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VEDICVEDIC
MATHEMATICSMATHEMATICS
-- By Prashanth K NBy Prashanth K N
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What is Vedic Mathematics ?
Vedic mathematics is the namegiven to the ancient system ofmathematics which wasrediscovered from the Vedas.
Its a unique technique ofcalculations based on simpleprinciples and rules , with whichany mathematical problem - be itarithmetic, algebra, geometry ortrigonometry can be solvedmentally.
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Who revived this?Who revived this?
The subject was revived largely due to the efforts ofJagadguru Swami Bharathikrishna Tirthajiof GovardhanPeeth, Puri Jaganath (1884-1960). Having researched the
subject for years, even his efforts would have gone invain but for the enterprise of some disciples who tookdown notes during his last days. That resulted in thebook, Vedic Mathematics, in the 1960s.
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Why Vedic Mathematics?Why Vedic Mathematics? It helps a person to solve problems 10-15 times faster.
It reduces burden (Need to learn tables up to nine only)
It provides one line answer.
It is a magical tool to reduce scratch work and finger counting. It increases concentration.
Time saved can be used to answer more questions.
Improves concentration.
Logical thinking process gets enhanced.
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Base of Vedic MathematicsBase of Vedic Mathematics
Vedic Mathematics
now refers to a set of
sixteen mathematical
formulae or sutras and
their corollaries
derived from the Vedas.
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Base of Vedic MathematicsBase of Vedic Mathematics
Vedic Mathematicsnow refers to a set
of sixteenmathematicalformulae or sutrasand theircorollaries derived
from the Vedas.
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EKDHIKENA PRVEAEKDHIKENA PRVEA
The Sutra
(formula)
Ekdhikena
Prvena means:
By one more than
the previous one.
This Sutra is
used to the
Squaring ofnumbers ending
in 5.
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Squaring of numbers ending in
5.
Conventional Method
65 X 65
6 5
X 6 5
3 2 53 9 0 X
4 2 2 5
Vedic Method
65 X 65 = 4225
( 'multiply the
previous digit 6 byone more thanitself 7. Than write25 )
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Corollary to Ekadhikena purvenaCorollary to Ekadhikena purvena
Squaring a number that does not end in 5.
This method requires rounding a number up and down based on thenearest base of 10 or 100, multiplying the two numbers, then addingthe square of the number added and subtracted. I'll explain with two examples:
Rounding to base-100: To find the square of 96, you would round up to 100.Since you added 4, you now subtract 4 from 96 to yield 92. Multiply 92 and100. This can be easily done in one's head: 9200. Since you added andsubtracted 4, square the 4 to yield 16. Now add 16 to 9200. Thus, 96 squared is9216.
Rounding to base-10: To find the square of 57, you would round up to 60. Sinceyou added 3, you now subtract 3 from 57 to yield 54. Multiply 60 by 54 (somepeople can do this in their head). The answer is 3240. Since you added andsubtracted 3, square it. That makes 9. Add 9 to 3240. The square of 57 is 3249.
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NIKHILAM AVATASCHARAMAMNIKHILAM AVATASCHARAMAM
DASATAHDASATAHThe Sutra (formula)
NIKHILAM
NAVATASCHARAMAM DASATAH
means :
all from 9 and thelast from 10
This formula canbe very effectively
applied inmultiplication ofnumbers, which arenearer to bases like
10, 100, 1000 i.e., tothe powers of 10(eg: 96 x 98 or 102x 104).
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Case I :When both the numbers are lower
than the base.
Conventional Method
97 X 94
9 7
X 9 4
3 8 8
8 7 3 X
9 1 1 8
Vedic Method
9797 33
XX 9494 669 1 1 89 1 1 8
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Case ( ii) : When both theCase ( ii) : When both the
numbers are higher than the basenumbers are higher than the base Conventional
Method
103 X 105
103
X 105
5 1 5
0 0 0 X1 0 3 X X
1 0, 8 1 5
Vedic Method
For Example103 X 105For Example103 X 105
103103 33
XX 105 55
1 0, 8 1 5
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Case III: When one number is moreCase III: When one number is more
and the other is less than the base.and the other is less than the base.
Conventional Method
103 X 98103
X 98
8 2 4
9 2 7 X
1 0, 0 9 4
Vedic Method
103103 33
XX 98 -2
1 0, 0 9 4
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NURPYENA
The Sutra (formula)
NURPYENA
means :
'proportionality'
or'similarly'
This Sutra is highlyThis Sutra is highlyuseful to finduseful to find
products of twoproducts of twonumbers whennumbers whenboth of them areboth of them arenear the Commonnear the Common
bases like 50, 60,bases like 50, 60,200 etc (multiples200 etc (multiplesof powers of 10).of powers of 10).
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NURPYENA
Conventional Method
58 X 485 8
X 4 8
4 6 42 4 2 X
2 8 8 4
Vedic Method
5858 88XX 4848 --22
2 82 8 8 4
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NURPYENA
Conventional Method
46 X 43
4 6
X 4 3
1 3 81 8 4 X
1 9 7 8
Vedic Method
4646 --44
XX 43 --77
1 9 7 8
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URDHVATIRYAGBHYAM
The Sutra (formula)
URDHVA
TIRYAGBHYAMmeans :
Vertically and cross
wise
This the generalThis the generalformula applicableformula applicable
to all cases ofto all cases ofmultiplication andmultiplication andalso in the divisionalso in the divisionof a large numberof a large number
by another largeby another largenumber.number.
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Two digit multiplication byby
URDHVA TIRYAGBHYAMThe Sutra (formula)
URDHVA
TIRYAGBHYAMmeans :
Vertically and cross
wise
Step 1Step 1: 5: 52=10, write2=10, writedown 0 and carry 1down 0 and carry 1
Step 2Step 2: 7: 72 + 52 + 53 =3 =
14+15=29, add to it14+15=29, add to itprevious carry over valueprevious carry over value1, so we have 30, now1, so we have 30, nowwrite down 0 and carry 3write down 0 and carry 3
Step 3Step 3: 7: 73=21, add3=21, add
previous carry over valueprevious carry over valueof 3 to get 24, write itof 3 to get 24, write itdown.down.
So we have 2400 as theSo we have 2400 as theanswer.answer.
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Two digit multiplication byby
URDHVA TIRYAGBHYAMVedic Method
4 6
X 4 3
1 9 7 8
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Three digit multiplication by
URDHVATIRYAGBHYAMVedic Method
103
X 105
1 0, 8 1 5
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YAVDUNAM TAAVDUNIKRITYA
VARGANCHAYOJAYETThis sutra means
whatever the extent
of its deficiency,lessen it stillfurther to that veryextent; and also set
up the square ofthat deficiency.
This sutra is veryhandy in
calculating squaresof numbersnear(lesser) topowers of 10
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YAVDUNAM TAAVDUNIKRITYAVARGANCHAYOJAYET
982
= 9604
The nearest power of 10 to 98 is 100.The nearest power of 10 to 98 is 100.Therefore, let us take 100 as our base.Therefore, let us take 100 as our base.
Since 98 is 2 less than 100, we call 2 as theSince 98 is 2 less than 100, we call 2 as thedeficiency.deficiency.
Decrease the given number further by anDecrease the given number further by anamount equal to the deficiency. i.e.,amount equal to the deficiency. i.e.,perform ( 98perform ( 98 --2 ) = 96. This is the left side2 ) = 96. This is the left sideof our answer!!.of our answer!!.
On the right hand side put the square ofOn the right hand side put the square of
the deficiency, that is square of 2 = 04.the deficiency, that is square of 2 = 04.
Append the results from step 4 and 5 toAppend the results from step 4 and 5 toget the result. Hence the answer is 9604.get the result. Hence the answer is 9604.
NoteNote :: While calculating step 5, the number of digits in the squared number (04)While calculating step 5, the number of digits in the squared number (04)
should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).
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YAVDUNAM TAAVDUNIKRITYAVARGANCHAYOJAYET
1032= 10609
The nearest power of 10 to 103 is 100.The nearest power of 10 to 103 is 100.Therefore, let us take 100 as our base.Therefore, let us take 100 as our base.
Since 103 is 3 more than 100 (base), weSince 103 is 3 more than 100 (base), wecall 3 as the surplus.call 3 as the surplus.
Increase the given number further by anIncrease the given number further by anamount equal to the surplus. i.e., perform (amount equal to the surplus. i.e., perform (103 + 3 ) = 106. This is the left side of our103 + 3 ) = 106. This is the left side of ouranswer!!.answer!!.
On the right hand side put the square ofOn the right hand side put the square of
the surplus, that is square of 3 = 09.the surplus, that is square of 3 = 09.
Append the results from step 4 and 5 toAppend the results from step 4 and 5 toget the result.Hence the answer is 10609.get the result.Hence the answer is 10609.
NoteNote :: while calculating step 5, the number of digits in the squared number (09)while calculating step 5, the number of digits in the squared number (09)
should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).
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YAVDUNAM TAAVDUNIKRITYA
VARGANCHAYOJAYET
10092
= 1018081
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Multiplying numbers by 11
To multiply any 2-figure number by 11 we just
put the total of the two figures between
the 2 figures.
26 X 11 = 286
72 X 11 = 792
77 X 11 = 847
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Multiplying numbers by 12
To multiply any 2-figure number by 12 we
double the last digit, put the total of the
twice the first digit and the 2nd
digit.
13 X 12 = 156
14 X 12 = 168
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SAKALANA
VYAVAKALANBHYAMThe Sutra (formula)
SAKALANA
VYAVAKALANBHYAMmeans :
'by addition and by
subtraction'
It can be applied inIt can be applied insolving a special typesolving a special typeof simultaneousof simultaneous
equations where theequations where thexx -- coefficients andcoefficients andthe ythe y -- coefficientscoefficientsare foundare found
interchanged.interchanged.
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SAKALANA
VYAVAKALANBHYAMExample 1:
45x 23y = 113
23x 45y = 91
Firstly add them,Firstly add them,
( 45x( 45x 23y ) + ( 23x23y ) + ( 23x 45y ) = 113 + 9145y ) = 113 + 91
68x68x 68y = 20468y = 204
xx y = 3y = 3
Subtract one from other,Subtract one from other,
( 45x( 45x 23y )23y ) ( 23x( 23x 45y ) = 11345y ) = 113 9191
22x + 22y = 2222x + 22y = 22
x + y = 1x + y = 1
Rrepeat the same sutra,Rrepeat the same sutra,
we getwe getx = 2x = 2 andand y =y = -- 11
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SAKALANA
VYAVAKALANBHYAMExample 2:
1955x 476y =2482
476x 1955y = - 4913
Just add,Just add,
2431( x2431( x y ) =y ) = -- 24312431
xx y =y = --11 Subtract,Subtract,
1479 ( x + y ) = 73951479 ( x + y ) = 7395
x + y = 5x + y = 5
Once again add,Once again add,2x = 42x = 4 x = 2x = 2
subtractsubtract
-- 2y =2y = -- 66 y = 3y = 3
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ANTYAYORDAAKE'PI
The Sutra (formula)
ANTYAYOR
DAAKE'PI
means :
Numbers of which
the last digits addedup give 10.
This sutra is helpful inThis sutra is helpful inmultiplying numbers whose lastmultiplying numbers whose lastdigits add up to 10(or powers ofdigits add up to 10(or powers of10). The remaining digits of the10). The remaining digits of the
numbers should be identical.numbers should be identical.
For ExampleFor Example: In multiplication: In multiplicationof numbersof numbers
25 and 25,25 and 25,
2 is common and 5 + 5 = 102 is common and 5 + 5 = 10
47 and 43,47 and 43,
4 is common and 7 + 3 = 104 is common and 7 + 3 = 10
62 and 68,62 and 68,
116 and 114.116 and 114.
425 and 475425 and 475
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ANTYAYORDAAKE'PI
Vedic Method
6 7
X 6 3
4 2 2 1
The same rule works whenThe same rule works whenthe sum of the last 2, lastthe sum of the last 2, last3, last 43, last 4 -- -- -- digits addeddigits added
respectively equal to 100,respectively equal to 100,1000, 100001000, 10000 ---- -- -- ..
The simple point toThe simple point toremember is to multiplyremember is to multiplyeach product by 10, 100,each product by 10, 100,1000,1000, -- -- as the case mayas the case may
be .be .
You can observe that this isYou can observe that this ismore convenient whilemore convenient whileworking with the productworking with the productof 3 digit numbersof 3 digit numbers
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ANTYAYORDAAKE'PI
892 X 808
= 720736
Try Yourself :Try Yourself :
A)A) 398 X 302398 X 302= 120196= 120196
B)B) 795 X 705795 X 705
= 560475= 560475
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LOPANASTHPANBHYM
The Sutra (formula)
LOPANASTHPANBHYM
means :
'by alternate
elimination and
retention'
Consider the case ofConsider the case offactorization of quadraticfactorization of quadraticequation of typeequation of type
axax22 + by+ by22 + cz+ cz22 + dxy + eyz + fzx+ dxy + eyz + fzx
This is a homogeneousThis is a homogeneousequation of second degreeequation of second degree
in three variables x, y, z.in three variables x, y, z.
The subThe sub--sutra removes thesutra removes thedifficulty and makes thedifficulty and makes thefactorization simple.factorization simple.
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LOPANASTHPANBHYM
Example :
3x 2 + 7xy + 2y 2+ 11xz + 7yz + 6z 2
Eliminate z and retain x, y ;factorize3x 2 + 7xy + 2y 2 = (3x + y) (x + 2y)
Eliminate y and retain x, z;factorize3x 2 + 11xz + 6z 2 = (3x + 2z) (x + 3z)
Fill the gaps, the given expression
(3x + y + 2z) (x + 2y + 3z)
Eliminate z by putting z = 0Eliminate z by putting z = 0and retain x and y andand retain x and y andfactorize thus obtained afactorize thus obtained aquadratic in x and y by meansquadratic in x and y by means
ofof AdyamadyenaAdyamadyena sutra.sutra.
Similarly eliminate y andSimilarly eliminate y andretain x and z and factorizeretain x and z and factorizethe quadratic in x and z.the quadratic in x and z.
With these two sets of factors,With these two sets of factors,fill in the gaps caused by thefill in the gaps caused by theelimination process of z and yelimination process of z and yrespectively. This gives actualrespectively. This gives actual
factors of the expression.factors of the expression.
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GUNTASAMUCCAYAH -
SAMUCCAYAGUNTAHExample :
3x 2 + 7xy + 2y 2+ 11xz + 7yz + 6z 2
Eliminate z and retain x, y ;factorize3x 2 + 7xy + 2y 2 = (3x + y) (x + 2y)
Eliminate y and retain x, z;factorize3x 2 + 11xz + 6z 2 = (3x + 2z) (x + 3z)
Fill the gaps, the given expression
(3x + y + 2z) (x + 2y + 3z)
Eliminate z by putting z = 0Eliminate z by putting z = 0and retain x and y andand retain x and y andfactorize thus obtained afactorize thus obtained aquadratic in x and y by meansquadratic in x and y by means
ofof AdyamadyenaAdyamadyena sutra.sutra.
Similarly eliminate y andSimilarly eliminate y andretain x and z and factorizeretain x and z and factorizethe quadratic in x and z.the quadratic in x and z.
With these two sets of factors,With these two sets of factors,fill in the gaps caused by thefill in the gaps caused by theelimination process of z and yelimination process of z and yrespectively. This gives actualrespectively. This gives actual
factors of the expression.factors of the expression.
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ReferencesReferences
y "VEDICMATHEMATICS"by H.H. Jagadguru SwamiSri Bharati Krishna TirthajiMaharaj.
Publishers Motilal Banarasidass, BunglowRoad, JawaharNagar,Delhi 110 007; or
Chowk, Varanasi (UP); or Ashok Raj Path, Patna, (Bihar)
y
www.vedicmaths.org
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Vedic Mathematics
And, you thought biggest
contribution of Indians to field of
Mathematics was Zero??
- Thank You !