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

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	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.

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.

Aśvinī	Netra	Ākāśa	Yamala	aṅkānāṃ vāmato gatiḥ
2	2	0	2	2022

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

विबुध	नेत्र	गज	अहि	हुताशन	त्रि	गुण	वेद	भ	वारण	बाहवः
vibudha	netra	gaja	ahi	hutāśana	tri	guṇa	veda	bha	vāraṇa	bāhavaḥ
33	2	8	8	3	3	3	4	27	8	2

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

${\frac {Circumference}{Diameter}}={\frac {2827433388233}{9\quad X\quad 10^{11}}}=3.14159265359$

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)

Katapayadi-system

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.

1	2	3	4	5	6	7	8	9	0
क	ख	ग	घ	ङ	च	छ	ज	झ	ञ
ट	ठ	ड	ढ	ण	त	थ	द	ध	न
प	फ	ब	भ	म
य	र	ल	व	श	ष	स	ह	ळ
									अ, आ, इ, ई, उ, ऊ, ऋ, ॠ, ए, ऐ, ओ, औ

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

र्	घ्	व्	य्	अङ्कानाम् ‌वामतो गतिः
2	4	4	1	→ 1442

भवति (bhavati) (Indian Antiquary vol.2.p.60) भ् + अ + व् + अ + त् + इ

भ्	व्	त्	अङ्कानाम् ‌वामतो गतिः
4	4	6	→ 644

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)

भ्	र्	ब्	ध्	स्	ध्	ज्	म्	ग्	ण्	त्	र्	ध्	म्	य्	भ्	प्	ग्	अङ्कानाम् ‌वामतो गतिः
4	2	3	9	7	9	8	5	3	5	6	2	9	5	1	4	1	3	→ 314159265358979324

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

See Also

संख्यान प्रणाली

External Links

• Bhutasamkhya_system
• Katapayaadi