Vedic Mathematics Dr. SUDHA GUPTA Department of 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 Ekadhikena Purvena (vkuq:I;s) ’kwU;eU;r~ (Anurupye) Sunyamanyat O;f"Vlef"V% Vyastisamastih fuf[kya uor’pjea n’kr% Nikhilam Navatascaramam Dasatah LkadyuO;odyukH;ke~ Sankalana –vyavakalanabhyam ’ks"kk.;³~dsu pjes.k Sesanyankena Caramena Å/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.k Ekanyunena Purvena ’kwU;a lkE;leqPp;s Sunyam Samyasamuccaye ;konwue~ Yavadunam Xkqf.krleqPp;% Gunitasamuccayah Xkq.kdleqPp;% Gunakasamuccayah

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 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. 5 3 Continue to add adjacent digits, that is 4+2 = 6, 3+4=7, 1+3 = 4, and 0+1=1 1 4 7 6 5 3 The answer is 1 4 7, 6 5 3 2

Multiplication by 12 LkksikUR;};eUR;e~ (Sopantyadvayamantyam) 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 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 4+2= 6. 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. 7 8 2 5 6 8 1 1 Ø The answer is 7 8 2 5 6 8

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