Transcript Vedic Mathematics Final

Vedic Mathematics
By
Dr. SUDHA GUPTA
Department of Mathematics
Lakshmibai College, University of Delhi
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 Mathematics
to Limelight ?
The ancient systems of Mathematics
was rediscovered from Vedas by
Jagadguru Swami
Bharathikrishna Tirthaji of
Govardhan Peeth, Puri Jaganath
(1884-1960)
What is the basis of Vedic
Mathematics ?
16 Sutras
&
13 Sub-Sutras
Vedic Mathematical Sutras
,dkf/kdsu iwosZ.k
(vkuq:I;s) ’kwU;eU;r~
O;f"Vlef"V%
Ekadhikena Purvena
(Anurupye) Sunyamanyat
Vyastisamastih
fuf[kya uor’pjea n’kr%
LkadyuO;odyukH;ke~
’ks"kk.;³~dsu pjes.k
Nikhilam
Navatascaramam Dasatah
Sankalana –
vyavakalanabhyam
Sesanyankena
Caramena
Å/oZfr;ZXHk;ke~
Ikwj.kkiwj.kkH;ke~
LkksikUR;};eUR;e~
Urdhva-tiryagbhyam
Puranapuranabhyam
Sopantyadvayamantyam
IkjkoR;Z ;kst;sr~
PkyudyukH;ke~
,dU;wusu iwosZ.k
Paravartya Yojayet
Calana-Kalanabhyam
Ekanyunena Purvena
’kwU;a lkE;leqPp;s
;konwue~
Xkqf.krleqPp;%
Sunyam Samyasamuccaye
Yavadunam
Xkq.kdleqPp;%
Gunakasamuccayah
Gunitasamuccayah
Multiplication of Numbers
The sutra which is used for
multiplication is:
fuf[kya uor’pjea n’kr%
Which literally translated, means ;
“All from 9 and the last from 10”
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 the
numbers to be multiplied. In this case 10
itself is that power;
 Put the two numbers 9 and 7 above and
below on the left hand side.
 Subtract each of them from the base (10) and
write down the remainders (1 and 3) on the
right hand side with a connecting minus sign
( - ) between them to show that the numbers
to be multiplied are both of them less that 10.
 The product will have two parts one on the
left side and one on the right. A vertical
dividing line may be drawn for the purpose of
demarcation of the two parts.
 Now, the left hand side digit (of the answer)
can be arrived at in one of 4 ways:-
v Subtract the base 10 from the sum of the
given numbers (9 and 7 i.e. 16) and put
(16-10) i.e. 6 as the left hand part of the
answer.
9 + 7 – 10 = 6
v or Subtract the sum of the two
deficiencies (1+3=4) from the base (10)
10 – 1 – 3 = 6
v or Cross – subtract deficiency (3) on the
second row from the original number (9)
in the first row.
9–3=6
v or Cross –subtract in the converse way
(i.e. 1 from 7) .
7–1=6
 Now, Vertically mulitply the two deficit
figures (1 and 3) . The product is 3 . And this
is the right hand side portion of the answer.
 Thus 9 x 7 = 63
Multiplication of Numbers
Next Sutra is Å/oZfr;ZXHk;ke~
(Urdhvatriyagbhayam)
which means
“Vertically and Crosswise”
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 digits
1 of the multiplier, get their product 1 and set it
down as the left hand most part of the answer.
 We then multiply 1 and 3 ; 1 and 2 crosswise, add
the two, get 5 as the sum and set it down as the
middle part of the answer.
 We multiply 2 and 3 vertically, get 6 as their
product and put it down as the last (the right hand
most) part of the answer.
 Thus 12 x 13 = 156
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 number with naught placed at both ends. This is a
naught sandwich
0134230
 Add the final two digits, 3 + 0 = 3 and write the answer below 0 .
0134230
3
 For the tens digit, add the final two digits to that point, that is 2 + 3 = 5.
0134230
53
 Continue to add adjacent digits, that is 4+2 = 6, 3+4=7, 1+3 = 4,
 and 0+1=1
0134230
147653
 The answer is 1 4 7, 6 5 3 2
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 digit to the left
before adding
Multiplication by 12
For example : 6 5 2 1 4 x 12
 we start with the nought
sandwich
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 The ultimate digit is 0 and the
penultimate digits is 4, so the
ultimate plus twice the
penultimate is 0 + 8 = 8.
0652140
8
 For the tens column, the
ultimate is 4 and the
penultimate is 1, so 4+2= 6.
0652140
68
 Likewise, 1 + 4 = 5, and 2 +
10 = 12. With 12 we set
down 2 and carry 1.
0652140
2568
1
 5 + 12 + Carry 1 = 18 and
again we carry 1.
 The final step is 6 + 0 + Carry
1 = 7.
0652140
782568
11
 The answer is 7 8 2 5 6 8
Thank you