Vedic Mathematics

1
Vedic Mathematics

• By
• Dr. SUDHA GUPTA
• Department of Mathematics
• Lakshmibai College, University of Delhi

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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.

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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)

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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 (vkuqIs) kwUeUr (Anurupye) Sunyamanyat Of"Vlef"V Vyastisamastih
fufkya uorpjea nkr Nikhilam Navatascaramam Dasatah LkadyuOodyukHke Sankalana vyavakalanabhyam ks"kk.³dsu pjes.k Sesanyankena Caramena
Å/oZfrZXHkke Urdhva-tiryagbhyam Ikwj.kkiwj.kkHke Puranapuranabhyam LkksikUReURe Sopantyadvayamantyam
IkjkoRZ kstsr Paravartya Yojayet PkyudyukHke Calana-Kalanabhyam ,dUwusu iwosZ.k Ekanyunena Purvena
kwUa lkEleqPps Sunyam Samyasamuccaye konwue Yavadunam Xkqf.krleqPp Gunitasamuccayah
Xkq.kdleqPp Gunakasamuccayah Xkq.kdleqPp Gunakasamuccayah Xkq.kdleqPp Gunakasamuccayah
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Multiplication of Numbers

• The sutra which is used for multiplication
  isfufkya uorpjea nkr
• Which literally translated, means All from 9
  and the last from 10

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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 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
  (134) 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

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Multiplication of Numbers

• Next Sutra is Å/oZfrZXHkke (Urdhvatriyagbhayam)
  
• which means
• Vertically and Crosswise

9
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

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Special Sub-Sutra for Multiplication by
11vURksjso (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 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 42 6,
  347, 13 4,
• and 011
• 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
• LkksikUReURe (Sopantyadvayamantyam)
• which means
• The ultimate and twice the penultimateThis is
  very similar to multiplication by 11 but we just
  double the digit to the left before adding

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Multiplication by 12

• For example 6 5 2 1 4 x 12
• Ø  we start with the nought sandwich 0 6 5
  2 1 4 0
• Ø  The ultimate digit is 0 and the penultimate
  digits is 4, so the ultimate plus twice the
  penultimate is 0 8 8.
• 0 6 5 2 1 4 0
• 8
• Ø  For the tens column, the ultimate is 4 and the
  penultimate is 1, so 42 6.
• 0 6 5 2 1 4 0
• 6 8

Ø  Likewise, 1 4 5, and 2 10 12. With 12
we set down 2 and carry 1. 0 6 5 2 1 4 0
2 5 6 8 1 Ø 5 12 Carry 1 18
and again we carry 1. Ø The final step is 6 0
Carry 1 7. 0 6 5 2 1 4 0 7 8 2 5 6 8
1 1 Ø The answer is 7 8 2 5 6 8
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• Thank you