Our Mathematical Heritage By Dr T H Chowdary
Our Mathematical Heritage By Dr T. H. Chowdary * Director, Center for Telecom Management & Studies * Chairman, Pragna Bharati (Intellect India), AP * Former Information Technology Adviser, Government of A. P Chairman & Managing Director Videsh Sanchar Nigam Ltd. , Bombay T: +91 (40) 2784 -6137/6667 -1191(O) F: +91 (40) 2789 -6103/6667 -1111 (O) thc@satyam. com Talk @RBVR Reddy Women’s College, Hyderabad: 8 Sept 2006 THC_CTMS S 312_Sept 06
Numeration &. • • • Roman: I, III, IV……X…. XV…. XX…. C. . Indian : Place value 1, 2, 3… 10. . 1000. . Zero or Sunya Negative numbers Surds THC_CTMS S 312_Sept 06 2
Numeration & Names for Numbers 102 103 106 109 1012 1015 1018 Sata Sahasra Dasa Laksha Sata Koti Khara Maha Padma Sanka Million Billion Trillion Zillkion ? 1026 1030 1032 1035 Madhya Ananta Achintya Hundred 1021 1024 Maha Brinda Kshobha Nidhi THC_CTMS S 312_Sept 06 3
Vastness of Space • Our Solar System extends into space more than 1512 = 1 Brahmanda • 7 Brahmanda= 1 Jagat • 1000 Jagats= 1 Viswa • 15 X 1016 Viswas = Maha Viswa • 2 X 1018 Mahaviswas=1 Loka • 1019 Lokas= 1 Maha Loka • 1016 Mahalokas= Samsara i. e In one Samsara = 21 X 1063 Brahmandas (1062= Mahogha) • Immense vastness of space • Vishnu Purana say that Brahmandas are countless in space THC_CTMS S 312_Sept 06 4
Concept of Infinity Om, poornamadah; poornamidah Pornath poornamudachyate Poornasya poornamaadaaya Poornamevaavasishyate That is infinity; this is infinity Infinity arises from infinity If you subtract infinity from infinity What remains is also infinity THC_CTMS S 312_Sept 06 5
The Concept of Time- Wheel of Time (1) Time is awake when all things sleep Time stands straight when all things fall Time shuts in all and will not be shut Time is, was and shall be there Time’s children be witness and be stable - Mahabharata THC_CTMS S 312_Sept 06 6
The Concept of Time- Wheel of Time (2) q Time: Surya Siddhanta 6 Pranas= 1 Vinadi ( 24 Secs. ) 60 Vinadis= 1 Nadi (24 mnts. ) 60 Nadis=1 Day • Prana-time taken to pronounce 10 long syllables THC_CTMS S 312_Sept 06 7
The Concept of Time- Wheel of Time (3) q Cosmic Cycles (1) • 60 y Jovian ( related to Jupiters movement) • 2700 y cycle of Sapta Mndala ( Great Bear) • 27, 000 y cycle of the (27) Nakshatra ( asterisms) the 27 fixed stars relative to the motion of the Solar System, situated equidistant around the Jyothis-Chakra (zodiac) • 432, 000 -year cycle called Yuga • 4. 32 mil(4. 32 X 106)= year cycle called Maha Yuga • One day of Brahma = 8. 64 X 109 Years • One Brahma Year = 8. 64 X 360 X 109 Years • Life period of Brahma = 100 Brahma Years = 8. 64 X 360 X 1011= 3. 1104 X 10 14 human year THC_CTMS S 312_Sept 06 8
The Concept of Time- Wheel of Time (5) q Cosmic Cycles (2) v Present Brahma 3. 1104 Padmas • 50 Brahma years gone • We are in the 1 st day of the 51 st year of Brahma in the present kalpa. Sweta Varaha kalpa • During this Kalpa, 6 Manvantaras are over • We are in the 7 th Manvantara – Vaivaswata • In this manvantara, 27 Maha Yugas are completed • We are in the 28 th Mahayuga ( M. Y) • In the 28 th M. Y, Krita, Treta, Dwapara are gone • We are in Kali Yuga, started 3102 BC • Kali has 4, 32, 000 Years; we completed just 5105 Years! THC_CTMS S 312_Sept 06 9
The Concept of Time- Wheel of Time (4) q Cosmic Cycles (3) • A Maha Yuga consists of 10 Yugas divided into 4 parts – – Krita Yuga : 1, 728, 000 Y Treta Yuga : 1, 296, 000 Y Dwapara : 864, 000 Y Kali : 432, 000 Y • A 306, 720, 000 cycle called Manvantara (=71 Maha Yugas) • A 4. 32 X 109 Year cycle called Kalpa called daytime of Brahma= High time of Brahma THC_CTMS S 312_Sept 06 10
The Concept of Time- Wheel of Time (5) q Cosmic Cycles (4) • Life period of Brahma = 1 day time of Vishnu • 360 days and nights of Vishnu= 1 Year of Vishnu • 1 billion ( 100 crore) Vishnu Years= Life of Vinshu = X 72 X 1024 Years • Life period of Vishnu= 226 Khobha Years THC_CTMS S 312_Sept 06 11
Encryption q Katapayaadi-Sankhya Gopibhagya madhuvraata sringisodadhisandhiga Khalajivitakhaataava galahaalaarasandhara (Oh Krishna, fortune of the Gopis, the destroyer of the demon Madhu, protector of cattle, the one who ventured the ocean depths, destroyer of evil doers, one with plough on the shoulder and the bearer of the nector, may (you) protect(us). • Katapayadi Sankhya (protocol), alphabets have numerical values. Eg: ka, a, pa, ya mean • If you apply the key to the Sloka, the number is 3. 1415, 9265, 3589, 7932, 3846, 2643, 3832, 792 • This is correct to 31 digits THC_CTMS S 312_Sept 06 12
Mathematics (1) q Brahma Gupta’s Lemmas • (2 degree indeterminate equations) • (Bija Ganitam, Ch. 6, Sl: 71&72 Bhaskaracharya II. 1150 AD) THC_CTMS S 312_Sept 06 13
Mathematics (2) q. Bhaskara’s Chakravala Method • Equation NX 2= Y 2 • Bhaskara claimed that finite number of chakravala will give the root of the equation • Fermat’s problem of solving the equation for N=61 resisted the integer solution for 100 years • Eular ( 1764) & La Grange ( 1768) solved the problem by different method tan Bhaskara (1150 AD) – 6000 years ahead THC_CTMS S 312_Sept 06 14
Mathematics (3) q Permutations of given things • Lilavati Ankapasah, Ch 33, Sloka 1 Ch 3, Sloka 4 Bhaskaracharya II ( 1150 AD) • Combinations of given things • Ganita Sara Sangraha Ch 6, Sl 218 • Mahavicharyah 850 AD • The concept of “Niruddha” (LCM) • Ganita Sara Sangraha Ch 3 Sl 56 • Sun of the Squares of n terms in A P • Ganita Sara Sangraha Ch 6, Sl 298 THC_CTMS S 312_Sept 06 15
Mathematics (4) • Sum of Cubes fo First ‘n’ terms of A P -Ganita Sara Sangraha, Ch 6, Sl. 304 • (also discussed by Sridharacharya, 750 AD & Narayana 1356 AD) THC_CTMS S 312_Sept 06 16
Geometry (1) • • Baudhayana ( 800 BCE) So called Pythogorus theorem Sulba Sutram, Ch 1, Slokah 12 Diagnals of a cyclic quadrilateral Bamha Sphuta Siddhana= Ch 12, Sl 28 • (W. Snell’s (1619) theorem • Construction of rational quadrilateral Ch. 12, Sl. 38 (Eular ( 1707 -1783) Is the 1 st Western mathematician who discussed construction of such quadrilaterals) THC_CTMS S 312_Sept 06 17
Geometry (2) • Area of cyclic quadrilateral & Triangle Ch 12, Sl 21 • Circumradius of cyclic quadrilateral Commentary on Lilavati Sl 191, 192 Parameswara 1430 Known as Simon A. J. Lhuilier’s (1782 AD) theorem q Value of • Aryabhatiyam, CH 2, Sl 10= 499 AD THC_CTMS S 312_Sept 06 18
Geometry (3) q Indian attempts • Mahavira ( 850 AD) = 10 • Bhaskara II ( 1150 AD) = 3927/1250 • Nilakanta Somayaji (1444 -1545 AD) 28, 27, 43, 33, 88, 233/9 X 1011 • Sankara Variyar (1732 AD) = 104348/33215 • Sankara Varma N ( 1823 AD) • Putumana Somayaji ( 1732 AD), • Ramanujan (1887 -1920 AD) THC_CTMS S 312_Sept 06 19
Geometry (4) q 3 Diagonal theorem • ( Ganita Sara Sangraha, Narayana 1356 AD) Rational Scalene Triangle, Mahaviracharya 850 AD • Pairs of Rational Isoceless Triangles • Construction of Rational Right Angled triangles THC_CTMS S 312_Sept 06 20
Trigonometry • Madhava’s sine & cosine series • Yuktibhaska, Ch 6, Sl 12, 13 by Jysthadeva ( 1500 to 1610) • First tangent series (Kriyakramakari, ch 2, SL 40 Putumana Somayaji 1350 -1410) (James Gregory 1638, Leibnitz 1644) THC_CTMS S 312_Sept 06 21
Our Mathematical Heritage (1) • Bhaskaracharya (5 th century AD) calculated the time taken by the earth to orbit the sun hundreds of years before the astromer Smart. Time taken by earth to orbit the sun: 365. 258756484 days. THC_CTMS S 312_Sept 06 22
Our Mathematical Heritage (2) • Budhayana first calculated the value of pi, and he explained the concept of what is known as the Pythagorean Theorem. He discovered this in the 6 th century long before the European mathematicians. Algebra, trigonometry and calculus came from India. Sridharacharya propounded quadratic equations in the 11 th century. THC_CTMS S 312_Sept 06 23
Our Mathematical Heritage (3) • The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 1053 with specific names as early as 5000 BCE during the Vedic period. Even today, the largest used number is Peta: 1015. The place value system, the decimal system was developed in India in 100 BC. India invented the Number System. Zero was invented by Aryabhatta THC_CTMS S 312_Sept 06 24
Our Mathematical Heritage (4) “India was China’s teacher in trigonometry, quadratic equations, grammar, phonetics…. ” - Lin Uting, Chinese Scholar and Author THC_CTMS S 312_Sept 06 25
Tribute to Bharat (1) If there is an country on earth which can justly claim the honour of having been the cradle of the human race are at least the seen of primitive civilization, a successive developments of which carried into all parts of the ancient world and even beyond the blessings of knowledge which is the second life of man, that country is assuredly India -Friedrich Creuzer (1771 -1858), German philologist and archeologist THC_CTMS S 312_Sept 06 26
Tribute to Bharat (2) Everything has come down to us from the banks of the Ganges -Francois Voltaire THC_CTMS S 312_Sept 06 27
Tribute to Bharat (3) • When Alexander set out to invade India, he went to his teacher, Aristotle and asked what Guru Dakshina he would want to be brought from India. Aristotle told him to bring a pail of Ganga Jel and the instrument Venu, the flute of Lord Krishna THC_CTMS S 312_Sept 06 28
Tribute to Bharat (4) • Vedic Indians solved square roots to build sacrificial altas of the proper size. This was science in the service of religion but science nandiless -Lost Discoveries, Dic. K Teresi (Page No: 26) THC_CTMS S 312_Sept 06 29
Dhanyawad: Thank You THC_CTMS S 312_Sept 06 30
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