Aryabhatta’s System of Numeration

Aryabhattiya, written by the great Aryabhatta, is a magnificent work in the field of Mathematics, Astronomy and Geometry. Written in poetic Sanskrit verses, Aryabhattiya has 121 verses organized in 4 chapters (or padas). These four chapters are —

1. Gitika
2. Ganita (Mathematics)
3. Kaalakriya (Reckoning of Time)
4. Gola (Celestial Sphere)

In the first chapter (Gitika Pada) of Aryabhattiya, Aryabhatta explains the system he has used in order to express numbers using alphabets. This system of numeration has been used to express large numbers and express those numbers in such a way that it perfectly fits in the poetic meter used for the composition. For this, Aryabhatta uses the alphabets of the Sanskrit language.

In Sanskrit, the consonants क (ka) to म (ma) are Varga consonants and they are grouped into 5 vargas, namely — क (ka) varga, च (ca) varga, ट (ṭa) varga, त (ta) varga and प (pa) varga. The remaining consonants from य (ya) to ह (ha) are called Avarga consonants (not belonging to any varga).

The vowels (अच्) in Sanskrit language are अ, इ, उ, ऋ, ऌ, ए, ओ, ऐ and औ.

In the second sutra of the first chapter of Aryabhattiya (named, Gitika Pada), Aryabhatta explains on how these alphabets can be used to represent any number. Let us look into this sutra.

Consonants

Aryabhatta states that the varga consonants, i.e., all alphabets from क (ka) to म (ma) each take a numerical value of 1 to 25 respectively. For the avarga consonants, य takes the numerical value of ङम (ṅama) (which is ङ (ṅ = 5) + म (m = 25) = 30). The following avarga consonants are the next multiples of ten. We get summarize this in the following format —

Recall the place value system in number representation. Aryabhatta differentiates between the odd places (even powers of 10 - ones, hundreds, ten thousands, etc.) and the even places (odd powers of 10 - tens, thousands, lakh, etc.) The odd places which are also perfect squares are also called varga and the even places are avarga. So the place value system starts with a varga on the right and alternates between varga (V) and avarga (A).

The varga consonants occupy the varga positions in the place value system while the avarga consonants occupy the avarga positions.

Vowels

In Aryabhatta’s system of numeration, the 9 vowels are duplicated to make varga — avarga pairs. Each vowel, has a varga and avarga form and these 9 vowels denote the places in the place value system.

The दीर्घ (deergha = long) form of the vowels hold the same value. That is, आ, ई, ऊ, ॠ, ॡ are the same as अ, इ, उ, ऋ, ऌ in Aryabhatta’s sytem. The long forms of these vowels are used only to maintain the poetic meter of the composition.

ग = गा, य = या, सि = सी, etc.

Each of these vowels represents a multiplicative factor of the order 10.

Examples

Let’s decipher some words into their numerical values.

Example 1

Consider the word गृ (gɽ̩). This is a combination of the consonant ग and the vowel ऋ. Since ग is a varga consonant, the value of ग ( = 3) wilil be placed in the varga place of the vowel ऋ.

So the word गृ = 3 * 10⁶= 3000000.

Consider the word सृ (sɽ̩). सृ(sɽ̩) = स (s) + ऋ (ɽ). Since स (s) is an avarga consonant, its value ( = 90) is placed in the avarga column of the vowel ऋ. Hence, सृ = 90 * 10⁶ = 90,000,000.

Example 2

Consider the following number-chronogram — ङिशिबुण्ऌष्खृ (ṅiśibuṇl̩ṣkhɽ̩)

Let’s break this as follows ङि (ṅi) शि (śi) बु (bu) ण्ऌ (ṇl̩) ष्खृ (ṣkhɽ̩). Why? Split it as the consonant and the added vowel. Further, we can break the individual vowel consonant pair.

1. ङि (ṅi) = ङ (ṅ) + इ (i)
2. शि (śi) = श (ś) + इ (i)
3. बु (bu) = ब (b) + उ (u)
4. ण्ऌ (ṇl̩) = ण (ṇ) + ऌ (l̩)
5. ष्खृ (ṣkhɽ̩) = ष (ṣ) + ऋ (ɽ̩) and ख (kh) + ऋ (ɽ̩)

Place the varga consonants in the varga positions of the attached vowel and avarga consonants in the avarga positions of the corresponding vowel.

ङ (ṅ), ब (b), ण (ṇ) and ख (kh) are written in the varga positions of the इ (i), उ (u), ऌ (l̩) and ऋ (ɽ̩) vowels respectively, while, श (ś) and ष(ṣ) are placed in the avarga positions of इ (i) and ऋ (ɽ̩) respectively. This can be clearly explained in the following image —

Hence, the above number-chronogram holds the value —

(15 * 10⁸) + ((80 + 2) * 10⁶) + (23 * 10⁴) + ((70+5) * 10²) = 1582237500

Other Examples

In the third verse of the Gitika Pada, Aryabhatta states the following —

A. One yuga consists of 43,20,000 eastwards revolutions of the Sun.

B. Number of eastward revolutions of the moon in one yuga (43,20,000 years) = चयगियिङुशुछ्लृ (cayagiyiṅuśuchlɽ). Let’s translate this to its numerical value.

First, place the consonants in the appropriate positions.

So, चयगियिङुशुछ्लृ (cayagiyiṅuśuchlɽ) = ((50+7) * 10⁶) + ((70+5) * 10⁴) + ((30 + 3) * 10²) + ((30 +6) * 10⁰) = 57753336 (=number of eastward revolutions of moon in one yuga).

Can you try finding the number of revolutions of Saturn (Shani)? It is written as ढुङ्विघ्व which is the same as ढुङिविघव numerically. (Ans — 14656400 revolutions.)

References

1. ARYABHATTA — The Forgotten Genius | Project SHIVOHAM | https://www.youtube.com/watch?v=jgjcy04PDRM
2. Aryabhattiya of Aryabhatta — English Commentary by Kripa Shankar Shukla with K V Sarma | https://ia803001.us.archive.org/10/items/AryabhatiyaWithEnglishCommentary/Aryabhatiya-with-English-commentary_text.pdf