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Vedic Mathematics
By
Dr. SUDHA GUPTA
Department of Mathematics
Lakshmibai College, University ofDelhi
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What is Vedic Mathematics ?
It is an ancienttechnique, which
simplifies multiplication, divisibility,
complex numbers, squaring, cubing,
square and cube roots. Even recurring
decimals and auxiliary fractions can
be handled by Vedic Mathematics.
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Who Brought Vedic Mathematics
to Limelight ?
Theancientsystems of Mathematics
wasrediscovered from VedasbyJagadguru Swami
Bharathikrishna Tirthaji of
Govardhan Peeth, Puri Jaganath
(1884-1960)
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What is the basis of Vedic
Mathematics ?
16 Sutras&
13 Sub-Sutras
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Vedic Mathematical Sutras
,dkf/kdsu iwosZ.k
Ekadhikena Purvena
(vkuq:I;s)kwU;eU;r~
(Anurupye) Sunyamanyat
O;f"Vlef"V%
Vyastisamastih
fuf[kya uorpjea nkr%
Nikhilam
NavatascaramamDasatah
LkadyuO;odyukH;ke~
Sankalana
vyavakalanabhyam
ks"kk.;~dsu pjes.k
Sesanyankena
Caramena
/oZfr;ZXHk;ke~
Urdhva-tiryagbhyam
Ikwj.kkiwj.kkH;ke~
Puranapuranabhyam
LkksikUR;};eUR;e~
Sopantyadvayamantyam
IkjkoR;Z ;kst;sr~Paravartya Yojayet
PkyudyukH;ke~Calana-Kalanabhyam
,dU;wusu iwosZ.kEkanyunena Purvena
kwU;a lkE;leqPp;s
Sunyam Samyasamuccaye
;konwue~
Yavadunam
Xkqf.krleqPp;%
Gunitasamuccayah
Xkq.kdleqPp;%
Gunakasamuccayah
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Multiplication of Numbers
Thesutra whichis used for
multiplicationis:fuf[kya uorpjea nkr%
W
hich literally translated,means ;All from9andthe last from10
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Procedure for Multiplication
Suppose we have to multiply 9 by 7
We should take, as base for our calculations,that power of 10 which is nearest to thenumbers to be multiplied. In this case 10itselfis that power;
Put t h e two numbers 9 and 7 above andbelow on the left hand side.
Subtract each ofthem from the base (10) andwrite down the remainders (1 and 3) on theright hand side with a connecting minus sign( - ) between them to show that the numbers
to be multiplied are both ofthem less that 10. The product will have two parts one on the
left side and one on the right. A verticaldividing line may be drawn for the purpose ofdemarcation ofthe two parts.
Now, the left hand side digit (ofthe answer)can be arrived at in one of4 ways:-
Subtractthebase10 fromthesum ofthegivennumbers(9and 7 i.e.16)andput(16-10)i.e.6asthe lefthandpart oftheanswer.
9 + 7 10 = 6
orSubtractthesum ofthetwodeficiencies(1+3=4) fromthebase(10)
10 1 3 = 6
orCross subtractdeficiency (3) onthesecondrow fromthe original number(9)inthe firstrow.
9 3 = 6 orCross subtractintheconverse way
(i.e.1 from 7).
7 1 = 6
Now, Vertically mulitply thetwo deficitfigures(1and3).Theproductis3.Andthis
istherighthandsideportion oftheanswer. Thus9 x 7 = 63
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Multiplication of Numbers
Next Sutrais
/oZfr;ZXHk;ke~(Urdhvatriyagbhayam)
whichmeansVertically and Crosswise
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12 X 13
Suppose we have to multiply 12 by 13
We multiply the left hand most digits 1 of the
multiplicand vertically by the left hand most digits1 of the multiplier, get their product 1 and set itdown as the left hand most part ofthe answer.
We then multiply 1 and 3 ; 1 and 2 crosswise, addthe two, get 5 as the sum and set it down as themiddle part ofthe answer.
We multiply 2 and 3 vertically, get 6 as theirproduct and put it down as the last (the right handmost) part ofthe answer.
Thus 12 x 13 = 156
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Special Sub-Sutra for Multiplication by 11
vUR;;ksjso (Antyayoreva)
which means Only the last two digits
The following example illustrate this very easy methods.13 423 x 11
Write down the numberwith naught placed at both ends. This is a
naught sandwich 0 1 3 4 2 3 0
Add the final two digits, 3 + 0 = 3 and write the answer below 0 .0 1 3 4 2 3 0
3
For the tens digit, add the final two digits to that point, that is 2 + 3 = 5.
0 1 3 4 2 3 0
5 3 Continue to add adjacent digits, that is 4+2 = 6, 3+4=7, 1+3 = 4,
and 0+1=1
0 1 3 4 2 3 0
1 4 7 6 5 3
The answer is 1 4 7, 6 5 3 2
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Multiplication by 12
The sutra used to obtained the product of any
number with 12 is
LkksikUR;};eUR;e~(Sopantyadvayamantyam)
which means
The ultimate and twice the penultimate
This is very similar to multiplication by 11
but we just double the digitto the left
before adding
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Multiplication by 12
For example : 6 5 2 1 4 x 12
we start with the noughtsandwich 0 6 5 2 1 4 0
The ultimate digit is 0 andthe penultimate digits is 4, so
the ultimate plus twice thepenultimate is 0 + 8 = 8.
0 6 5 2 1 4 0
8
For the tens column, theultimate is 4 and thepenultimate is 1, so 4+2= 6.
0 6 5 2 1 4 0
6 8
Likewise, 1 + 4 = 5, and 2 +10 = 12. With 12 we setdown 2 and carry 1.
0 6 5 2 1 4 0
2 5 6 8
1
5 + 12 + Carry 1 = 18 andagain we carry 1.
The final step is 6 + 0 + Carry1 = 7.
0 6 5 2 1 4 0
7 8 2 5 6 8
1 1
The answer is 7
82 5 6
8
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Thankyou