VEDIC MATHEMATICS : Arithmetic Operations

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

2. Positional Number System 43210 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 + 0 Arithmetic Operations

3. Adding Numbers • Unary System !!! + !!!!!!!! = !!!!!!!!!!! • Roman System III + VIII = XI Too laborious and unreadable! Not uniform, Not incremental! Arithmetic Operations

4. Arabic System 45 + 31 = 4 * 10 + 5 + 3 * 10 + 1 = 7 * 10 + 6 = 76 Arithmetic Operations

5. Arabic System (introducing carry) 45 + 36 = 4 * 10 + 5 + 3 * 10 + 6 = 7 * 10 + 11 = 7 * 10 + 1 * 10 + 1 = 8 * 10 + 1 = 81 Arithmetic Operations

6. Subtracting Numbers • Unary System !!!!!!!!!!! - !!!! = !!!!!!! • Roman System XI - IV = VII Arithmetic Operations

7. Arabic System 45 – 31 = 4 * 10 + 5 – [3 * 10 + 1] = [4 – 3] *10 + [5 – 1] = 1 * 10 + 4 = 14 Arithmetic Operations

8. Arabic System (introducing borrow) 65 - 36 = 6 * 10 + 5 – [ 3 * 10 + 6 ] = [6 – 3] * 10 + [5 – 6] = 5 * 10 + 1 * 10 + 5 – [ 3 * 10 + 6 ] = [5 – 3] * 10 + [10+ 5 – 6] = 2 * 10 + [15 – 6] = 29 Arithmetic Operations

9. Recap: Positional Number System 43210 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 + 0 Arithmetic Operations

10. Prerequisite / Background • Single Digit Addition • E.g., 4 + 5 = 9 • With carry • E.g., 4 + 8 = 12 • Single Digit Subtraction • E.g., 9 - 5 = 4 • With borrow • E.g., 12 - 04 = 08 Arithmetic Operations

11. Multiplying Single Digit Numbers • 1 * 2 = 2 • 1 x 2 = 2 • 3 * 4 = • 4 * 3 = 12 • 5 * 5 = 25 Arithmetic Operations

12. Single Digit Multiplication (of Large Digits in terms of Small Digits) using Vedic Approach Method : Vertically and Crosswise Sutra Correctness and Applicability Arithmetic Operations

13. TC(1) = 9 TC(3) = 7 TC(4) = 6 TC(6) = 4 TC(8) = 2 TC(9) = 1 10’s Complement 10’s complement of a digit d is (10 – d). Arithmetic Operations

14. Method: Multiply 7 * 8 • Write the first digit to be multiplied and its 10’s complement in the first row, and the second digit to be multiplied and its 10’s complement in the second row. 7 3 8 2 Arithmetic Operations

15. 7 3 8 2 • To determine the 2-digit product: • subtractcrosswise to obtain the left digit • (7 – 2) = (8 – 3) = 5 • and • multiply the complements vertically to obtain the right digit. • (3 * 2) = 6 • 7 * 8 = 56 Arithmetic Operations

16. Another Example • 8 * 9 = • 8 2 • 9 1 • 7 2 • 8 * 9 = 72 Arithmetic Operations

17. 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? Arithmetic Operations

18. Proof Sketch • (8 – 3) = (7 – 2) = 5 • Why are they same? • That is, the difference between first digit and 10’s complement of the second digit. • (8 – (10 – 7)) = (8 + 7 – 10) = (15 – 10) = 5 • (7 – (10 – 8)) = (7 + 8 – 10) = (15 – 10) = 5 Arithmetic Operations

19. 8 = (10 – 2) 7 = (10 – 3) 8 * 7 = (10 – 2) * 7 = 10 * 7 – 2 * 7 = 10 * 7 – 2 * (10 – 3) = 10 * 7 – 2 * 10 + (2 * 3) = 10 * (7– 2) + 6 = 10 * 5 + 6 = 56 8 = (10 – 2) 7 = (10 – 3) 8 * 7 = 8 * (10 – 3) = 8 * 10 – 8 * 3 = 8 * 10 – (10 – 2) * 3 = 8 * 10 – 10 * 3+ (2 * 3) = 10 * (8– 3) + 6 = 10 * 5 + 6 = 56 Correctness Argument:Two possibilities Right digit [Vertical Product] Right digit [Vertical Product] Left digit [Crosswise Subtraction] Left digit [Crosswise Subtraction] Arithmetic Operations

20. Another Example • 5 * 8 5 5 8 2 3 10 4 0 Arithmetic Operations

21. Yet Another Example • 3 * 2 3 7 2 8 – 5 56 – 5+5 6 0 6 Pointless in practice but need proof to feel comfortable! Arithmetic Operations