The “वेद” (Veda) which in Sanskrit literally means knowledge or wisdom has its origin in the root to know “विद्” (Vid). It is but natural that the Vedas are full of knowledge. As we are aware, knowledge includes scientific knowledge also. Ancient Samskrit and Vedic texts consist of several scientific and mathematical observations. Important scientific branches like mathematics, astronomy, metallurgy, medicine, health science, sculpture, civil engineering, agriculture, horticulture, water management, forestry, and astrology etc are found in ancient Sanskrit texts. Similarly mention should also be made about the medicinal subjects like Ayurveda, siddha. Another most important subject is Yoga Sastra. Acharya Patanajali’s Yoga Sutra consists of several scientific and mathematical studies. Varahamihira’s Brahat Saṃhitā talks about modern subjects like cloud seeding and water management.
NOTES ON VEDAS
(i) Vedas, four in number ऋग्, यजुः, साम, and अथर्वण ।
The names of Sastras or Scriptures are given below:
(i) उपनिषदः, more than one hundred and eight, all affiliated to one Veda or other
(iii) उपवेदाः (Upa-Vedas - subordinate Vedas) four in number Ayur-veda, (ऋग्) Gandharva Veda, (यजुः) Dhanur Veda (साम) and Stapathya Veda (अथर्वण)
(iv) वेदाङ्गानि (Vedangas - auxiliary sciences) six in number, शिक्षा, व्याकरणम्, छन्दः, ज्योतिषम्, निरुक्तम्, कल्पः, (siksa, vyakarna, chandas, jyothisa, niruktha and kalpa)
(v) वेद-उपांगानि (Veda Upangas - dharshans) six in number: न्याय, वैशेषिक, सांख्य, योग, मीमांस, वेदान्त (Nyaya, Vaiśeśika, Sankya, Yoga, Mimamsa, and Vedanta)
(i) इतिहास (Ithihas - epics) two in number
1. The Mahabharatha and 2. The Ramayana
पुराणा (Puranas - ancient legends) Eighteen Puranas are considered as Major which divided in three categories of six each.
उपपुराणानि (Upa Puranas). They are also eighteen in number
स्मृति (Smrtis - code of law) a host of them. Important ones are मनुस्मृतिः, याज्ञवल्क्यस्मृतिः, बृहस्पतिस्मृतिः, विष्णु-स्मृतिः, परासरस्मृतिः (Manu-Smrti, Yajñavalkya-Smrti/ Brihaspati-Smriti, Viśnu-Smrti, Parasara-Smrti, etc.,)
सूत्र (Sutras - smrtis codified in aphorisms) a host of them वेदान्तसूत्र (Vedanta Sutra), गृह्यसूत्र (Grihya Sutra), नारद-भक्ति-सूत्र (Narada Bhakthi Sutras) etc., are some of them.
These constitute sastras or scriptures of Sanatana Dharma or Vedic Religion.
All the sacred books mentioned above are divided into two broad categories ; The श्रुति (Sruthis - all of A above) and the स्मृति (Smrtis -all of B) above.
Hindu Culture and Mathematics
Mathematics is one of the very interesting subjects found in ancient Vedic literature. Number theory, cube roots, square roots, algebra, geometry, analytical geometry, calculus, trigonometry, astronomy, arithmetic and interest calculations are explained in great detail in Vedic literature. Vedas and Brahmanas are the excellent sources of astronomical and mathematical information during Vedic period. Yajur-veda also contains systematic mathematical presentations. Vedas and Brahmanas deserve scientific research for number theory, progressions, products, sums, differences, digits and other mathematical calculations. “It is said that in ancient India no science ever attained an independent existence and was cultivated for its own sake. Whatever science is found in Vedic India is supposed to have originated and grown as the handmaid of one or the other of the six “members of the Vedas” that too with the primary object of helping the Vedic rituals. It is also supposed sometimes, that any further culture of the science was somewhat discouraged by the Vedic Hindus in suspicion that it might prove a hindrance to their great quest of the knowledge of the Supreme by diverting the mind to other external channels. That is not indeed a correct view on the whole. It is perhaps true that in the earlier Vedic age, sciences grew as a help to religion. Mathematical applications in ancient India are not confined to Sanskrit alone. Such things are observed in other languages also. For instance, in Tamil there are smaller units of numbers (the decimals), having specific names as follows:
1/8 = Araikaal, 1/16 = maakaani,
1/32 = arai veesam, 1/64 = Kaal Veesam,
1/80 = kaani, 1/320 = munthiri
Numbers in Sanskrit:
A few large numbers used in India by about 5th century BCE
लक्ष (lakṣá) —
कोटि (kōṭi) —
आयुक्त (ayuta) —
नियुक्त (niyuta) —
पकोटि (pakoti) —
विवार (vivara) —
क्षोभ्य (kṣobhya) —
विवाह (vivaha) —
कोटिप्पकोटी (kotippakoti) —
बहुला (bahula) —
नागबला (nagabala) —
नाहुत (nahuta) —
तितलाम्भ (titlambha) —
व्यवस्थानपज्ञापति (vyavasthanapajñapati) —
हेतुहिला (hetuhila) —
निन्नहुता (ninnahuta) —
हेत्विन्द्रीय (hetvindriya) —
समाप्तलाम्भा (samaptalambha) —
गणणगति (gananagati) —
अक्खोभिनी (akkhobini) —
निरवाद्य (niravadya) —
मुद्राबला (mudrabala) —
सर्वाबला (sarvabala) —
बिन्दु (bindu) —
सर्वाज्ञ (sarvajña) —
विभूतगाम (vibhutangama)
अब्बुद (abbuda) —
निराब्बुद (nirabbuda) —
अहाहा (ahaha) —
अबाबा (ababa) —
अटाटा (atata) —
सोगगन्घिका (soganghika) —
उप्पाला (uppala) —
कुमुदा (kumuda) —
पुण्डरीका (pundarika) —
पदुमा (paduma) —
कथाना (kathana) —
महा कथाना (mahakathana) —
आसंक्षेया (asaṃkhyeya) —
ध्वजग्रनिषमणी (dhvajagraniśamani) —
Algebra
It is generally accepted that the technique of algebra and the concept of zero originated in India. In India around the 5th century A.D., a system of mathematics that made astronomical calculations easy was developed. In those times its application was limited to astronomy as its pioneers were astronomers. Astronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature it scores over conventional arithmetic.
Conventional mathematics termed Ganitam and algebra was referred to as Bījaganitam. Bījaganitam means 'the other mathematics' (Bīja means 'another', or 'second' and Ganitam means mathematics). The fact that this name was chosen for this system of computation implies that it was recognized as a parallel system of computation, different from the conventional one which was used since the past and was till then the only one. Some have interpreted the term Bīja to mean seed, symbolizing origin or beginning. And the inference that Bījaganitam was the original form of computation derived.
Credence is lent to this view by the existence of mathematics in the Vedic literature which was also shorthand method of computation. But whatever the origin of algebra, it is certain that this technique of computation originated in India. Aryabhata lived in the 5th century A.D. refers to Bījaganitam in his treatise on Mathematics, Aryabhatiya. Bhaskaracharya, a mathematician and astronomer authored a treatise in the 12th century AD entitled 'Siddhanta-Shiromani' containing a section on Bījaganitam.
Thus the technique of algebraic computation was known and was developed in India in earlier times. From the 13th century onwards, India was subject to invasions from the Arabs and other Islamized communities like the Turks and Afghans. Along with these invaders came chroniclers and critics like Al-beruni who studied Indian society and polity.
Value of Pi
Here is an actual sūtra of spiritual content, as well as secular mathematical significance:
गोपीभाग्य मधुव्रत शृंगिशो दधिसंधिगा। खल जीवति खाताव गलहाला रसन्दर॥
gopībhāgya madhuvrāta śṛngiso dadhisandhigā|
khalā jīvitakhaṭavā galā halā rasandra||
While this verse is a type of petition to Kriśna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.
The translation is as follows:
O Lord anointed with the yogurt of the milkmaids' worship (Kriśna), O savior of the fallen, O master, please protect me.
At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10:
pi/10 = 0.31
Thus one can memorize significant mathematical facts while offering praise to God in devotion.
Representation of Large Numbers
The place value system is built into the Sanskrit language, whereas in English we only use thousand, million, billion, etc. In Sanskrit, there are specific nomenclature for the powers of 10, the most used in modern times are दश daśa (10), शत śata (100=102), सहस्र sahaśra (1,000 = 103), अयुत ayuta (10,000 = 104), लक्ष lakśa (100,000 = 105), नियुत niyuta (1 million = 106), कोटि koti (10 million = 107), व्यर्बुद vyarbuda (100 million = 108), परार्ध parārdha (1 trillian = 1012) etc.
The removal of special importance of numbers - Instead of naming numbers in groups of three, four or eight orders of units, one could use the necessary name for the power of 10.
The notion of the term "of the order of" - To express the order of a particular number, one simply needs to use the nearest two powers of 10 to express its enormity (ie: koti koti (107 * 107 = 1014)).
1 ऋगवेद (Rig Veda)
Geometry
Error Correction & Detection Codes
First usage of Pi
2 यजुर्वेद (Yajur Veda)
2.1 Big Numbers
2.2 Concept of Infinity
3 अथर्वणवेद (Atharva Veda)
3.1 Concept of Infinity
3.2 Concept of Śunya (Zero)
Yajur Veda
Big numbers
In Yajurveda, numbers starting from four and with a difference of four forming an arithmetic series is discussed. The Yajurveda also mentions the counting of numbers up to 10^18, the highest being named parardha.
In the Taittiriya Upaniśad, there is an anuvaka (section), that extols the "Beatific Calculus" or a quasi-mathematical relationship between the highest possible bliss of a human being to the bliss of the Brahman.
Translated roughly it will be as follows: It continues by assuming that a young, good man who is fit, healthy and strong, and has all the wealth in the world, is one unit of human bliss. The anuvaka provides a precise calculation of a series of multiplications by 100 to give number 10010 units of human bliss that can be had when one attains Brahman. The previous anuvaka exhorts the aspirants to be fearless and strong, as only such a person may realize the absolute within.
INDIAN MATHEMATICIANS
In ancient India supreme importance was given to Mathematics. Following YajurVedic śloka highlight this point:
यथा शिखा मयूराणां नागाणां मणयो यथा ।
तद्वद्वेदाङ्गशास्त्राणां गणितं (ज्योतिषं) मूर्धनि स्थितम्।।
yathā śikhā mayūrāṇāṁ nāgāṇāṁ maṇayo yathā |
tadvadvedāṅgaśāstrāṇāṁ gaṇitaṁ (jyotiṣaṁ) mūrdhani sthitam||
The meaning of the above śloka runs as follows: śloka
“Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics (astronomy) is at the top of the vedanga-sastras”
In Yajurveda and Rgveda the words Ganitham and jyotisam are used. Therefore it is evident that the germs of the Indian mathematics dates back to the Vedic period. Another example of ancient mathematics is the Nakśtra (stars) system. The nakśatra system consists of 27 nakśatras dates back to Vedic period. Similarly yajurveda talks about 12 rtus (tropical) months.
ARYABHATA I
ARYABHATA I
Aryabhata was born on March 21, 476 A.D. He had early education at Kusumpura, a village near present Patna. There is a belief that Aryabhata I was born in Asamaka country. There are divergent views about the location of this place. His Aryabhatiya or Aryasiddhanta (laghu) are very popular among astronomers. It consists of four parts (i) Dasagitika (ii) Ganitapada (Mathematics), Kalakriya (reckoning of time), and Gola (sphere) deals with arithmetic and astronomy.
There is a belief that Aryabhata I has written another bigger work Aryasiddhanta (maha). This work is not available today. Aryabhata I revolutionized the study of astronomy in India. He found out that the earth is round in shape. The value of Pi was found to be of a high degree of correctness by him. He also gave rules for computing the numbers and their squares and cubes. The direct disciples of Aryabhata are Panduranga Swami, Latadeva, and Nisanku.
The system of astronomy explained in Arysabhatiyam is called Audayika system since the day begins from the sunrise. Later Mathematicians like Varahamihira and Brahmagupta talks Ardharatrika system expounded by the Aryabhata I. The Ardharatrika system written by him is not available to us.
His major contributions are:
Alphabetical representation of numbers
The trigonometric tables and ratios
Earth’s shape and rotation
Yuga theory
Arithmetic and Geometric progressions.
Aryabhata I is known for his innovation in astronomical studies which made him pioneer in Indian Astronomy. Later Mathematicians like Varahamihira, Brahmagupta and Baskara I glorified his work and praised for his innovative contribution to the Indian Astronomical studies. In fact Baskara I says, “None except Aryabhata has been able to know the motion of the heavenly bodies; others merely move in the ocean of utter darkness of ignorance.”
BHASKARA I
BHASKARA I
Bhaskara I was the great admirer of Arayabhata. Even though he was not a direct disciple of Aryabhata I, he made detailed commentary on Aryabhatiyam. Aryabhata’s birth place and personal details were explained by Bhaskara I. The personal details like his place of birth, his parents’ name, etc., are not available to us now. With the references he made, it is presumed that Bhaskara I originates from the Vallabhi which is located near Kathiawar and Bhaunagar of the present day Gujarat.
The renowned author Śri Kuppanna Sastri has concluded in his research that BhaskaraI flourished in 600 A.D.
Bhaskara I have composed three renowned works (1) Maha Bhaskariyam (2) Laghu Bhaskariyam and (iii) Aryabhatiya Bhasya.
Maha-bhaskariyam is an astronomical work with eight chapters, Laghu-bhaskariyam is an abridged and simplified version of Mahabaskariyam, and Aryabhatiya bashya is a commentary on the Aryabhatiya of Aryabhata I. His three books were regarded as ideal textbooks on the subject of astronomy for several centuries by students.
VARAHAMIHIRA
YAJUR VEDA (xvii, 24,25) CHAMAKAM
Chamakam is yet another Yajur Veda chapter. The arithmetical numbers in Arithmetic and Geometric progressions are explained in systematic manner. Coding is also being used in certain places. Various numbers starting from one to forty eight are discussed.
एका च मे तिस्रश्च मे पञ्च च मे सप्त च मे नव च म एकदश च मे त्रयोदश च मे पंचदश च मे सप्तदश च मे नवदश च मे एक विशतिश्च मे त्रयोविशतिश्च मे पंचविशतिश्च मे सप्तविशतिश्च मे नवविशतिश्च मे एकत्रिशच्च मे त्रयस्त्रिशच्च मे
Ĕkā chame thiśraschame pancha chame saptha chame navachame Ĕkādasa cha me thrayodasachame panchadasa chame sapthadasacha me, navadasachame Ĕkāvimsathischame, thrayovimsathischa me panchavimsathi schame sapthavimsatischame navavimsathischame Ĕkā thrimsaschame thryasthrimscha me ……..
The panchadhi given above is the arithmetic progression of odd numbers starting from 1 and ending in 33, proportionately with common difference of 2.
एका च दश च दश च शतंचशतं च सहस्रं च सहस्रं चायुतं चायुतं च नियुतं च नियुतं च प्रयुतं चार्बुदं च समुद्रश्च मध्यं चान्तश्च परार्धश्चैता…
Ĕkā cha dasam cha dasam cha satamchasatam cha sahaśram cha sahaśram chayutham cha ayutam cha niyutham cha niyutham cha prayutham cha prayutham chaarbudam cha samudrascha madhyam chaanthascha paraardhaschaithaa …
चतस्रश्च मेऽष्टौ च मे द्वादश च मे षोडश च मे विशतिश्च मे चतुर्विशतिश्च मेऽष्टाविशतिश्च मे
Chathaśras cha may, aśtou cha may, dwadasa cha may, śodasa cha may, Vimsathis cha may, chatur vimsathis cha may, aśtaa vimsathis cha may,
द्वात्रिशच्च मे, षट्त्रिशच्च मे, चत्वरिशच्च मे,
चतुश्चत्वारिशच्च मे, ऽष्टाचत्वारिशच्च मे,
वाजश्च प्रसवश्चापिजश्च क्रतुश्च सुवश्च मूर्धा च
व्यश्नियश्चान्त्यायनश्चान्त्यश्च भौवनश्च
भुवनश्चाधिपतिश्च ||११||
Dwathrimasthis cha may, śat trimsas cha may, chatvarimsa cha may,
Chathus chathvarimsa cha may, aśta chatvarimsa cha may,
Vaajas cha prasavas cha pijascha kradis cha suvas cha moordha cha
Vyasniyas cha anthyayanas cha anthyas cha
Bhouvans cha bhuvanas chadhipadhis cha., 11
Following numbers or Arithmetic Progression are explained in the above ślokas:
1, 3, 5,7,9,111315,17,19, 21,23,25,27,29,31,33
Again the following numbers of Arithmetic progression are also explained
4,8,12,16,20,24,28,32,36,40,44,48,
Ĕkā cha śatam cha sahaśram chayutham cha niyutam cha , prayutham charbudam cha nyaarbudam chasamudram cha madhyam cha antah cha parardha cha
One, and a hundred, and a thousand, and ten thousand (ayuta) and a hundred thousand (niyuta), and a million (arbuda) and a hundred million (nyarbuda), and a thousand million and ten thousand million, and a hundred thousand million, and ten hundred thousand million, and a hundred thousand million (parardha).
The geometric progression of numbers can be seen in the above śloka. Chamakam of Yajur veda highlights the maximum number of hundred thousand million (parardha),
Another śloka of Geometric Progression of Chamakam is given below:
एका चमे, तिस्रश्चमे, पंच चमे, सप्त चमे, नवचमे एकादश चमे, त्रयोदश चमे, पंचदश चमे,सप्तदशा चमे नवदशा चमे एकविंशातिश्चमे त्रयोविंशातिश्च मे पंचविंशतिश्चमे सप्तविंशतिश्चमे नवविंशातिश्चमे एकत्रिंश्चमे त्रयस्त्रिंशच्चमे…
Ĕkā cha me, tiśrah cha me, pancha cha me, saptha cha me, nava cha me Ĕkādeśa cha me, trayodaśa cha me, navadaśa cha me, Ĕkāvimsatiś cha me, trayovimśatiś cha me, panchavimśatish cha me, saptavimshatish cha me, navvimshatish cha me, Ĕkātrimshatish cha me, trayastrish cha me…
The brief meaning of the above śloka can be given as “may the powers associated with these integers manifest in me”. For example, one stands for the absolute, three stands for the power of three lower planes, three upper planes etc. There is another list of integers, the highest being parardha, ten hundred thousand million i.e. ten raised to the power of 12.
YAJUR VEDA – SAPTHA KANDA (Kanda-Part) 7.1.15AnukkandakandaVI
In Kandam 4.4.11 (Anuvaka 11) various timings, Gods, and numbers are discussed. The divine timing (rtavya) including twelve lunar months, six seasons, the role of Agni and other Gods, and the numbers measuring the greatness yajña are discussed. Even though the list of numbers is brief, the highest number here is parardha, a million million, i.e. ten raised to the power of 12. In Kandam 6 and 7 various rites discussed. It starts from two night prayer to forty nine night prayers for various diseases, social evils. In Kandam 5,(5.7.26) the number hundred and its increasing are discussed. Various measurements are discussed in Kandam 6.6.4.
7.1.15 Anuvaka 15 highlights various directions and seasons of nature.
7.2.11 of Anuvaka (chapter) 11 of Kriśna Yajur Veda talks about odd numbers. The anuvaka 11 through 20 mention talks about various integers. Each number stands for the cosmic power signified by that number. For instance one stands for the absolute. The largest integer parardha (ten raised to the power of twelve ) is mentioned in Anuvaka 20.
Anuvaka 11 talks about Arithmetic progression:
(i) एकास्मै Ĕkāsmai 1 (ii) द्वाभ्यम dvabhyam 2
(iii) त्रिभ्य: tribhyah 3 (iv) चतुर्भ्य: chaturbhyah 4
(v) पंचभ्य: Panchabhyah 5 (vi) षदभ्यः śadbhyah 6
(vii) सप्तभ्य: saptabhyah 7 (viii) अष्टभ्य: Aśtabhyah 8
(ix) नवभ्य Navabhyah 9
Like this it talks up to 19. Afterwards the numbers are written in terms of tens up to two hundred.
Anuvaka 12 talks about odd numbers.
(i) एकास्मै Ĕkāsmai 1 (ii) त्रिभ्य: tribhyah 3
(v) पंचभ्य: Panchabhyah 5 (vii) सप्तभ्य: saptabhyah 7
The odd numbers up to 99 are explained in the above anuvaka. Anuvaka 14 also talks about odd numbers.
Anuvaka 13 talks about even numbers
(i) द्वाभ्यम dvabhyam 2 (ii) चतुर्भ्य: chaturbhyah 4
(iii) षदभ्यः śadbhyah 6 (iv) अष्टभ्य: Aśtabhyah 8
(v) दसभ्य: Daśabhyah 10 (vi) द्वादसभ्य: Dwaaśabhyah 12
The even numbers upto 100 are explained in the above anuvaka.
Anuvakam 15 talks about the multiples of four
(i) चतुर्भ्य: chaturbhyah 4 (ii) अष्टभ्य: Aśtabhyah 8
(iii) द्वादसभ्य: Dwaaśabhyah 12
The multiples of four upto one hundred are explained in the above anuvakam.
Anuvakam 16 talks about multiples of five.
(i) पंचभ्य: Panchabhyah 5 (ii) दसभ्य: Daśabhyah 10
The multiples of five upto one hundred are explained in the Anuvakam 16.
Anuvakam 17 talks about multiples of 10
(i) दसभ्य: Daśabhyah 10 (ii) विम्शत्यै vimśatyai 20
(iii) त्रिमशत्यै Trimśate 30
The multiples of ten upto one hundred are explained in the above anuvakam.
Anuvakam 19 talks about fifty and multiples of one hundred
(i) पंचशते Panchaśate 50 (ii) शतायया śatayaya 100
The numbers in terms of 100 up to one thousand are explained in the above anuvaka.
Anuvakam 20 talks about large numbers. The Yajurveda is known for the large numbers and infinity.
(i) शतया śataya 1000
(ii) सहस्रया sahaśraya 1000
(iii) आयुतया Ayutaya 10,000
(iv) नियुतया niyutaya 1,00,000
(v) प्रयुतयाprayutaya 10,00,000
(vi) अर्बुदया arbudaya 1,00,00,000
(vii) न्यार्बुदया nyarbudaya 10,00,00,000
(viii) समुद्रया samudrayaya 100,00,00,000
(ix) मध्यया madhyaya 1,00,00,00,00,00
(x) अन्तया antaya 100,00,00,00,00
(xi) पररहया Pararahaya 1000,00,00,00,00,
(i.e. ten raised to the power of twelve)
Decimals and Fractions
In the Vedic literature fractions are called as अर्ध (Ardha), पाद (Pada), सफ (Sapha), and कला (kala). (Ardha) means ½, (Pada) means ¼, सफा (sapha) means 1/8, and कला (kalaa) means 1/16. In addition to this पंचाम्भागा (panchama-bhaga) is called as 1/5, दसम्भागा (dasam-bhaga) is called 1/10, अर्धचतुर्था (ardhachathurtha) is 3 ½, अर्धपंचसत (ardhapanchasat) is 49 ½ .
The great Adi Sankaracharya in His Vedanta Sutra Bhaśya had given explanations for writing numbers giving decimal place values. He explained in detail about the glory of Ganitha sastra
यथा चैकापी रेखा स्थानान्यत्वेन निविशामानैक
दशा शत सहस्रादी शब्द प्रत्यय भेदामनुभवती
Yathaachaikaapi rekha sthaananyathvena nivismaanaika dasa satha sahaśraadi sabda prathyaya bhedhamanubhavathi
One and the same numerical sign when occupying different places is
conceived as measuring 1,10,100,1000 etc.
Similarly in Manavala Sulbasutra (9.4) the Sanskrit number is given
नवाम्गुलसहस्राणि द्वे सहते षोडशोत्तरे
Navaangula sahaśrani dve sathe śodasotthare
The fire altar measures 9216 angula, 9thousand 2 hundred and 16
एकैकस्य सहस्रं स्याच्छते षण्णवति: परा
Ĕkāikasya sahaśram syacchatam śannavaithi para
The area is 1196, one thousand, one hundred and ninety six.
Permutations and Combinations
Permutation and combination is yet another one where the greatness of Indian mathematicians can be found. In Vedic literature we can find out several permutations and combinations. Susrutha Saṃhitā there are 63 permutations and combinations from a mixture of six rasas (flavours)
Pasa, ankusa, serpant, damaru, kapala, soola, khatvanga, sakti, chapa, are the ten items of weapons Lord Siva. With these several combinations were worked out.
Similarly sanku, chakra, gadha, and padma, are the items Lord Viśnu is holding. With these things several combinations were worked out. Gayathri Mantra which has 24 syllables is the fittest example of Indian system of permutation and combination.
Percentage and Interest calculations:
Interest calculations are yet strength of Indian mathematicians. The loan documentation, various types of interest that are to be charged, and the periodicity of the loan and their repayment etc were highlighted in the various smrtis. This financial and mathematical knowledge helped to establish leading financial institutions, and banks in India. This helped the smooth flow of funds from the lender to the borrower.
Viśnu Smriti, one of the well known Dharma Sastra, talks in detail about the need to avail loans, rate of interest, periodicity of repayment and related calculations in a detailed manner. It also talked about the properties that to be mortgaged. The differential rate of interest for secured and unsecured loans was also discussed. The punishments for defaulters were also highlighted.
It says that the interest can be charged as manual worek equivalent to the money, materials, depending, on the period of loan, money on fixed rates, and compound rate of interest.
Viśnu Smrti approves the interest on monthly basis. Similarly Narada Dharma advocates charging of interest on half yearly basis. Vyasa Dharma also approves the payment of interest.
Viśu Smrti talks about the methods of forming a partnership organization, method of profit sharing, method of appointing a new partner, profit calculations, and payment of taxes to the Government. It even talked about the method of spending by the Government. Similarly Yagnavalka Smrti talks about the loan documentation, method of recovery of the loans with other details.
Averages
The averaging is another area where our ancient Vedic system specialized. The sulba sutras talked about the system of averaging.
Square root
The Sulba Sutra is talking about the Square root and fractions of the Square root. Aryabhatta, the eminent mathematician, refers the Vedic literature for this purpose.
Infinity
Indian Vedas are known for mathematics in general and large numbers in particular. There are various Vedic scripts about infinity. Indian Mathematicians have written in volumes about Śunya (the zero) and infinity. Sri Srinivasa Ramanujan, the specialist number theory expert, had written elaborately about the term infinity.
Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito" means unending.
Our forefathers have a passion for infinity. The word Anantha has been used in all the four Vedas.
अनंता (Anantha) means endless. Anantha in Sanskrit is for infinity. It is equated with the Supreme Brahman. Infinitely powerful and so infinitely free. It is bigger than any quantity that can be imagined It is bigger than any infinite number.
They were aware of the basic mathematical properties of infinity and had several words for the concept-chief being अनंता (ananta), पूर्णं (purnam), अदिति (aditi), and असंख्यता (asamkhyata). असंख्यता (Asamkhyata) is mentioned in the Yajur Veda, and the Brihadaranyaka Upaniśad as describing the number of mysteries of Indra as ananta.
These two statements are elaborated in the opening lines of the Iśa Upaniśad (Śukla Yajur Veda). This ślōka is as much metaphysical as it is mathematical.
पूर्णमदः पूर्णमिदं पूर्णात् पूर्णमुदच्यते। पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते।।
pūrṇamadaḥ pūrṇamidaṁ pūrṇāt pūrṇamudacyate| pūrṇasya pūrṇamādāya pūrṇamevāvaśiṣyate||
on temporary mathematics is based.
From infinity is born infinity. When infinity is taken out of infinity,
only infinity is left over.
Taitriya Upaniśad says
सत्यं ज्ञानम् अनन्तं ब्रह्म
Satyam Gnanam Anantham Brahma
There is also another Yajur Vedic hymn
अनन्ता वै वेदाः।
Anantha vai Veda
The Vedas are said to be endless or infinite.
The scientific community is telling us that infinity is an unreal number, whereas in spirituality the words used for infinity are Anantham and Purnam. We all know Anantham means endless and Purnam is complete.
Ancient Indian Mathematicians have written extensively about the Anantham and Purnam. All the four Vedas, various Upaniśads, upavedas, and epics talk about Anantham and Purnam.
Another word used for infinity is असंख्यत Asamakyatha which means uncountable.
We simply cannot count infinity.
Sanskrit grammar and interpretation in ancient India were closely linked to the handling of high value numbers. Studies relating to poetry and metrics initiated sastragnaas or scientists to both arithmetic and grammar.
Grammarians were just as competent at calculations as professional mathematicians. Indian sastragnaas or scientists, philosophers, astronomers and cosmographers - in order to develop their arithmetical, metaphysical and cosmological speculations concerning ever higher numbers - became at once mathematicians, grammarians and poets.
Infinity is well known in ancient India. The word Anantham may be taken as infinity. Another word for Infinity is Kha hara. It is also referred as ananthō risi i.e. never ending number. It was the view of ancient mathematicians that nothing happens to the infinity when any number enters (added) or leaves (subtracted). The infinity remains unchanged. Indian mathematician s developed proficiency and expertise in the knowledge of infinity. Srinivasan Ramanujan, a number theory expert, is being called “the Man who knew Infinity”
The concept of Anantham, Purnam, and Asamkyatha need extensive study and research.
