http://www.ourkarnataka.com/vedicm/vedicms.htm
Vedic Math,
a Forgotten Science *
Some of the Sutras (Phrases) used
in Vedic Mathematics : *
1^{st} Example  1 Divided
by 19, 29, 39, …. 129 etc *
2^{nd} Example  Square of
Numbers ending in 5 *
3^{rd} Example 
Multiplierdigits consist entirely of nines
*
4^{th} Example General
Multiplication of any number by any number
*
5^{th} Example Algebraic
Divisions *
6^{th} Example Division
*
7^{th} Example
Factorisations of Quadratics *
8^{th} Example  Verifying
Correctness of answers *
9^{th} Example
Factorisations of Harder Quadratics
*
10^{th} 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.
 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.
 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 (18841960).
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)
 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, biquadratic
equations, higher degree equations, differential
calculus, Partial fractions, Integrations, Pythogorus
theoram, Apollonius Theoram, Analytical Conics and so
on.
 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.
 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 subsutras, 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.
 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 :
 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
Crosswise)
is useful in General
Multiplication of any number by any number.
4. Paravartya
Yojayet (P64)
(Transpose and Apply)
is useful in solving
Algebraic factors and divisions of some numbers etc.
5. Anurupyena
(P87) (Proportionately)
6.
AdhyamAdhyena, Antyamantyena (P87) (first
by the first and the last by the last)
are useful in solving
Quadratics
7.
Lopanastapanabhyam (P90)
(by (alternate) Elimination and
Retention)
is useful in
factorizing long and harder quadratics..
8.
GunitaSamuchhayah Samuchhayagunitah 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 Subsutra
of immense utility for the purpose of verifying the
correctness of our answers in multiplications,
divisions and factorisations:
9. Sunyam
Samya samuccaye P107 (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.
1^{st} Example  1
Divided by 19, 29, 39, …. 129 etc
 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
2^{nd} 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.
3^{rd} Example 
Multiplierdigits 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.
4^{th} Example General
Multiplication of any number by any number
The Sutra used is
UrdhvaTiryagbhyam which means, "Vertically
and crosswise" (P39)
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
5^{th} Example Algebraic
Divisions
Divide (12X^{2 }–8X32) by
(X2) using UrdhvaTiryak, (Vertically and
Crosswise)
Just by observation we
can say the quotient must be (12X+k)
We also know that –8x
= kx  24x Hence k = 16 crosscheck –2k = 32 third term
Divide (X^{3}
+ 7X^{2} +6X +5) by (X2)
X^{3} divided
by X gives X^{2} which is therefore the 1^{st}
term of the equation.
X^{2}
multiplied by –2 gives 2x^{2}, But
we have 7x^{2} in the Dividend. This
means we have to get 9X^{2} more.
This must result from the multiplication of
X by 9X. Hence the 2^{nd} 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
6^{th} Example Division
Sutra:
Paravartya Yojayet "Transpose and Apply"
(P63)
Divide
(12x^{2 }– 8x – 32) by (x 
2), factor is
+2
(X –
2)/_{2 }12x^{2 } 8x 32
+24x + 32

12x + 16 R = 0
Divide
7x^{2} + 5x + 3 by x – 1
Q = 7x + 12 R = 15
Divide
7x^{2} + 5x + 3 by x + 1
Q = 7x –2 R = 5
Divide
x^{3} + 7x^{2} + 6x + 5 by
x – 2 Q = x^{2} + 9x + 24 R =
53
Divide
x4 – x3 + x2 + 3x +5 by x^{2 }
– x – 1 Factors are x + 1
Q = x^{2}
+ 0x + 2 R = 5x + 7
Divide
6x^{4} + 13 x^{3} + 39 x^{2} + 37x +
45 by X^{2
}– 2x – 9
Factors 2x + 9 Q = 6x2
+ 25x + 143 R = 548x + 1332
Divide x^{4 }+
x^{2} + 1 by x^{2} – x
+1 (add 0x^{3 }&
0x)
X^{4 }+ 0x^{3}
+ x^{2} + 0x + 1 Factors are: x  1
Q
= x^{2} + 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
7^{th} Example
Factorisations of Quadratics
Sutras:
Anurupyena (P84)
"Proportionately"
AdhyamAdhyena, Antyamantyena
(P84)
"first by the first and the last
by the last"
8^{th} Example 
Verifying Correctness of answers
A Subsutra of
immense utility for the purpose of verifying the
correctness of our answers in multiplications,
divisions and factorisations:
GunitaSamuchhayah Samuchhayagunitah (P86) means
"The
product of the sum of the coefficients in
the factors is equal to
the
sum of the coefficients in the product"
S_{c}
of the product = Product of the S_{c}
in the factors
For example (x+7)
(x+9) = (x^{2} + 16x + 63)
(1+7) (1+9) = (1 + 16
+ 63) = 80
or (x+1) (x+2) (x+3) =
(x^{3} + 6X^{2} + 11x + 6)
(1+1) (1+2) (1+3) = (1
+ 6 + 11 + 6) = 24
9^{th} Example
Factorisations of Harder Quadratics
Lopanastapanabhyam (P87)
"by (alternate) Elimination and
Retention"
It is very difficult
to factorise the long quadratic
(2x^{2} + 6y^{2} + 3z^{2}
+ 7xy + 11yz + 7zx)
But "LopanaSthapana"
removes the difficulty. Eliminate z by putting z = 0.
Hence the given
expression E = 2x^{2} + 6y^{2} + 7xy =
(x+2y) (2x+3y)
Similarly, if y=0,
then E = 2x^{2} + 3z^{2} + 7zx = (x+3z)
(2x+z)
Hence E =
(x+2y+3z) (2x+3y+z)
Factorise 2x2
+ 2y2 + 5xy + 2x 5y –12 = (x+3) (2x4) and (2y+3)
(y4)
Hence, E =
(x+2y+3) (2x+y4)
* This "Lopanasthapana"
method (of alternate elimination and
retention) will be found highly useful in HCF, in
Solid Geometry and in Coordinate Geometry of the
straight line, the Hyperbola, the conjugate
Hyperbola, the Asymptotes etc.
10^{th} Example 
Factorisations of Harder Quadratics – Special Cases
Sunyam Samya samuccaye
P107 (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/ (x7)
+ 1/(x9) = 1/(x6) + 1/(x10)
Vedic Sutra says, (other elements being
equal), the sumtotal of the
denominators on LHS and the total on the
RHS are the same, then the total is
zero.
Here, D1 + D2 = D3 +
D4 = 2x16 =0 hence x = 8
Examples 1/(x+7) +
1/(x+9) = 1/(x+6) + 1/(x+10) x =  8
1/(x7) + 1(x+9) =
1/(x+11) + 1/(x9) x =  1
1/(x8) + 1/(x9) =
1/(x5) + 1/(x12) x = 81/2
1/(xb)  1/(xbd) =
1/(xc+d)  1/(xc) 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, (x3) + (x9) = 2x – 12
Hence x = 6
(x149)^{3 }+
(x51)^{3 }= 2(x100)^{3} Hence 2x200 =0 &
x = 100
(x+a+bc)^{3 }
+ (x+b+ca)^{3} = 2(x+b)^{3} x = b
Sixteen Simple
Mathematical Sutras (Phrases) From The Vedas
Sixteen Sutras and Their Corollaries

Sutras or
Formulae 
SubSutras or Corollaries 

Ekadhikena
Purvena
(One more than
the previous) P2
Division eg.
1/19 
Anurupyena
(Proportionately) P20, 87 Multiplications 

Nikhilam
Navatascaram Dashtah
(All from 9
last from 10) P13
Multiplication
eg 9927 X 9999 
Sishyate
Seshasanjnah 

Urdhvatiryaghyam P
(Vertically
and crosswise) 
Adhyamadhyena
antyamantyena P87
(First by the
first and last by the last) 

Paravartya
Yojayet P
(Transpose and
apply) 
Kevalaih
Saptakam Gunyat 

Sunyam Samya
samuccaye P107
(when
Samuccaya is the same, that Samuccaya is zero) 
Veshtanam 

(Anurupye)
Sunyamanyat 
Yavadunam
Tavadunam 

Sankalanavyavakalana 
Yavadunam Tavadunikrtya Vargamca
Yojayet 

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

CalanaKalanabhyam 
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 P89 *13 

Ekanyunena
purvena
(one less than
the previous) P35
one of the
multiplierdigits 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 storehouse 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) illinformed (d) biased
against India
 If you were to keep one rice
grain on the 1^{st} square of the chess board, 2
grains on the 2^{nd} square, 4 grains on the 3^{rd},
8 on the 4^{th}, 16 on the 5^{th}, 32 on
the 6^{th} 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
ax^{n} + bx^{n1} + cx^{n2}
+ dx^{n3} + ….. is divided by
x – p
The Reminder R is
R = ap^{n}
+ bp^{n1} + cp^{n2 }+ dp^{n3 }+
…..
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.
Socalled 'plantlike' and
'tridentshaped' 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 predate 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 carbondated to 33003200
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.