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== Introduction ==
== 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.
Ancient Indians  through the medium of poetry made [[Development of Mathematics|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.


# Bhūta-saṅkhyā system
# Bhūta-saṅkhyā system
Line 6: Line 6:


== Bhūta-saṅkhyā ==
== 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.
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<ref>{{Cite book|title=A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1.|publisher=Samskrit Promotion Foundation|year=2021|isbn=978-81-951757-2-7|location=Delhi}}</ref>.


====== Representation of Numbers ======
====== Representation of Numbers ======
Line 17: Line 17:
Number '3' is represented by the word guṇa (triguṇa - three qualities namely Sattva, Rajas, and Tamas).
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)
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 ======
{| class="wikitable"
|+
!Numeral
!Bhūta-saṅkhyā
!Meaning
!Details
|-
|0
|Kha , Ākāśa
|Space, Sky
|Space represents void meaning zero
|-
| rowspan="3" |1
|Candra
|Moon
|Earth has only one Moon
|-
|Rūpa
|Form
|Form is unique for each individual
|-
|Bhūmi
|Earth
|Earth is one
|-
| rowspan="3" |2
|Netra
|Eye
|Humans have two eyes
|-
|Yamala
|Twin
|Twins
|-
|Aśvinī
|Twin Devatās
|Nāsatya, Dasra - Twin Aśvinī  Devatas.
|-
| rowspan="2" |3
|Guṇa
|Quality
|Sattva, Rajas, Tamas - 3 Qualities
|-
|Rāma
|Rāma
|Paraśurāma,  Balarāma, Dāśarathi Rāma
|-
|4
|Veda
|Veda
|Ṛg, Yajur, Sāma and Atharva - 4 Vedas
|-
| rowspan="2" |5
|Bhūta
|Elements
|[[File:Prakrti.png|alt=Elements|thumb|Elements]]Earth, Water, Fire, Air and Ākāśa - 5 elements
|-
|Mahayajña
|Sacrifice
|Devayajña, Pitṛyajña, Manuyajña, Manuṣyayajña, Bhūtayajña - Pañcamahāyajña
|-
| rowspan="2" |6
|Vedāṅga
|6 auxiliary disciplines of Hinduism
|Śikṣā, Vyākaraṇa, Kalpa, Nirukta, Jyotiṣa, Candas
|-
|Rasa
|Taste
|Sweet, salt, sour, pungent, astringent, and bitter
|-
| rowspan="3" |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.
|-
| rowspan="2" |8
|Vasu
|A class of Devas
|Dharā, Anila, Anala, Āpa, Pratyūṣa, Prabhāsa, Soma, Dhruva.
|-
|Prakṛti
|Nature
|Pañcabhūtās, manas, buddhi,  Ahaṅkāra.
|-
| rowspan="2" |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.<ref>{{Cite web|title=List of Nakshatras|url=https://en.wikipedia.org/wiki/List_of_Nakshatras}}</ref>
|-
|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.
{| class="wikitable"
|+
!Aśvinī
!Netra
!Ākāśa
!Yamala
!aṅkānāṃ vāmato gatiḥ
|-
|2
|2
|0
|2
|2022
|}
2. [[Nilakantha Somasutvan|Nīlakaṇṭha Somasutvan]] quotes a śloka given by [[Madhava|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 ॥<ref>{{Cite book|title=Āryabhaṭīya-bhāśya by Nīlakaṇṭha Somasutvan|pages=42}}</ref>
 
"The Circumference of the circle with diameter 9 X 10<sup>11</sup> is 2827433388233."
 
As per Bhūta-saṅkhyā sytem
{| class="wikitable"
|+
|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 10<sup>11</sup>
 
<math>\frac{Circumference}{Diameter} =\frac{2827433388233}{9 \quad X  \quad10^{11}} = 3.14159265359</math>
 
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) ==
[[File:Katapayadi-eng.svg|alt=Katapayadi-system|thumb|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.
{| class="wikitable"
|+
!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
|
|-
| colspan="9" |
|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) (<small>Epigrpahia Indica Vol.6 p.121</small>)  र् + आ + घ् + अ + व् + आ + य् + अ
{| class="wikitable"
|+
!र्
!घ्
!व्
!य्
!अङ्कानाम् ‌वामतो गतिः
|-
|2
|4
|4
|1
|→ 1442 
|}
भवति (bhavati) (<small>Indian Antiquary vol.2.p.60</small>)  भ् + अ + व् + अ + त् + इ
{| class="wikitable"
|+
!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 ।<ref>{{Cite book|title=Sadratnamālā|pages=IV.2, p.26}}</ref> <small>(Sadratnamālā IV.2, p.26)</small>
 
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)
{| class="wikitable"
|+
!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 10<sup>17</sup>, we get the value of π (pi) as 3.14159265358979324
 
== <small>See Also</small> ==
[[संख्यान प्रणाली]]
 
== External Links ==
 
* [[:en:Bhutasamkhya_system|Bhutasamkhya_system]]
* [https://sanskrit.iitk.ac.in/jnanasangraha/sankhya/katapayaadi/help/sa/ Katapayaadi]
 
== References ==
 
<references />
 
[[Category:Arithmetic]]
[[Category:Mathematics]]
[[Category:Organic Articles English]]

Latest revision as of 15:21, 15 November 2022

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[1].

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
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.[2]
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 ॥[3]

"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 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

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
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
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 ।[4] (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

References

  1. A Primer to Bhāratīya Gaṇitam , Bhāratīya-Gaṇita-Praveśa- Part-1. Delhi: Samskrit Promotion Foundation. 2021. ISBN 978-81-951757-2-7.
  2. "List of Nakshatras".
  3. Āryabhaṭīya-bhāśya by Nīlakaṇṭha Somasutvan. p. 42.
  4. Sadratnamālā. pp. IV.2, p.26.