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 (Rig Veda, Yajur Veda , Atharva Veda, Sama Veda)

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.

aśvinī ākāśa guṇa kha candra rūpa bhūmi rāma bhūta bāṇa vāyu mahayajña ṛtu vedāṅga

List of words used as Bhūta-saṅkhyā

Numeral	Bhūta-saṅkhyā	Meaning	Details
0	Kha , Ākāśa	Space, Sky	Space represents void meaning zero
1	Candra	Moon	Earth has only one Moon
	Rūpa	Form	Form is unique for each individual
	Bhūmi	Earth	Earth is one
2	Netra	Eye	Humans have two eyes
	Yamala	Twin	Twins
	Aśvinī	Twin Devatās	Nāsatya, Dasra - Twin Aśvinī Devatas.
3	Guṇa	Quality	Sattva, Rajas, Tamas - 3 Qualities
	Rāma	Rāma	Paraśurāma, Balarāma, Dāśarathi Rāma
4	Veda	Veda	Ṛg, Yajur, Sāma and Atharva - 4 Vedas
5	Bhūta	Elements	Earth, Water, Fire, Air and Ākāśa - 5 elements
	Mahayajña	Sacrifice	Devayajña, Pitṛyajña, Manuyajña, Manuṣyayajña, Bhūtayajña - Pañcamahāyajña
6	Vedāṅga	6 auxiliary disciplines of Hinduism	Śikṣā, Vyākaraṇa, Kalpa, Nirukta, Jyotiṣa, Candas
	Rasa	Taste	Sweet, salt, sour, pungent, astringent, and bitter
7	Muni, Ṛsi	Sage	kaśyapa atri vasiṣṭha viśvāmitra gautama jamadagni bhāradhvāja
	Parvata	Mountain	Mahendra, Malaya, Sahya, Vindhya, Śaktimān, Ṛkṣa, Pāriyātra
	Svara	Musical Note	Seven notes in Indian music - ṣaḍja, ṛśabha, gāndhāra, madhyama pañcama, dhaivata, niśāda.
8	Vasu	A class of Devas	Dharā, Anila, Anala, Āpa, Pratyūṣa, Prabhāsa, Soma, Dhruva.
	Prakṛti	Nature	Pañcabhūtās, manas, buddhi, Ahaṅkāra.
9	Śarīra-dvāra	Apertures in the body	Two eyes, two ears, two nostrils, one mouth and organs of excretion and generation.
	Nanda	Nanda dynasty kngs	Famous Nanda dynasty kings - Ugrasena, Pāṇḍuka, Pāṇḍugati, Bhūtapāla, Rāṣṭrapāla, Goviśaṇaka, Daśasiddhaka,  Kaivarta, Dhanānanda.
10	Dik	Direction	The ten directions are East, West, North, South, North-East, North-West, South- East, South-West, Upwards and downwards.
11	Rudra	Devatās	Eleven Rudras - Manyu, Manu, Mahīna, Mahān, Śiva, Ṛtudhvaja, Ugraceta, Bhava, Kāma, Vāmadeva, Dhṛtavrata.
12	Sūrya	Sun	Mitra, Ravi, Sūrya, Bhānu, Khaga, Pūśa, Hiraṇyagarbha, Marīci, Āditya, Savitṛ, Arka, Bhāskara.
13	Viśvedeva	A class of Devas	Special class of Devas who are 13 in number.
14	Manu	Heads of kalpa	14 Manus are mentioned in the Purāṇic literature. They are Svāyambhuva, Svārociṣa, Uttama, Tāpasa, Raivata, Cākṣuṣa Vaivasvata, Sāvarṇī, Dakṣa-Sāvarṇī, Brahma-Sāvarṇī, Dharma-Sāvarṇī, Deva-Sāvarṇī, Rudra-Sāvarṇī, Indra-Sāvarṇī
15	Thithi	Days of a pakṣa	15 thithis from prathamā to amāvāsyā (new moon day) or paurṇamī (full moon day)
16	Atyaṣṭi	Type of chandas	A poetic metre of 17 letters in each of its quarter.
18	Dhṛti	Type of chandas	A poetic metre of 18 letters in each of its quarter.
27	Nakṣatra	Constellation of Stars	In Indian astronomy, there are 27 Nakṣatra or sectors along the ecliptic.
32	Danta	Teeth	Number of teeth in humans.
33	Deva, Devata	Celestial beings	33 types of devas referred in purāṇās.