Systems of Numerations

Introduction
Ancient Indians through the medium of poetry made Mathematics simplified and interesting. Through poetry it is easy to remember theorem or formulae and interesting also. The Science of Indian Mathematics was composed in many meters(Chandas) of poetry. Ancient Indians found a unique way to express numbers in poetry. Hence the following two systems of number representation were born.


 * 1) Bhūta-saṅkhyā system
 * 2) Kaṭapayādi system

Bhūta-saṅkhyā
Bhūta-saṅkhyā is a system of representation of numbers where the numerals are expressed by certain words. The meanings of these words indicate numbers either naturally or by well known references in popular Indian literature. Bhūta-saṅkhyā is found in Chandassūtra of Piṅgala which was composed in 3rd Century BCE.

Representation of Numbers
Number '1' is represented by Candra which means moon. Moon is the only natural satellite of the earth which is unique. Any such unique object can be used for representing the number '1'. The other words popularly used to represent "1" are  bhūmi (earth), rūpa (form) etc.

Number '2' is represented by any popular pairs, like the eyes, the hands, the Aśvinī twins.

Number '0' is represented by the word ākāśa (space) which stands for void.

Number '3' is represented by the word guṇa (triguṇa - three qualities namely Sattva, Rajas, and Tamas).

Number '4' is represented by the word Veda (Ṛgveda, Yajurveda, Atharvarveda, Sāmaveda)

A common convention is used while using the words for representation of numbers. The Convention is to follow the rule "अङ्कानां वामतो गतिः". It means that the numbers are to be read from right to left. Numbers expressed through the words are to be formed from the units place onwards.

List of words used as Bhūta-saṅkhyā
Any of the synonyms of the above words can also be used to represent the respective numeral. For example: moon (Candra) - Soma, Indu, Śaśi.

Examples of Bhūta-saṅkhyā
1.The year 2022 can be expressed as follows. 2. Nīlakaṇṭha Somasutvan quotes a śloka given by Mādhava in his Āryabhaṭīya-bhāśya. This śloka provides the value of π (pi which is the ratio of the circumference to the diameter of the circle) in Bhūta-saṅkhyā system as follows:

विबुधनेत्रगजाहिहुताशनत्रिगुणवेदभवारणबाहवः ।                                                                                                                                                                 नवनिखर्वमिते वृतिविस्तरे परिधिमानमिदं जगदुर्बुधाः ॥

"The Circumference of the circle with diameter 9 X 1011 is 2827433388233."

As per Bhūta-saṅkhyā sytem The numbers are 33 2 8 8 3 3 3 4 27 8 2.

As per aṅkānāṃ vāmato gatiḥ the number is reverse of the above number which is 2827433388233.

Circumference of the circle = 2827433388233   Diameter of the circle = 9 X 1011

$$\frac{Circumference}{Diameter} =\frac{2827433388233}{9 \quad X \quad10^{11}} = 3.14159265359$$

The Value of π is obtained upto 11 decimal places from this śloka. By way of śloka student remembers important numbers with ease.

Kaṭapayādi System ( Letter Numerals)
Kaṭapayādi is a notational system for representing numbers  using Samskrit varṇamālā. In this system, the consonants are used in place of digits to express numbers. Important points to be noted:

नञौ अचः च शून्यानि सङ्ख्याः कटपयादयः                                                                                                                                                             मिश्रे तु उपान्त्यहल् सङ्ख्या न च चिन्त्यः हलः स्वरः।

Above verse from Sadratnamālā (सद्रत्नमाला) by Śaṅkaravarman, describes the method to be followed.


 * Consonants are assigned values as given in the table above.
 * Vowels, such as अ, आ, इ, .. etc., are assigned the value 0.
 * In a joint letter, only the last consonant, the one that appears with a vowel, is to be considered.
 * Consonants without a vowel (हलन्त) are to be ignored.
 * A general rule is - "अङ्कानाम् ‌वामतो गतिः", that is, numbers go from right to left.

Examples of Kaṭapayādi System
राघवाय (rāghavāya) ( Epigrpahia Indica Vol.6 p.121 ) र् + आ + घ् + अ + व् + आ + य् + अ भवति (bhavati) ( Indian Antiquary vol.2.p.60 ) भ् + अ + व् + अ + त् + इ Sadratnamālā text mentions the value of π (pi) upto 17 decimal places.

स्याद् भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगीः । (Sadratnamālā IV.2, p.26)

The number for भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगीः using Kaṭapayādi system.

(भ्+अ+द्+र्+आ+म्+ब्+उ+ध्+इ+स्+इ+द्+ध्+अ+ज्+अ+न्+म्+अ+ग्+अ+ण्+इ+त्+अ+श्+र्+द्+ध+आ) (स्+म्+अ) (य्+अ+द्) (भ्+ऊ+प्+अ+ग्+ईः) As per the text by dividing this number (circumference of a circle) with the diameter of the circle as 1017, we get the value of π (pi) as 3.14159265358979324