Mar 25, 2014 - grammar was discovered in far away America by NASA scientist called .... however big the coefficients may
VEDIC MATHEMATICS AND ENGINEERING APPLICATIONS - Dr. KISHORE SONTI 3/25/2014
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Scaling down Surface tension force
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F = γL
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Scaling down Mass m = ρV
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A water bug can walk on water but man cannot
Gravitational force of attraction
F=g
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S4, S
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Our holy India is the entrance of the heaven and salvation. Those who are born in this country are fortunate than gods, that is what gods sing; Vishnu Puran 2:3:34 Meaning of word BHARAT BHA = Light, Splendor, Luster, Beauty, Knowledge, Asterisms RATA= Pleased, Delighted, Gratified, Engaged in, Devoted to
Veda Garbbha- Bharat Veda word originated from ‘Vid’ Dhatu; means ‘that which educates’ (knowledge)
Vedas were divided into two parts: Nigama & Aagama Nigama- Science and Aagama- Technology 3/25/2014
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What are in Veda: ‘Anoraneeyan Mahato Maheyaan’ From atom (matter)to universe (Energy).
How Vedas were deciphered: Every Vedic word has six meanings, the sixth one is Oshadhi (Material)meaning.
How Vedas (Jnana) are Classified: Samjnana-Purity: Vijnana- Utility: Prajnana: Spirutuality How Universe was described: There are only two domains i.e., Matter and Energy basis on Time and Space
The Mahabharata says: "Time is the seed of the Universe.” 3/25/2014
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Nigama (Science) & Aagama (Technologies)
Padaartha
Shakthi
(Matter)
(Energy)
Sthavara
Jangama
Immovable
(Movable)
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Jyothirbrah ma
Sabdabrah ma
(Light energy)
(Sound energy) 12
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HOW, WE LOST OUR KNOWLEDGE
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Anka Ganitha
- by Kasyapa Maharshi
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Beeja Ganitha
- by GanapathiMaharshi
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Rekha Ganitha
- by Chayapurusha Maharshi
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Vata Ganita
– by Jaimini Muni
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Chakra Ganita
– by Anjaneya
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Mandala Ganita
– by Anjaneya
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Vaastu Ganita
– by Garga
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Shakthi Ganitha
- by Brihaspathi
• Bhava Ganith - by Surya Maharshi • Shakthi Ganitha 3/25/2014
- by Brihaspathi 15
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VEDIC MATHEMATICS Vedic Mathematics discovered by H.H. Jagadguru Swami Bharathi krishna Tirthaji of Govardhan Peeth, Puri, between the period 1911 to 1918 at Shrungeri. He was born in 1884 in Tamilnadu state of India.
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• Founder of Vedic Mathematics H.H. Jagadguru Swami Bharathi krishna Tirthaji was an exceptional scholar. • By age twenty he had studied at a number of colleges and universities throughout the country, been awarded the title of ‘Saraswati’ by the Madras Sanskrit Association for his remarkable proficiency in Sanskrit. • He had completed seven masters degrees, including Sanskrit, Philosophy, English, Mathematics, History and Science, with the American College of Sciences. 3/25/2014
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VEDIC MATHEMATICS • According to Swamiji, the sutras (16 Main sutras & 13 Sub-sutras) cover every branch of mathematics, from arithmetic to spherical conics, and that “there is no mathematics beyond their jurisdiction
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VEDIC MATHEMATICS • Only 8 Chapters of Vedic Mathematics came into light out of 100 chapters. • From Mineralogy to space includes interstellar travel, what not! All about existing. • Entire matter and Energy, all disciplines like Mathematics and so on were described in various chapters basing on Time & Space. 3/25/2014
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VEDIC APPLICATIONS AREA
• • • • • • • • • • •
Toxicology Engineering Robotics Computer Science Artificial Intelligence Physics (all branches) Chemistry (all branches) Biology (all branches) Metallurgy (all branches) Nano Technology Aeronautics (all branches)
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Vedic text
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Agada Tantra Aparajitha Priccha Samarangana Sutradhara Veda Ganitha Panini’s Vyakarana Amsu Bodhini All Rasa Sastra Grandhas No. Ayurveda Grandhas Dhatuvada like 100’s of Gr 100’s of Alchemy books Vymanika Sastra 21
ANCIENT INDIAN MATHEMATICAL MODEL
“The Ancient Indian Mathematicians had no computer but some of the techniques they developed are precisely the ones used in solving problems with today’s computers.” “Rules are so scientific and logical in manner that they closely resemble structures used by computer scientists through out the world.” “Sadly, the link between Artificial intelligence and Panini’s grammar was discovered in far away America by NASA scientist called Mr. Rick Briggs.” 3/25/2014
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Panini has made Sanskrit Precise, concise and complete. It is like a set of condensed codes for the entire language with some rules attached. It is a terse, very condensed form of Sanskrit, which paradoxically at times becomes so obtuse that a commentatory is necessary to clarify it.”
“Interestingly, many scientists are tempted to speculate why and how Panini developed his rules in so concise and precise a manner without a computer……” - The Times of India, 22-3-1992 3/25/2014
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Engineering & Technologies • • • • • • • • • • • • •
Pata Samskara Pradipaka Tamo Yantra Darpana Sastram Yantra Sarvaswa Asana Kalpataru Jeeva Sarvaswam Mulikarka Prakasa Niryasa Chandrika Kheta Sarvasvam Taila Prakaranam Twak Nirnayadhikara Kriya Sara Paramkushah
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- Textile Engineering - Spectroscopy - Opto-Electronics - Mechanical Engineering - Food Technologies - Life Sciences - Botany -Solvents preparation - Astronomy - Oil Extract Machinery - Leather Technologies - Processing Technologies - Weaponry Technologies 24
Brief survey on Vimana Sastra “Many researchers into the UFO enigma tend to overlook very important fact. While it assumed that most flying saucers are of alien, or perhaps governmental Military origin, another possible origin UFO’s is Ancient India and Atlantic. What we know about Ancient Indian flying vehicles comes from Ancient Indian sources; written texts that have come down to us through the centuries.”
“There is no doubt that most of these texts are authentic; many are the well known Ancient Indian Epics themselves, and there are literally hundreds of them. Most of them have not even been translated into English from the old Sanskrit.” - JOHN BURROWS 3/25/2014
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“Ashoka was also aware of devastating wars such advanced vehicles and other “futuristic weapons” that had destroyed the Ancient Indian “Rama Empire” several thousand years before”. - JOHN BURROWS
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• Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). • According to his research all of mathematics is based on sixteen Sutras, or word-formulae. • The term Vedic Mathematics now refers to a set of sixteen mathematical formulae or sutras and thirteen sub-sutras 3/25/2014
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Uses of vedic mathematics • Solving mathematical problems 10 to 15 times faster. • It reduces burden(tables upto 9 is enough for calculations) • It increases concentration • Helps reduce mistakes 3/25/2014
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Applications • • • • • • •
Arithmetic computations Algebraic operations Factorisations Simple quadratic and higher order equations Partial fractions Squaring,square root Cubing,cube root
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SUTRAS 1) (Anurupye) Shunyamanyat – If one is in ratio, the other is zero. 2) Chalana-Kalanabyham – Differences and Similarities. 3) Ekadhikina Purvena – By one more than the previous one. 3/25/2014
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4) Ekanyunena Purvena – By one less than the previous one. 5) Gunakasamuchyah – The factors of the sum is equal to the sum of the factors. 6) Gunitasamuchyah – The product of the sum is equal to the sum of the product. 3/25/2014
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7) Nikhilam Navatashcaramam Dashatah – All from 9 and the last from 10. 8) Paraavartya Yojayet – Transpose and adjust. 9) Puranapuranabyham – By the completion or noncompletion.
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10) Sankalana-vyavakalanabhyam – By addition and by subtraction. 11) Shesanyankena Charamena – The remainders by the last digit. 12) Shunyam Saamyasamuccaye – When the sum is the same that sum is zero. 3/25/2014
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13) Sopaantyadvayamantyam – The ultimate and twice the penultimate. 14) Urdhva-tiryakbyham – Vertically and crosswise.
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15) Vyashtisamanstih – Part and Whole.
16) Yaavadunam – Whatever the extent of its deficiency. 3/25/2014
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Principle • Urdhva – tiryagbhyam is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number.
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URDHVA TRIYABYAM Steps in Urdhva sutra
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Implementation with example
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The Sutra (formula) Ekādhikena Pūrvena means: “By one more than the previous one”.
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Ekadhikena purvena
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When sum of the last digits is the base(10) and previous parts are same
• 44 x 46 = (4 x (4+1)) (4 x 6) = (4 x 5) (4 x 6) = 2024 • 37 x 33 = (3 x (3+1)) (7 x 3) = (3 x 4) (7 x 3) = 1221 • 11 x 19 = (1 x (1+1)) (1 x 9) = (1 x 2) (1 x 9) = 209
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Sankalana - Vyavakalanabhyam • This Sutra means 'by addition and by subtraction'. It can be applied in solving • a special type of simultaneous equations where the x - coefficients and the y • - coefficients are found interchanged. • Example 1: • 45x – 23y = 113 • 23x – 45y = 91 3/25/2014
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• • • • • • • • •
From Sankalana – vyavakalanabhyam add them, i.e., ( 45x – 23y ) + ( 23x – 45y ) = 113 + 91 i.e., 68x – 68y = 204 x – y = 3 subtract one from other, i.e., ( 45x – 23y ) – ( 23x – 45y ) = 113 – 91 i.e., 22x + 22y = 22 x + y = 1 and repeat the same sutra, we get x = 2 and y = - 1 Very simple addition and subtraction are enough, however big the coefficients may be.
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• • • • • • • •
Example 2: 1955x – 476y = 2482 476x – 1955y = -4913 Oh ! what a problem ! And still just add, 2431( x – y ) = - 2431 x – y = -1 subtract, 1479 ( x + y ) = 7395 x + y = 5 once again add, 2x = 4 x = 2 subtract - 2y = - 6 y = 3
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Ekanyunena Purvena • The Sutra Ekanyunena purvena comes as a Sub-sutra to Nikhilam which gives the meaning 'One less than the previous' or 'One less than the one before'.
• 1) The use of this sutra in case of multiplication by 9,99,999.. is as follows . 3/25/2014
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• • • • • •
Method a) The left hand side digit (digits) is ( are) obtained by applying the ekanyunena purvena i.e. by deduction 1 from the left side digit (digits) . e.g. ( i ) 7 x 9; 7 – 1 = 6 ( L.H.S. digit ) b) The right hand side digit is the complement or difference between the multiplier and the left hand side digit (digits) . i.e. 7 X 9 R.H.S is 9 - 6 = 3. c) The two numbers give the answer; i.e. 7 X 9 = 63.
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Example 1
• 8 x 9 Step ( a ) gives 8 – 1 = 7 ( L.H.S. Digit ) • Step ( b ) gives 9 – 7 = 2 ( R.H.S. Digit ) • Step ( c ) gives the answer 72
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Example 2 • • • •
15 x 99 Step ( a ) : 15 – 1 = 14 Step ( b ) : 99 – 14 = 85 ( or 100 – 15 ) Step ( c ) : 15 x 99 = 1485
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Second Segment - Verification
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Nikhilam navatascaramam Dasatah • “All from 9 and the last from 10”
• The formula can be very effectively applied in multiplication of numbers, which are nearer to bases like 10, 100, 1000i.e., to the powers of 10.
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• The difference between the number and the base is termed as deviation. • Deviation may be positive or negative. • Positive deviation is written without the positive sign and the negative deviation, is written using Rekhank (a bar on the number). 3/25/2014
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Both the numbers are higher than the base
• The only difference is the positive deviation. Instead of cross – subtract, we follow cross – add.
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1275X1004. Base is 1000
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One number is more and the other is less than the base. • In this situation one deviation is positive and the other is negative.
• So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted.
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Nikhilam in Division two digit numbers (dividends) and same divisor 9
• Split each dividend into a left hand part for the Quotient and right - hand part for the remainder by a slant line or slash.
• Put the first digit of the dividend as it is under the horizontal line. 3/25/2014
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• Put the same digit under the right hand part for the remainder, add the two and place the sum i.e., sum of the digits of the numbers as the remainder.
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13 as 1 / 3
13 ÷ 9 gives Q = 1, R = 4
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3 digit numbers
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1204 ÷ 9 132101 ÷ 9
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Puranapuranabhyam
• The Sutra can be taken as Purana Apuranabhyam which means by the completion or non - completion
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Gunita Samuccayah Gunitah
:
Samuccaya
• It is intended for the purpose of verifying the correctness of obtained answers in multiplications, divisions and factorizations
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• The product of the sum of the coefficients sc in the factors is equal to the sum of the coefficients sc in the product
• sc of the product = product of the sc (in the • factors)
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Beejank • The Sum of the digits of a number is called Beejank • Beejank of 27 is 2 + 7 = 9. • Beejank of 348 is 3 + 4 + 8 = 15 • Further 1 + 5 = 6. i.e. 6 is Beejank.
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• • • • •
64 + 125 = 189 134 – 49 = 85 376 – 284 = 92 24 X 16 = 384 237 X 18 = 4266
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Gunita by beejank
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2 24
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Beejank -Division
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General numbers numbers
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into
vinculum
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Viniculum – S&V
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Gunakasamuchyah • The factors of the sum is equal to the sum of the factors
• C*- ADYAMADYENANTYA - MANTYENA
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the first by the first and the last by the last
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Factorization of quadratics
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2 4x
+ 12x + 5
• Split 12 into 2 and 10 so that as per rule 4 : 2 = 10 : 5 = 2 : 1 i.e.,, 2x + 1 is first factor.
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Sunyam SamyaSamuccaye • The Sutra 'Sunyam Samyasamuccaye' says the 'Samuccaya is the same, that Samuccaya is Zero.' i.e., it should be equated to zero. • It has got different meanings acc to situation.
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'Samuccaya‘ - common factor • The equation 7x + 3x = 4x + 5x has the same factor ‘ x ‘ in all its terms. Hence by the sutra it is zero,i.e., x = 0.
»or • • • •
7x + 3x = 4x + 5x 10x = 9x 10x – 9x = 0 x=0
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5(x+1) = 3(x+1) • 5x + 5 = 3x + 3 • 5x – 3x = 3 – 5 • 2x = -2 or x = -2 ÷ 2 = -1 • Samuccaya is ( x + 1) • x + 1 = 0 gives x = -1
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Samuccaya' as product of independent terms in expressions like (x+a) (x+b) • ( x + 3 ) ( x + 4) = ( x – 2) ( x – 6 ) • Here Samuccaya is 3 x 4 = 12 = -2 x -6 • Since it is same , we derive x = 0
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Samuccaya ‘ - the sum of the denominators of two fractions having the same numerical numerator.
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• Samuccaya - sum of the denominators • i.e., 3x – 2 + 2x - 1 = 5x - 3 = 0 • 5x = 3 • x=3/5
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'Samuccaya' - combination or total.
• If the sum of the numerators and the sum of the denominators be the same, then that sum = 0.
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N1 + N2 = K (D1 + D2 )
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Two values of x • If N1 + N2 = D1 + D2 and also the differences • N1 ~ D1 = N2 ~ D2 then both the things are equated to zero • The solution gives the two values for x.
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‘Samuccaya’ - the same sense but with a different context and application
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• Now ‘Samuccaya’ sutra, tell us that, • if other elements being equal, • sumtotal of the denominators on the L.H.S. and their total on the R.H.S. be the same, that total is zero.
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This is not in the expected form. But a little work regarding transposition
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Sunyam Samya Samuccaye in Certain Cubes
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But once again observe the problem in the vedic sense
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The traditional method will be horrible even to think of. • ( x – 249 ) + ( x + 247 ) = 2x – 2 = 2 ( x – 1 ). And x – 1. on R.H.S. • x – 1 = 0 by the ‘sutra’. • x = 1 is the solution. • No cubing or any other mathematical operations. 3/25/2014
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• Observe that ( N1 + D1 ) with in the cubes on L.H.S. is x + 2 + x + 3 = 2x + 5 and N2 + D2 on the right hand side is x + 1 + x + 4 = 2x + 5. • By vedic formula we have 2x + 5 = 0 • x = - 5 / 2.
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Anurupye Sunyamanyat
• 'If one is in ratio, the other one is zero'.
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• We use this Sutra in solving a special type of simultaneous simple equations in which the coefficients of 'one' variable are in the same ratio to each other as the independent terms are to each other.
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• Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is same as the ratio of independent terms • i.e., 2 : 6 i.e., 1 : 3. • Hence the other variable x = 0 • and 7y = 2 or 21y = 6 • y=2/7
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• 323x + 147y = 1615 969x + 321y = 4845 • The very appearance of the problem is frightening. • But just an observation and anurupye sunyamanyat give the solution x = 5, because coefficient of x ratio is
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• 323 : 969 = 1 : 3 and constant terms ratio is 1615 : 4845 = 1 : 3.
• y = 0 and 323 x = 1615 or 969 x = 4845 gives x = 5.
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Yavadunam Tavadunikrtya Varganca Yojayet • What ever the deficiency subtract that deficit from the number and write along side the square of that deficit
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Method-1 : Numbers near and less than the bases of powers of 10.
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2 96
Here base is 100.
• Since deficit is 100-96=4 • square of it is 16 • deficiency subtracted from the number 96 gives 96-4 = 92, • we get the answer 92 / 16 • Thus 962 = 9216. 3/25/2014
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2 994
Base is 1000
• Deficit is 1000 - 994 = 6. • Square of it is 36. • Deficiency subtracted from 994 gives • 994 - 6 = 988 • Answer is 988 / 036 [since base is 1000] 3/25/2014
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99882 Base is 10,000 • Deficit = 10000 - 9988 = 12. • Square of deficit = 122 = 144. • Deficiency subtracted from number = 9988 12 = 9976. • Answer is 9976 / 0144 [since base is 10,000]. 3/25/2014
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2 88
Base is 100
• Deficit = 100 - 88 = 12. • Square of deficit = 122 = 144. • Deficiency subtracted from number = 88 - 12 = 76. • Now answer is 76 / 144 =7744 [since base is 100] 3/25/2014
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Algebraic proof
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• a2 - b2 = (a + b) ( a - b). • a2 = (a + b) ( a - b) +b2 • Thus for a = 985 and b = 15; • a2= (a + b) ( a - b) + b2 • 9852 = ( 985 + 15 ) ( 985 - 15 ) + (15)2 = 1000 ( 970 ) + 225 = 970225. 3/25/2014
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Numbers near and greater than the bases of powers of 10.
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Antyayor Dasakepi • The Sutra signifies numbers of which the last digits added up give 10. • Sutra works in multiplication of numbers for example: • 25 and 25, 47 and 43, 62 and 68, 116 and 114.
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• Further the portion of digits or numbers left wards to the last digits remain the same. • At that instant use Ekadhikena on left hand side digits. Multiplication of the last digits gives the right hand part of the answer.
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127 x 123 • As antyayor dasakepi works, • we apply ekadhikena • 127 x 123 = 12 x 13/ 7 x 3 = 156 / 21 = 15621.
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• It is further interesting to note that the same rule works when the sum of the • last 2, last 3, last 4 - - - digits added respectively equal to 100, 1000, 10000 --• The simple point to remember is to multiply each product by 10, 100, 1000, - - as the case may be . 3/25/2014
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292 x 208 • Here 92 + 08 = 100, L.H.S portion is same i.e. 2 • 292 x 208 = ( 2 x 3 )/ 92 x 8 • 60 / =736 ( for 100 raise the L.H.S. product by 0) • = 60736. 3/25/2014
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Antyayoreva
'only the last terms'
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• The type of equations are those whose numerator and denominator on the L.H.S. bearing the independent terms stand in the same ratio to each other as the entire numerator and the entire denominator of the R.H.S. stand to each other.
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Algebraic Proof
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• Cross–multiplying • 28x + 42 = 30x + 40 • 28x – 30x = 40 – 42 • -2x = -2 • x = -2 / -2 = 1. 3/25/2014
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(x + 1) (x + 2) (x + 9) = (x + 3) (x + 4) (x + 5)
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Himanshu Thapliyal, Hamid R. Arabnia • A Time-Area- Power Efficient Multiplier and Square Architecture Based On Ancient Indian Vedic Mathematics • For low power and high speed applications. It is based on generating all partial products and their sums in one step.
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• Urdhva Tiryakbhyam” algorithm of Ancient Indian Vedic Mathematics which is utilized for multiplication to improve the speed, area parameters of multipliers. • Vedic Mathematics suggests one more formula for multiplication of large number i.e. “Nikhilam Sutra” which can increase the speed of multiplier by reducing the number of iterations 3/25/2014
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Ramesh pushpangadan, vineeth sukumaran , Rino, Dinesh,Sukumaran2011 • High speed vedic multiplier for DSP • Combinational delay obtained after synthesis is compared with booth wallace multiplier.
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Sumit Vaidya and Deepak Dandekar 2010 • DELAY-POWER PERFORMANCE COMPARISON OF MULTIPLIERS IN VLSI CIRCUIT DESIGN • Comparative study of different multipliers is done for low power requirement and high speed.
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• In FPGA implementation it has been found that the proposed Vedic multiplier and square are faster than array multiplier and Booth multiplier.
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• Vedic mathematics is the ancient mathematics which contain 16 sutras. There is a large scope to reduce the size and delay in electronic gadgets by using these sutras.
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CSE AND IT APPLI • DATA PATH • ALU DESIGN • PROCESSING OF INFORMATION • CRYPTOGRAPHY 3/25/2014
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CIRCUIT BRANCHES APPLI • DSP • CO PROCESSOR DESIGN • VLSI
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Mechanical Appli • ECU in Automobile • Motor control – Timing Variations • Control system simplifications • Quadratic equations simplification 3/25/2014
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Vedic engineering
Gurukula Kangri Vishwavidyalaya , Haridwar
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ACKNOWLEDGEMENTS
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When you change your thinking You change your beliefs. When you change your beliefs You change your expectations. When you change your expectations You change your attitude. When you change your attitude You change your behavior. When you change your behavior You change your performance. When you change your performance You change your LIFE. 3/25/2014
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Thank you 3/25/2014
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