1. PrasadArithmetic Operations1 VEDIC MATHEMATICS : Arithmetic Operations T. K. Prasad http://www.cs.wright.edu/~tkprasad

2. PrasadArithmetic Operations2 Multiplying Numbers 45 * 63 = (4 * 10 + 5) * (6 * 10 + 3) = 4 * 10 * 6 * 10 + 4 * 10 * 3 + 5 * 6 * 10 + 5 * 3 = 4 * 6 * 100 + 4 * 3 * 10 + 5 * 6 * 10 + 5 * 3 = 24 * 100 + (12 + 30) * 10 + 15

3. PrasadArithmetic Operations3 Multiplying Numbers = 24 * 100 + (12 + 30) * 10 + 15 = 24 * 100 + 42 * 10 + 15 = 1 5 1 2 x 3 0 x 2 4 x x = 2 8 3 5

4. PrasadArithmetic Operations4 Two Digit Multiplication (of Large Digits in terms of Small Digits) using Vedic Approach 1.Method : Vertically and Crosswise Sutra 2.Correctness and Applicability

5. PrasadArithmetic Operations5 Alternatively, 100’s complement of d 1 d 2 (d 2 ≠ 0) is (9  d 1 ) (10  d 2 ). HC(35) = 9 10 -3 -5 6 5 100’s Complement HC(11) = 89 HC(35) = 65 HC(94) = 6 100’s complement of a 2-digit number n is (100 – n).

6. PrasadArithmetic Operations6 Method: Multiply 79 * 97 Write the first number to be multiplied and its 100’s complement in the first row, and the second number to be multiplied and its 100’s complement in the second row. 79 21 97 03

7. PrasadArithmetic Operations7 79 21 97 03 To determine the 4-digit product: –subtract crosswise to obtain the left digits (79 – 03) = (97 – 21) = 76 –and –multiply the complements vertically to obtain the right digits. (21 * 03) = 63 79 * 97 = 7663

8. PrasadArithmetic Operations8 Another Example 89 * 91 = 89 11 91 09 80 99 89 * 91 = 8099

9. PrasadArithmetic Operations9 Questions Why do both crosswise subtractions yield the same result? Why does this method yield the correct answer for this example? Does this method always work for any pair of digits?

10. PrasadArithmetic Operations10 Proof Sketch (89 – 9) = (91 – 11) = 80 Why are they same? (89 – (100 – 91)) = (89 + 91 – 100) = 80 (91 – (100 – 89)) = (91 + 89 – 100) = 80

11. PrasadArithmetic Operations11 Right digit [Vertical Product] Right digit [Vertical Product] Left digit [Crosswise Subtraction] Left digit [Crosswise Subtraction] Correctness of Product : Two possibilities 89 = (100 – 11) 91 = (100 – 9) 89 * 91 = 89 * (100 – 9) = 89 * 100 – (100 – 11) * 9 = 89 * 100 – 100 * 9 + (11 * 9) = 100 * (89 – 9) + 99 = 100 * 80 + 99 = 8099 89 = (100 – 11) 91 = (100 – 9) 89 * 91 = (100 – 11) * 91 = 100 * 91 – 11 * (100 – 9) = 100 * 91 – 11 * 100 + (11 * 9) = 100 * (91 – 11) + 99 = 100 * 80 + 99 = 8099

12. PrasadArithmetic Operations12 69 * 88 69 31 88 12 57 372 60 72 Note that, the product of the 100’s complements has more than two digits (exceeds 100). However, the weight associated with 57 and 3 is 100, and hence they can be combined. Another Example

13. PrasadArithmetic Operations13 30 * 21 30 70 21 79 – 49 5530 – 49+55 30 6 30 This approach is valid This approach is valid with suggested modifications but not very useful! Yet Another Example Breakdown?!