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VEDIC MATHEMATICS : Arithmetic Operations

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

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VEDIC MATHEMATICS : Arithmetic Operations

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

  2. 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 Multiplying Numbers Arithmetic Operations

  3. = 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 Multiplying Numbers Arithmetic Operations

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

  5. Alternatively, 100’s complement of d1 d2 (d2≠ 0) is (9 - d1) (10 - d2). HC(35) = 9 10 -3 -5 6 5 HC(11) = 89 HC(35) = 65 HC(94) = 6 100’s Complement 100’s complement of a 2-digit number n is (100 – n). Arithmetic Operations

  6. 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 Arithmetic Operations

  7. 79 21 97 03 • To determine the 4-digit product: • subtractcrosswise 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 Arithmetic Operations

  8. Another Example • 89 * 91 = • 89 11 • 91 09 • 80 99 • 89 * 91 = 8099 Arithmetic Operations

  9. 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

  10. 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 Arithmetic Operations

  11. 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 Correctness of Product :Two possibilities Right digit [Vertical Product] Right digit [Vertical Product] Left digit [Crosswise Subtraction] Left digit [Crosswise Subtraction] Arithmetic Operations

  12. Another Example • 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. Arithmetic Operations

  13. Yet Another Example • 30 * 21 30 70 21 79 – 49 5530 – 49+55 30 6 30 This approach is validwith suggested modifications but not very useful! Breakdown?! Arithmetic Operations

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