The Āryabhaṭīya numeration system, introduced by the ancient Indian mathematician Āryabhaṭa in the early 6th century, is an alphasyllabic numeral system that assigns numerical values to syllables. This system was outlined in the first chapter of Āryabhaṭa's seminal work, the Āryabhaṭīya, and it represents one of the earliest known examples of using letters to denote numbers in a systematic way.
The Āryabhaṭīya system is primarily used today in historical, cultural, and academic contexts, particularly in the study of ancient Indian mathematics and astronomy. Āryabhaṭa used it in his work Āryabhaṭīya to denote numbers. It serves as a foundational reference in traditional Indian astronomical texts and is studied by scholars of history of science, Sanskrit, and Indology. While it is not used in modern practical astronomy or science, its influence persists in educational materials in India and in comparative analyses of early scientific traditions.
The system is defined using the verse:
~ आर्यभटीय गीतिकापाद। (आर्यभटः)
| Rule | Meaning |
|---|---|
| वर्गाक्षराणि वर्गे | Varga letters (i.e., क to म) appear in the varga places (even powers of 10 like units, hundreds, etc.) |
| अवर्गे अवर्गाक्षराणि | Avarga letters (य to ह) appear in avarga places (odd powers of 10 like tens, thousands, etc.) |
| कात् | Letters are assigned values 1, 2, ... starting from क |
| ङ्मौ यः | ङ and म are equivalent to य (i.e., value 30) |
|
खद्विनवके स्वरा नव वर्गे अवर्गे नव अन्त्यवर्गे वा |
The nine vowels अ, इ, उ, ऋ, लृ, ए, ऐ, ओ, औ represent powers of ten and combine with consonants. They appear in both varga and avarga places and beyond if needed. |
In this system, consonants represent digits and vowels represent
powers of ten. For example, अ = 100 = 1,
इ = 102 = 100, up to औ =
1016. Consonants क-म
are valued 1-25, while य-ह are 30-100.
Each consonant-vowel combination corresponds to a specific
numerical value. The consonants represent digits and vowels
represent powers of 10. A number is constructed by multiplying
the numerical value of the consonant with the power of ten
represented by the vowel. For instance, the syllable चि is
decoded as 6 × 102 = 600.
You can read more about the Āryabhaṭīya system in Āryabhaṭīya or History of Hindu Mathematics by B. B. Datta.
The Decode tab will let you calculate the Āryabhaṭīya number for any Devanagari text.
To encode a number in Āryabhaṭīya system, first break it down in groups of two digits, e.g. 12345 as 1-23-45. Process each group, starting from the left to right and associate one vowel with each group in the order अ, इ, उ, ..., औ. For each group, if the two-digit number is less than or equal to 25, find a direct consonant representing the number with it and combine it with the associated vowel, else find the appropriate pair of varga and avarga consonants and combine the associated vowel with both the consonants. If any group is "00", skip it. Once all the groups are processed, the Āryabhaṭīya representation of the number is obtained.
The work, Saṅkhyāpaddhatiḥ, consisting of Kaṭapayādi Saṅkhyā, Āryabhaṭīya Saṅkhyā, and Bhūtasaṅkhyā encoding-decoding systems has been accepted in The First Annual Academic Conference on Indian Knowledge Systems, New Delhi, 2025.
BibTeX Citation: to be updated soon
Yes! The code is available at hrishikeshrt/sankhya.
If you enjoy using Saṅkhyāpaddhatiḥ, please star the project on GitHub.
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Contributions are welcome.