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http://www.ourkarnataka.com/vedicm/vedicms.htm

Vedic Math, a Forgotten Science *

Some of the Sutras (Phrases) used in Vedic Mathematics : *

1st Example - 1 Divided by 19, 29, 39, …. 129 etc *

2nd Example - Square of Numbers ending in 5 *

3rd Example - Multiplier-digits consist entirely of nines *

4th Example General Multiplication of any number by any number *

5th Example Algebraic Divisions *

6th Example Division *

7th Example Factorisations of Quadratics *

8th Example - Verifying Correctness of answers *

9th Example Factorisations of Harder Quadratics *

10th Example - Factorisations of Harder Quadratics – Special Cases *

Sixteen Simple Mathematical Sutras (Phrases) From The Vedas *

Notes and Quiz *

 

 

 

 

Vedic Maths, a Forgotten Science

It is being taught in some of the most prestigious institutions in England and Europe. NASA scientists applied its principles in the area of artificial intelligence. And yet, in the country of its birth it languishes as a forgotten science.

Vedic mathematics, which simplifies arithmetic and algebraic operations, has increasingly found acceptance the world over. Experts suggest that it could be a handy tool for those who need to solve mathematical problems faster by the day.

 

  1. What is Vedic Mathematics?

    It is an ancient technique, which simplifies multiplication, divisibility, complex numbers, squaring, cubing, square and cube roots. Even recurring decimals and auxiliary fractions can be handled by Vedic mathematics.

     

  2. Who Brought Vedic Maths to limelight?

    The subject was revived largely due to the efforts of Jagadguru Swami Bharathikrishna Tirthaji of Govardhan Peeth, Puri Jaganath (1884-1960). Having researched the subject for years, even his efforts would have gone in vain but for the enterprise of some disciples who took down notes during his last days. That resulted in the book, Vedic Mathematics, in the 1960s. These are now available in a book called

    "VEDIC MATHEMATICS"

    by H.H. Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaj.

    Publishers Motilal Banarasidass, Bunglow Road, Jawahar Nagar, Delhi –110 007; or

    Chowk, Varanasi (UP); or Ashok Raj Path, Patna, (Bihar)

  3. What is the basis of Vedic Mathematics?

    The basis of Vedic mathematics, are the 16 sutras, which attribute a set of qualities to a number or a group of numbers. The ancient Hindu scientists (Rishis) of Bharat in 16 Sutras (Phrases) and 120 words laid down simple steps for solving all mathematical problems in easy to follow 2 or 3 steps.

    Vedic Mental or one or two line methods can be used effectively for solving divisions, reciprocals, factorisation, HCF, squares and square roots, cubes and cube roots, algebraic equations, multiple simultaneous equations, quadratic equations, cubic equations, bi-quadratic equations, higher degree equations, differential calculus, Partial fractions, Integrations, Pythogorus theoram, Apollonius Theoram, Analytical Conics and so on.

  4. What is the Speciality of Vedic Mathematics?

    Vedic scholars did not use figures for big numbers in their numerical notation. Instead, they preferred to use the Sanskrit alphabets, with each alphabet constituting a number. Several mantras, in fact, denote numbers; that includes the famed Gayatri mantra, which adds to 108 when decoded.

  5. Is it useful today?

Given the initial training in modern maths in today's schools, students will be able to comprehend the logic of Vedic mathematics after they have reached the 8th standard. It will be of interest to every one but more so to younger students keen to make their mark in competitive entrance exams.

India's past could well help them make it in today's world.

It is amazing how with the help of 16 Sutras and 16 sub-sutras, the Vedic seers were able to mentally calculate complex mathematical problems.

1. Why did the Vedic Seers need Vedic Mathematics or Mental Mathematics?

The Vedic Seers were highly ritualistic in practice. All the four Vedas namely Rig Veda, Yajur Veda, Sama Veda and Atharva Veda, consist of Samhitas, Brahmanas, Aranyakas and Upanishads. Of these four, the first three namely Samhitas, Brahmanas and Aranyakas contain several thousand Mantras or Hymns, Ritual practices and their interpretations. The Vedic Seers were very particular about the time of doing the rituals. Hence they needed a very scientific and accurate calendar and time measurements. Hence we have the adage IST which once meant Indian Standard Time. But in recent times due to our laziness and carelessness, it has come to mean Indian Stretchable Time.

The Vedic culture was that of Yajna and the Vedic Purohits or priests were also very particular about the shape and size of the Yajna Kund. For this they needed a very highly developed Geometry and Trigonometry.

     

  1. Where is Vedic Mathematics found?

     

Vedic Mathematics forms part of Jyotish Shastra which is one of the six parts of Vedangas. The Jyotish Shastra or Astronomy is made up of three parts called Skandas. A Skanda means the big branch of a tree shooting out of the trunk

 

 

 

 

 

 

 

Some of the Sutras (Phrases) used in Vedic Mathematics :

 

     

  1. Ekadhikena Purvena (One More than the Previous)

     

is useful in solving Special Multiplications like 25X25, 95X95, 105X105 etc

Special Divisons like 1 divided by 19, 29, 39, …. 199 etc.

 

2. Eka Nunena Purvena (One less than the Previous)

is useful in solving Special Multiplications like 777 X 999, 123456789X 999 999 999

 

3. Urdhva Tiryak bhyam (Vertically and Cross-wise)

is useful in General Multiplication of any number by any number.

 

4. Paravartya Yojayet (P-64) (Transpose and Apply)

is useful in solving Algebraic factors and divisions of some numbers etc.

 

5. Anurupyena (P-87) (Proportionately)

6. Adhyam-Adhyena, Antyam-antyena (P-87) (first by the first and the last by the last)

are useful in solving Quadratics

 

7. Lopana-stapana-bhyam (P-90) (by (alternate) Elimination and Retention)

is useful in factorizing long and harder quadratics..

 

8. Gunita-Samuchhayah Samuchhaya-gunitah which means

"The product of the sum of the coefficients in the factors is equal to

the sum of the coefficients in the product"

is a Sub-sutra of immense utility for the purpose of verifying the correctness of our answers in multiplications, divisions and factorisations:

 

9. Sunyam Samya samuccaye P-107 (when Samuccaya is the same, that Samuccaya is zero) Samuccaya is a technical term which has several meanings. This is useful in solving many complex factors and equations.

1st Example - 1 Divided by 19, 29, 39, …. 129 etc

 

     

  1. To Divide 1 by numbers ending in 9 like 1 divided by 19, 29, 39, ….. 119 etc.

     

Some of these numbers like 19, 29, 59 are prime numbers and so cannot be factorised and division becomes all the more difficult and runs into many pages in the present conventional method and the chances of making mistakes are many.

 

The Vedic Solution is obtained by applying the Sutra (theorem) Ekadhikena Purvena which when translated means "One more than the Previous"

Take for example 1 divided by 19. In the divisor 19, the previous is 1 and the factor is obtained by adding 1 to it which is 2. Similarly when we have to divide by 29, 39, … 119 the factors shall be 3,4,… 12 respectively. (Add 1 to the previous term in the divisor). After this divide 1 by the factor in a typical Vedic way and the answer is obtained in 1 step. Thus
1 divided by 19 = 0.0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1

1 divided by 29 = 0.0 3 4 4 8 2 7 5 8 6 2 0 6 8 9 6 5 5 1 7 2 4 1 3 7 9 3 1

 

2nd Example - Square of Numbers ending in 5

 

Squares of 25, 35, 45, 85, 95, 105, 195 etc can be worked out mentally

Again the Sutra used here is Ekadhikena Purvena which means, "One more than the previous."

The last term is always 5 and the Previous terms are 2, 3, 4, 8, 9, 10, 19 etc and we have to add 1 to them. Square of the last term 5 is always 25.

Thus the Square of 25 is 2x3/25 = 625

the Square of 35 is 3x4/25 = 1225

the Square of 45 is 4x5/25 = 2025

the Square of 85 is 8x9/25 = 7225

the Square of 95, 105, 195 can be obtained in the same way.

 

Use the above formula to find the products of

23 multiplied by 27; 44 multiplied by 46; 192 multiplied by 198 and so on.

 

 

 

3rd Example - Multiplier-digits consist entirely of nines

The Sutra (theorem) used here is Ekanyunena Purvena, sound as if it were the converse of Ekadhik Sutra ie "one less"

 

777 multiplied by 999 = 776,223

(776 is one less than multiplicand 777 and 223 is the compliment of 776 from 9)

120 35 79 multiplied by 999 99 99 = 120 35 78, 879 64 21

1234 5678 09 multiplied 9999 9999 99 = 1234 5678 08 8765 4321 91

Such multiplications come up in advanced astronomy.

 

 

4th Example General Multiplication of any number by any number

The Sutra used is Urdhva-Tiryagbhyam which means, "Vertically and cross-wise" (P-39)

 

To multiply 12 by 13 mentally multiply

1 by 1, 3 by 1 and 2 by 1 and finally 2 by 3 and write the answer as 1 56

 

To multiply 37 by 42 mentally multiply

3 by 4, 3 by 2 and 7 by 4 and finally 7 by 2 and the answer is 12/34/14 = 1554

 

To multiply 1021 by 2103 mentally multiply as follows

1 by2 1by1+0by2 1by0+2by2+1by0 1by3+0by0+2by1+1by2 0by3+2by0+1by1 2by3+1by0 1by3 = 2147163

 

Multiplying 8 7 2 6 5 by 3 2 1 1 7 gives 2 8 0 2 6 9 0 0 0 5

 

5th Example Algebraic Divisions

Divide (12X2 –8X-32) by (X-2) using Urdhva-Tiryak, (Vertically and Cross-wise)

Just by observation we can say the quotient must be (12X+k)

We also know that –8x = kx - 24x Hence k = 16 cross-check –2k = -32 third term

 

Divide (X3 + 7X2 +6X +5) by (X-2)

X3 divided by X gives X2 which is therefore the 1st term of the equation.

X2 multiplied by –2 gives -2x2, But we have 7x2 in the Dividend. This means we have to get 9X2 more. This must result from the multiplication of X by 9X. Hence the 2nd term of the divisor must be 9X. Hence the quotient Q= X2+ 9x + k…

For the third term we already have 6X = kX – 18X Hence k = 24

To find the Remainder R: 2k+5 = 53

 

6th Example Division

Sutra: Paravartya Yojayet "Transpose and Apply" (P-63)

 

Divide (12x2 – 8x – 32) by (x - 2), factor is +2

 

(X – 2)/2 12x2 - 8x -32

+24x + 32

--------------------------

12x + 16 R = 0

 

Divide 7x2 + 5x + 3 by x – 1 Q = 7x + 12 R = 15

Divide 7x2 + 5x + 3 by x + 1 Q = 7x –2 R = 5

Divide x3 + 7x2 + 6x + 5 by x – 2 Q = x2 + 9x + 24 R = 53

 

Divide x4 – x3 + x2 + 3x +5 by x2 – x – 1 Factors are x + 1

Q = x2 + 0x + 2 R = 5x + 7

 

Divide 6x4 + 13 x3 + 39 x2 + 37x + 45 by X2 – 2x – 9

Factors 2x + 9 Q = 6x2 + 25x + 143 R = 548x + 1332

 

Divide x4 + x2 + 1 by x2 – x +1 (add 0x3 & 0x)

X4 + 0x3 + x2 + 0x + 1 Factors are: x - 1

Q = x2 + x + 1 R = 0

 

Divide 1 2 3 4 by 1 1 2 Factors -1 –2 Q = 11 R = 2

 

Divide 1 3 4 5 6 by 1 1 2 3 Q = 12 R = - 20

The Reminder cannot be negative. Hence Q = 11 R = 1 1 0 3

 

Divide 1 3 9 0 5 by 1 1 3 Q = 1 24 R = -107

Q = 123 R = 6

7th Example Factorisations of Quadratics

Sutras: Anurupyena (P-84) "Proportionately"

Adhyam-Adhyena, Antyam-antyena (P-84) "first by the first and the last by the last"

 

8th Example - Verifying Correctness of answers

A Sub-sutra of immense utility for the purpose of verifying the correctness of our answers in multiplications, divisions and factorisations:

Gunita-Samuchhayah Samuchhaya-gunitah (P-86) means

"The product of the sum of the coefficients in the factors is equal to

the sum of the coefficients in the product"

Sc of the product = Product of the Sc in the factors

For example (x+7) (x+9) = (x2 + 16x + 63)

(1+7) (1+9) = (1 + 16 + 63) = 80

or (x+1) (x+2) (x+3) = (x3 + 6X2 + 11x + 6)

(1+1) (1+2) (1+3) = (1 + 6 + 11 + 6) = 24

9th Example Factorisations of Harder Quadratics

Lopana-stapana-bhyam (P-87) "by (alternate) Elimination and Retention"

 

It is very difficult to factorise the long quadratic (2x2 + 6y2 + 3z2 + 7xy + 11yz + 7zx)

But "Lopana-Sthapana" removes the difficulty. Eliminate z by putting z = 0.

Hence the given expression E = 2x2 + 6y2 + 7xy = (x+2y) (2x+3y)

Similarly, if y=0, then E = 2x2 + 3z2 + 7zx = (x+3z) (2x+z)

Hence E = (x+2y+3z) (2x+3y+z)

Factorise 2x2 + 2y2 + 5xy + 2x- 5y –12 = (x+3) (2x-4) and (2y+3) (y-4)

Hence, E = (x+2y+3) (2x+y-4)

 

* This "Lopana-sthapana" method (of alternate elimination and retention) will be found highly useful in HCF, in Solid Geometry and in Co-ordinate Geometry of the straight line, the Hyperbola, the conjugate Hyperbola, the Asymptotes etc.

 

 

10th Example - Factorisations of Harder Quadratics – Special Cases

Sunyam Samya samuccaye P-107 (when Samuccaya is the same, that Samuccaya is zero) Samuccaya is a technical term which has several meanings.

 

First Meaning: It is a term which occurs as a common factor in all the terms concerned

Thus 12x + 3x = 4x + 5x x is common, hence x = 0

Or 9 (x+1) = 7 (x+1) here (x+1) is common; hence x +1= 0

 

Second Meaning: Here Samuccaya means "the product of the independent terms"

Thus, (x +7) (x +9) = (x +3) (x +21)

Here 7 x9 = 3 x 21. Therefore x = 0

 

Third Meaning Samuccaya thirdly means the sum of the Denominators of two fractions

having the same numerical numerator

Thus, 1/(2x –1) + 1/(3x –1) = 0 Hence 5x – 2 =0 or x = 2/5

 

Fourth Meaning: Here Samuccaya means combination (or TOTAL).

In this context it is used in different contexts. These are

If the sum of the Numerators and the sum of the Denominators be the same, then that sum = 0

(2x +9)/ (2x +7) = (2x +7)/ (2x +9)

N1 + N2 = D1 + D2 = 2x + 9 + 2x + 7 = 0

Hence 4x + 16 = 0 hence x = -4

 

Note: If there is a numerical factor in the algebraic sum, that factor should be removed.

(3x +4)/ (6x +7) = (x +1)/ (2x +3)

Here N1 +N2 = 4x +5; D1 +D2 = 8x + 10; 4x +5 =0 x= -5/4

 

Fifth Meaning: Here Samuccaya means TOTAL ie Addition & subtraction

 

Thus, (3x +4)/ (6x +7) = (5x +6)/ (2x +3)

Here N1+N2 = D1 + D2 = 8x + 10 =0 hence x = - 5/4

D1 – D2 = N2 – N1 = 2x + 2 = 0 x = -1

 

Sixth Meaning: Here Samuccaya means TOTAL; used in Harder equations

 

Thus, 1/ (x-7) + 1/(x-9) = 1/(x-6) + 1/(x-10)

Vedic Sutra says, (other elements being equal), the sum-total of the denominators on LHS and the total on the RHS are the same, then the total is zero.

Here, D1 + D2 = D3 + D4 = 2x-16 =0 hence x = 8

 

Examples 1/(x+7) + 1/(x+9) = 1/(x+6) + 1/(x+10) x = - 8

 

1/(x-7) + 1(x+9) = 1/(x+11) + 1/(x-9) x = - 1

 

1/(x-8) + 1/(x-9) = 1/(x-5) + 1/(x-12) x = 8-1/2

 

1/(x-b) - 1/(x-b-d) = 1/(x-c+d) - 1/(x-c) x = 1/2(b+c)

 

Special Types of seeming Cubics (x- 3)3 + (x –9)3 = 2(x –6)3

current method is very lengthy, but Vedic method says, (x-3) + (x-9) = 2x – 12 Hence x = 6

 

(x-149)3 + (x-51)3 = 2(x-100)3 Hence 2x-200 =0 & x = 100

 

(x+a+b-c)3 + (x+b+c-a)3 = 2(x+b)3 x = -b

Sixteen Simple Mathematical Sutras (Phrases) From The Vedas

 

Sixteen Sutras and Their Corollaries

 

Sutras or Formulae

Sub-Sutras or Corollaries

 

Ekadhikena Purvena

(One more than the previous) P-2

Division eg. 1/19

Anurupyena

(Proportionately) P-20, 87 Multiplications

 

Nikhilam Navatascaram Dashtah

(All from 9 last from 10) P-13

Multiplication eg 9927 X 9999

Sishyate Seshasanjnah

 

Urdhva-tiryaghyam P-

(Vertically and cross-wise)

Adhyam-adhyena antyam-antyena P-87

(First by the first and last by the last)

 

Paravartya Yojayet P-

(Transpose and apply)

Kevalaih Saptakam Gunyat

 

Sunyam Samya samuccaye P-107

(when Samuccaya is the same, that Samuccaya is zero)

Veshtanam

 

(Anurupye) Sunyamanyat

Yavadunam Tavadunam

 

Sankalana-vyavakalana

Yavadunam Tavadunikrtya Vargamca Yojayet

 

Purna purnabhyam

Antyayor Dasakepi (also for two numbers whose last digits together total 10)

 

Calana-Kalanabhyam

Antyayoreva

 

Yavadunam

Samuccayagunitah

 

Vyashti samashti

Lopana Sthapanabhyam

"by (alternate) Elimination and Retention"

very useful in HCF, Solid Geometry, Coordinate Geometry,

 

Seshanyankena caramena

Vikolanam

 

Sopantya dvayamantyam

Gunita samuccayah Samucchaya gunitah P-89 *13

 

Ekanyunena purvena

(one less than the previous) P-35

one of the multiplier-digits is all 9

 

 

 

Gunita Samuccayah P-

 

16.

Gunaka Samuccayah

 

 

 

 

Notes and Quiz

 

 

To many Indians Vedic and Sanskrit Mantras are relevant only for religious occasions like weddings, Grahapraveshas and Upanayanams. Some who are aware of Upanishads and Bhagwad Gita, are fascinated by the depth of their philosophical teachings. But Vedas in fact are a store-house of knowledge - both secular and spiritual eg. Ayur Veda is a treatise on medicine and health. Dhanur Veda on war techniques and martial arts, Stapathya Veda on buildings and constructions, Gandharva Veda on Music and art, and Artha Shastra on political systems and economics.

 

     

  • According to scholars like Max Muller, Vedas are the oldest texts of mankind. They are the treasure house of both secular and spiritual knowledge.

     

 

     

  • Schaupenhopher, the German scholar said,

     

"The Upanishads are the solace of my life and they shall be the solace of my death too".

* Upanishads are the Philosophical portions of the Vedas.

 

     

  • Once a great Vedic scholar Mandana Mishra remarked, "If the Vedas are true nothing shall happen to me" and he jumped from a tall building. To every one’s surprise he landed safely but one of his legs was bruised little bit. He questioned the Ved Bhagwan why he was hurt and he got the answer why he was punished.

     

Can you guess the answer?

 

 

     

  • Once a German Indologist remarked "India’s contribution in Science in zero"

     

Do you think he was

(a) right (b) wrong (c) ill-informed (d) biased against India

 

 

     

  • If you were to keep one rice grain on the 1st square of the chess board, 2 grains on the 2nd square, 4 grains on the 3rd, 8 on the 4th, 16 on the 5th, 32 on the 6th and so on, … how many kilograms of rice would you need to fill all the squares in a chess board?

     

 

 

 

Page 65

The Reminder Theorem R = Ep where (x - p) is the divisor

(E = Q (x – p) + R Put x = p Hence, Ep = R)

 

Thus when axn + bxn-1 + cxn-2 + dxn-3 + ….. is divided by x – p

The Reminder R is R = apn + bpn-1 + cpn-2 + dpn-3 + …..

 

 

 

 

Vedic Sciences

 

As it is the Rgveda, which pointed to the Cosmic Law first, I feel aggrieved to note that all the ancient wisdom that the vedic seers discovered has been attributed to other nations. The discovery of the Natural Law is credited to the Stoics of Greece who existed circa 300 B.C. whereas the Rgveda was composed at least in the 3500 B.C., if not earlier. This is the price of political weakness, and the Hindus have still not stopped paying for it. Until they learn to be strong, the erosion of their cultural greatness will not only continue, but is likely to accelerate its pace.

 

 

From: Nilesh Solanki, Birmingham

  • Subject: 'Earliest writing' found

BBC News Online: Sci/Tech

Tuesday, May 4, 1999 Published at 01:04 GMT 02:04 UK

 

'Earliest writing' found

Exclusive by BBC News Online Science Editor Dr David Whitehouse

 

The first known examples of writing may have been unearthed at an archaeological dig in Pakistan.

So-called 'plant-like' and 'trident-shaped' markings have been found on fragments of pottery dating back 5500 years.

 

They were found at a site called Harappa in the region where the great Harappan or Indus civilisation flourished four and a half thousand years ago.

 

Harappa was originally a small settlement in 3500 BC but by 2600 BC it had developed into a major urban centre.

 

The earliest known writing was etched onto jars before and after firing. Experts believe they may have indicated the contents of the jar or be signs associated with a deity.

 

According to Dr Richard Meadow of Harvard University, the director of the Harappa Archaeological Research Project, these primitive inscriptions found on pottery may pre-date all other known writing.

 

Last year it was suggested that the oldest writing might have come from Egypt.

 

Clay tablets containing primitive words were uncovered in southern Egypt at the tomb of a king named Scorpion.

 

They were carbon-dated to 3300-3200 BC. This is about the same time, or slightly earlier, to the primitive writing developed by the Sumerians of he Mesopotamian civilisation around 3100 BC.

 

"It's a big question as to if we can call what we have found true writing," he told BBC News Online, "but we have found symbols that have similarities to what became Indus script.

 

"One of our research aims is to find more examples of these ancient symbols and follow them as they changed and became a writing system," he added.

 

One major problem in determining what the symbols mean is that no one understands the Indus language. It was unique and is now dead.

 

Dr Meadow points out that nothing similar to the 'Rosetta stone' exists for the Harappan text. The Rosetta stone contained Egyptian hieroglyphics and a Greek translation and thus helped early language researchers decipher the meaning.

 

The Harappan language died out and did not form the basis of other languages.

 

"So probably we will never know what the symbols mean," Dr Meadow told BBC News Online from Harappa.

 

What historians know of the Harappan civilisation makes them unique. Their society did not like great differences between social classes or the display of wealth by rulers. They did not leave behind large monuments or rich graves.

 

They appear to be a peaceful people who displayed their art in smaller works of stone.

 

Their society seems to have petered out. Around 1900 BC Harappa and other urban centres started to decline as people left them to move east to what is now India and the Ganges.

 

This discovery will add to the debate about the origins of the written word.

 

It probably suggests that writing developed independently in at least three places - Egypt, Mesopotamia and Harappa between 3500 BC and 3100 BC.

 

 

 






 

 

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