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8/12/2019 Vedic Mathematics Worlds Fastest Mental Math System
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By Abhishek Sharma
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What is Vedic Mathematics ?
Vedic mathematics is the namegiven to the ancient system ofmathematics in India.
It‟s a unique technique ofcalculations based on simpleprinciples and rules , with which anymathematical problem - be itarithmetic, algebra, geometry ortrigonometry can be solved
mentally.
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Why Vedic Mathematics?
It helps to solve problems much faster.
It reduces burden of long mental calculations.
It provides one line answer.
It is a magical tool to reduce scratch work and counting onfingers.
It increases concentration. Time saved can be used to answer more questions.
Logical thinking process gets enhanced.
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Base of Vedic Mathematics
Vedic Mathematicsnow refers to a setof sixteen
mathematicalformulae or sutrasand their corollariesderived from the
Vedas.
Text on left is in Sanskrit LanguageExplanation given on right.
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Base of Vedic Mathematics
Vedic Mathematicsnow refers to a setof sixteen
mathematicalformulae or sutrasand their corollariesderived from the
Vedas.
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Example: „Squaring of numbers ending in 5‟.
Conventional Method
65 X 656 5
X 6 5
3 2 5
3 9 0 X4 2 2 5
Vedic Method
65 X 65 = 4225
( 'multiply the previousdigit 6 by one morethan itself 7. Thanwrite 25 )
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NIKHILAM NAVA TAS‟ CHARA MAM DASA TAH“all from 9 and the last from 10”
The Sutra (formula)NIKHILAMNAVATAS‟CHARAMA
M DASATAH means :
“all from 9 and the lastfrom 10”
This formula can bevery effectivelyapplied in
multiplication ofnumbers, which arenearer to bases like10, 100, 1000 i.e.,
to the powers of 10(e.g.: 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 88 7 3 X
9 1 1 8
Vedic MethodBase is 100
97 3 (100- 97)X 94 6 (100 – 94)
9 1 1 894-3 3x6
94 6X 97 3
9 1 1 897-6 6x3
Subtract difference of upper number fromlower number and multiply both differences.
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Case ( ii) : When both the numbers are higher thanthe base
Conventional Method103 X 105
103
X 1055 1 5
0 0 0 X1 0 3 X X
1 0, 8 1 5
Vedic Method
For Example103 X 105
103 3 (100+3)X 105 5 (100+5)
1 0, 8 1 5105+3 3x5
Add lower number 105 and with uppernumber‟s difference 3 that‟s the left partand multiply both differences 3 and 5that‟s the right part
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Case III: When one number is more and the other isless than the base.
Conventional Method
103 X 98
103
X 98
8 2 4
9 2 7 X
1 0, 0 9 4
Vedic Method
103 3 (100+3)X 98 -2 (100-2)
98+3 / 3x2101 / 06( 101-1)/(100-06)Ans: 1 00 9 4
Steps:1. Add lower number and upper‟sdifference subtract one fromresult that‟s left part
2. Multiply both differences andsubtract it from 100 this is rightpart
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ĀNURŨPYENA' proportionality „ or 'similarly '
The Sutra (formula)ĀNURŨPYENA means :
' proportionality 'or
'similarly '
This Sutra is highlyuseful to findproducts of two
numbers when both ofthem are near theCommon bases like50, 60, 200 etc
(multiples of powersof 10).
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ĀNURŨPYENA' proportionality „ or 'similarly '
Conventional Method
46 X 434 6
X 4 3
1 3 8
1 8 4 X1 9 7 8
Vedic Method
Working base 100/2= 50
46 -4 (50-4)X 43 -7 (50-7)(43-4) (4x7)39 / 2819 ½ / 28 ( 39 divide by 2 )
19/ (50 + 28)( Half of 100 to be added to 28) since we have ½had it been divisible by 2 we would have left it
as is.)
19 /78Ans: 1 9 7 8
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URDHVA TIRYAGBHYAM“Vertically and cross wise”
The Sutra (formula)
URDHVATIRYAGBHYAMmeans :
“Vertically and crosswise”
This the general
formula applicable
to all cases of
multiplication and
also in the division
of a large number
by another largenumber.
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Two digit multiplication by URDHVA TIRYAGBHYAM
The Sutra (formula)
URDHVATIRYAGBHYAMmeans :
“Vertically and crosswise”
E.g. 75X 322400
Step 1: 5×2=10, write down
0 and carry 1 Step 2: 7×2 + 5×3 =
14+15=29, add to it previouscarry over value 1, so wehave 30, now write down 0
and carry 3 Step 3: 7×3=21, add
previous carry over value of3 to get 24, write it down.
So we have 2400 as the
answer.
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Two digit multiplication by URDHVA TIRYAGBHYAM
Vedic Method4 6
X 4 3
1 9 7 8
Step1: 6x3=18 write 8 carry 1
Step2:
4x6+4x3=24+12=36+carry =37
write 7 carry 3
Step3: 4x4 = 16 + carry 3 = 19
write 19
Ans : 19/7/8 or 1978
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Three digit multiplication byURDHVA TIRYAGBHYAM
Vedic Method
103
X 105
1 0, 8 1 5
Step1: 3x5=15 carry 1 write 5
Step2: 0x3=0x5 = 0 + carry 1
Write 1
Step3: 1x5+0x0+3x1=5+0+3=8
Write 8
Step4: 1x0+1x0=0 write 0
Step5: 1x1=1 write 1
Ans : 10815
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YAVDUNAM TAAVDUNIKRITYA VARGANCHA YOJAYET“Squares of numbers near(lesser) to powers of 10“
This formulae meanswhatever the extentof its deficiency,
lessen it still furtherto that very extent;and also set up thesquare of thatdeficiency.
This formulae is veryhandy in calculatingsquares of numbers
near(lesser) to powersof 10
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YAVDUNAM TAAVDUNIKRITYA VARGANCHA YOJAYETSquares of numbers near(lesser) to powers of 10
982
= 9604
Base 100100-2 = 98
(98-2)/2x2
96/04
Ans: 9604
The nearest power of 10 to 98 is 100. Therefore,let us take 100 as our base.
Since 98 is 2 less than 100, we call 2 as thedeficiency.
Decrease the given number further by anamount equal to the deficiency. i.e., perform ( 98-2 ) = 96. This is the left side of our answer!!.
On the right hand side put the square of thedeficiency, that is square of 2 = 04.
Append the results from step 4 and 5 to get theresult. Hence the answer is 9604.
Importan t : Whi le calculat ing step 5, the num ber of d ig i ts in the s quared
number (04) shou ld be equal to number of zeroes in the base(100).
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YAVDUNAM TAAVDUNIKRITYA VARGANCHA YOJAYETSquares of numbers near(lesser) to powers of 10
1032
= 10609
Base 100
(100+3)
103+3= 106
3x3= 09
10609
Ans: 10609
The nearest power of 10 to 103 is 100.Therefore, let us take 100 as our base.
Since 103 is 3 more than 100 (base), we call 3as the surplus.
Increase the given number further by anamount equal to the surplus. i.e., perform ( 103+ 3 ) = 106. This is the left side of our answer!!.
On the right hand side put the square of thesurplus, that is square of 3 = 09.
Append the results from step 4 and 5 to getthe result. Hence the answer is 10609.
Important: while calculating step 5, the number of digits in the squarednumber (09) should be equal to number of zeroes in the base(100).
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YAVDUNAM TAAVDUNIKRITYA VARGANCHA YOJAYETSquares of numbers near(lesser) to powers of 10
Another example:
10092= 1018081
Base 1000
1000+9
1009+9=1018
9x9=081 (since base has three zeros)
1018/081Ans: 1018081
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SAŃKALANA – VYAVAKALANĀBHYAM'by addition and by subtraction'
The Sutra (formula)
SAŃKALANA –VYAVAKALANĀBHYA
Mmeans :
'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.
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SAŃKALANA – VYAVAKALANĀBHYAM'by addition and by subtraction'
Example 2:
1955x – 476y =
2482476x – 1955y = -
4913
Just add,
2431( x – y ) = - 2431
x – y = -1
Subtract,
1479 ( x + y ) = 7395
x + y = 5
Once again add,
2x = 4 x = 2subtract
- 2y = - 6 y = 3
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ANTYAYOR DAŚAKE'PI„ Numbers of which the last digits added up give 10.‟
The Sutra (formula)
ANTYAYORDAŚAKE'PImeans :
„ Numbers of which thelast digits added up
give 10.‟
This sutra is helpful inmultiplying numbers whose lastdigits add up to 10(or powers of10). The remaining digits of the
numbers should be identical.For Example: In multiplication of
numbers 25 and 25,
2 is common and 5 + 5 = 10 47 and 43,4 is common and 7 + 3 = 10
62 and 68, 116 and 114.
425 and 475
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ANTYAYOR DAŚAKE'PI„ Numbers of which the last digits added up give 10.‟
Vedic Method6 7X 6 3
4 2 2 1
The same rule works whenthe sum of the last 2, last 3,last 4 - - - digits addedrespectively equal to 100,
1000, 10000 -- - - . The simple point to
remember is to multiply eachproduct by 10, 100, 1000, - -as the case may be .
You can observe that this ismore convenient whileworking with the product of 3digit numbers
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ANTYAYOR DAŚAKE'PI„ Numbers of which the last digits added up give 10.‟
892 X 808= 720736
Try Yourself :
A) 398 X 302
= 120196
B) 795 X 705
= 560475
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LOPANA STHÂPANÂBHYÂM'by alternate elimination and retention'
The Sutra (formula)
LOPANASTHÂPANÂBHYÂMmeans :
'by alternateelimination andretention'
Consider the case offactorization of quadraticequation of type
ax2 + by2 + cz2 + dxy + eyz + fzx
This is a homogeneous equationof second degree in threevariables x, y, z.
The sub-sutra removes thedifficulty and makes thefactorization simple.
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LOPANA STHÂPANÂBHYÂM'by alternate elimination and retention'
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;factorize
3x 2 + 11xz + 6z 2 = (3x + 2z) (x+ 3z)
Fill the gaps, the givenexpression
(3x + y + 2z) (x + 2y + 3z)
Eliminate z by putting z = 0 andretain x and y and factorize thusobtained a quadratic in x and yby means of Adyamadyena
sutra.
Similarly eliminate y and retain xand z and factorize thequadratic in x and z.
With these two sets of factors,fill in the gaps caused by theelimination process of z and yrespectively. This gives actual
factors of the expression.
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Thank you