Āryabhaṭa Numeration
How a 6th-century astronomer packed numbers into verse, one Sanskrit syllable at a time.
Āryabhaṭa (b. 476 CE) opens his Āryabhaṭīya, the foundational Sanskrit text on astronomy and mathematics (written around 499 CE), with a chapter called the Gītikāpada, "the verse of song." Its second stanza doesn't teach astronomy yet. It teaches a code. Sanskrit consonants and vowels, combined as they are in any ordinary syllable, stand for digits and decimal places. So a number of any size can be spelled as a pronounceable word and dropped straight into a metrical verse. Āryabhaṭa used the code for the astronomical constants in the rest of the book, from small counts up to the billions.
The core rule
Every syllable is a consonant plus a vowel, and each half does a different job:
Consonant → the digit (1–100)
The 8 "non-class" (avarga) consonants, य र ल व श ष स ह, count by tens, 30 to 100.
Vowel → the place (a power of 100)
Put together: value(consonant + vowel) = consonant's number × 100(vowel's position, 0–8). ख (kha = 2) alone is 2. खि (khi), which is kha with the second vowel, i, is 2 × 100¹ = 200. खु (khu) is 2 × 100² = 20,000, and so on up to खौ (khau) = 2 × 10¹⁶.
Consonant values
The 25 varga consonants, in the same grid as the Characters page, now carry a number as well as a sound:
| Velar | Palatal | Retroflex | Dental | Labial | |
|---|---|---|---|---|---|
| Unaspirated voiceless | क ka 1 |
च ca 6 |
ट ṭa 11 |
त ta 16 |
प pa 21 |
| Aspirated voiceless | ख kha 2 |
छ cha 7 |
ठ ṭha 12 |
थ tha 17 |
फ pha 22 |
| Unaspirated voiced | ग ga 3 |
ज ja 8 |
ड ḍa 13 |
द da 18 |
ब ba 23 |
| Aspirated voiced | घ gha 4 |
झ jha 9 |
ढ ḍha 14 |
ध dha 19 |
भ bha 24 |
| Nasal | ङ ṅa 5 |
ञ ña 10 |
ण ṇa 15 |
न na 20 |
म ma 25 |
The 8 avarga consonants count on by tens to 100. They are the same semivowel and sibilant sounds from the Characters page, with ह moved to the end as the highest single-consonant value:
य ya 30 |
र ra 40 |
ल la 50 |
व va 60 |
श śa 70 |
ष ṣa 80 |
स sa 90 |
ह ha 100 |
Vowel places
The same nine vowels from the Characters page (monophthongs, syllabic liquids and diphthongs), used here as a base-100 place-value ladder:
अ a 100⁰ 1 |
इ i 100¹ 100 |
उ u 100² 10,000 |
ऋ ṛ 100³ 10⁶ |
ऌ ḷ 100⁴ 10⁸ |
ए e 100⁵ 10¹⁰ |
ऐ ai 100⁶ 10¹² |
ओ o 100⁷ 10¹⁴ |
औ au 100⁸ 10¹⁶ |
The Āryabhaṭa system doesn't distinguish long vowels from short ones: आ counts as अ, ई as इ, and so on. Only the short forms are needed here. (Sanskrit has no separate long ऌ̄ in ordinary use, so the ladder never needs that distinction.)
Building a two-digit value
A single place (one vowel) can hold any value from 0 to 99, more than the 25 limit of a lone varga consonant, because more than one consonant can share the same vowel and their values are simply added. ग (3) alone gives 3. But 58 needs two syllables: ल (50, avarga) plus ज (8, varga), both spoken with the vowel अ, which gives ja-la. A value like 24 fits inside a single varga consonant's 1–25 range, so it can be written as one syllable (भ, bha = 24) or as two added together (घ 4 + न 20). Both are valid, and a real verse often chooses between them for the meter, not by a fixed rule.
Reading order
The one habit to unlearn: Āryabhaṭa writes the smallest place first. A number is spelled ones pair, then hundreds pair, then myriads pair, ascending from left to right, the reverse of the modern habit of writing the largest digit first. If you read any example on this page as ordinary left-to-right place value, you will get the number backwards.
Worked example
299,792,458 is the speed of light in metres per second. It is used here only because it is a familiar number; it is not from the original text. Split it into base-100 pairs from the ones end, give each pair one shared vowel, and read in Āryabhaṭa's ascending order:
| Digits | Place | Syllables | Value |
|---|---|---|---|
| 58 | 100⁰ | जल ja-la |
8 + 50 = 58 |
| 24 | 100¹ | घिनि ghi-ni |
4 + 20 = 24, ×100 |
| 79 | 100² | झुशु jhu-śu |
9 + 70 = 79, ×100² |
| 99 | 100³ | झृसृ jhṛ-sṛ |
9 + 90 = 99, ×100³ |
| 2 | 100⁴ | खॢ khḷ |
2, ×100⁴ |
Strung together in that same ascending order: जल घिनि झुशु झृसृ खॢ (ja-la ghi-ni jhu-śu jhṛ-sṛ khḷ). That is a five-syllable word encoding a nine-digit number.
A real example survives too: ङिशिबुणॢष्खृ (ṅiśibuṇḷṣkhṛ) encodes 1,582,237,500 in a historical source, though its 100⁶ and 100⁸ syllables are swapped and merged into a ligature for reasons that aren't fully understood. Āryabhaṭa's own verse for a table of sines is a real, though far more intricate, use of the system. The system may also have been able to express fractions and mixed fractions in some contexts. Wikipedia's account of this is incomplete, so it is not reproduced here as settled.
Two related but separate Sanskrit number-encoding systems exist. The kaṭapayādi system is a later cipher that maps consonants to the digits 0–9 for cyclic, calendar and memory uses (see Kaṭapayādi). The bhūtasaṃkhya system writes numbers as ordinary words for things conventionally linked with that count, like "eyes" for two. Neither is Āryabhaṭa's alphasyllabic system.