kaṭapayādi system. The first verse of Chapter 4 of the Sad·ratna·mālā ends with the line: (स्याद्) भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी: Transliteration (syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ Splitting the consonants in the relevant phrase gives, Reversing the digits to modern-day usage of descending order of decimal places, we get 314159265358979324 which is the value of pi (π) to 17 decimal places, except the last digit might be rounded off to 4. This verse encrypts the value of pi (π) up to 31 decimal places. गोपीभाग्यमधुव्रात-शृङ्गिशोदधिसन्धिग॥ खलजीवितखाताव गलहालारसंधर॥ ಗೋಪೀಭಾಗ್ಯಮಧುವ್ರಾತ-ಶೃಂಗಿಶೋದಧಿಸಂಧಿಗ || ಖಲಜೀವಿತಖಾತಾವ ಗಲಹಾಲಾರಸಂಧರ || This verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792 గోపీభాగ్యమధువ్రాత-శృంగిశోదధిసంధిగ | ఖలజీవితఖాతావ గలహాలారసంధర || Traditionally, the order of digits are reversed to form the number, in katapayadi system. This rule is violated in this sloka. Carnatic music The melakarta ragas of the Carnatic music is named so that the first two syllables of the name will give its number. This system is sometimes