Our Mathematical Heritage By Dr T.H. Chowdary * Director, Center for Telecom Management & Studies *...

Our Mathematical Heritage

ByDr T.H. Chowdary

* Director, Center for Telecom Management & Studies* Chairman, Pragna Bharati (Intellect India), AP

* Former Information Technology Adviser, Government of A.PChairman & Managing Director

Videsh Sanchar Nigam Ltd., BombayT: +91 (40) 2784-6137/6667-1191(O) F: +91 (40) 2789-6103/6667-1111 (O)

[email protected]

Talk @RBVR Reddy Women’s College,Hyderabad: 8 Sept 2006

THC_CTMS S312_Sept06 2

Numeration &.

• Roman: I,II,III,IV……X….XV….XX….C..

• Indian :Place value 1,2,3…10..100..1000..

• Zero or Sunya

• Negative numbers

• Surds


Numeration &Names for Numbers

102 103 106 109 1012 1015 1018

Sata Sahasra Dasa Laksha

Sata Koti

Khara Maha Padma

Sanka

Hundred Million Billion Trillion Zillkion ?

1021 1024 1026 1030 1032 1035

Maha Brinda

Kshobha Nidhi Madhya Ananta Achintya


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• 2X1018 Mahaviswas=1Loka• 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


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


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



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



Cosmic Cycles (1)• 60y Jovian ( related to Jupiters movement)• 2700y cycle of Sapta Mndala ( Great Bear)• 27,000y 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(4.32X106)= 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 360X

1011= 3.1104X 10 14 human year



Cosmic Cycles (2)

Present Brahma 3.1104 Padmas• 50 Brahma years gone• We are in the 1st day of the 51st year of Brahma in the present kalpa-

Sweta Varaha kalpa • During this Kalpa, 6 Manvantaras are over• We are in the 7th Manvantara – Vaivaswata• In this manvantara, 27 Maha Yugas are completed• We are in the 28th Mahayuga ( M.Y)• In the 28th M.Y, Krita, Treta, Dwapara are gone• We are in Kali Yuga, started 3102BC• Kali has 4,32,000 Years; we completed just 5105 Years!



Cosmic Cycles (3)

• A Maha Yuga consists of 10 Yugas divided into 4 parts– Krita Yuga : 1,728,000 Y– Treta Yuga : 1,296,000Y– Dwapara : 864,000 Y– Kali : 432,000Y

• 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



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


Encryption

Katapayaadi-SankhyaGopibhagya madhuvraata sringisodadhisandhigaKhalajivitakhaataava 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


Mathematics (1)

Brahma Gupta’s Lemmas

• (2 degree indeterminate equations)

• (Bija Ganitam, Ch.6, Sl:71&72 Bhaskaracharya II. 1150 AD)


Mathematics (2)

Bhaskara’s Chakravala Method• Equation NX2= Y2

• 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 (1150AD) – 6000 years ahead


Mathematics (3)

Permutations of given things• Lilavati Ankapasah, Ch 33, Sloka 1

Ch 3, Sloka 4Bhaskaracharya 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


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)


Geometry (1)

• Baudhayana ( 800 BCE)• So called Pythogorus theorem• Sulba Sutram, Ch1, 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 1st Western mathematician who discussed

construction of such quadrilaterals)


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 (1782AD)

theorem Value of • Aryabhatiyam, CH2, Sl 10= 499 AD


Geometry (3)

Indian attempts• Mahavira ( 850 AD) = 10• Bhaskara II ( 1150 AD) = 3927/1250• Nilakanta Somayaji (1444-1545AD)

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 (1887-1920AD)


Geometry (4)

3 Diagonal theorem

• ( Ganita Sara Sangraha, Narayana 1356AD) Rational Scalene Triangle, Mahaviracharya 850AD

• Pairs of Rational Isoceless Triangles

• Construction of Rational Right Angled triangles


Trigonometry

• Madhava’s sine & cosine series

• Yuktibhaska, Ch 6, Sl 12,13 by Jysthadeva ( 1500 to 1610)

• First tangent series (Kriyakramakari, ch2, SL 40 Putumana Somayaji 1350-1410)

(James Gregory 1638,Leibnitz 1644)


Our Mathematical Heritage (1)

• Bhaskaracharya (5th 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.



• 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 6th century long before the European mathematicians.

Algebra, trigonometry and calculus came from India. Sridharacharya propounded quadratic equations in the 11th century.



• 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



“India was China’s teacher in trigonometry, quadratic equations, grammar, phonetics….”

- Lin Uting, Chinese Scholar and

Author


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



Everything has come down to us from the banks of the Ganges

-Francois Voltaire



• 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



• 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, DicK Teresi (Page No: 26)


Dhanyawad:Thank You

Our Mathematical Heritage By Dr T.H. Chowdary * Director, Center for Telecom Management & Studies *...

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