VEDIC MATHEMATICS : Arithmetic Operations

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VEDIC MATHEMATICS Arithmetic Operations

• T. K. Prasad
• http//www.cs.wright.edu/tkprasad

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Positional Number System
TEN-THOUSANDS THOUSANDS HUNDREDS TENS UNITS
43210 4 10,000 3 1,000 2 100 1
10 0
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Adding Numbers

• Unary System
• !!! !!!!!!!!
• !!!!!!!!!!!
• Roman System
• III VIII
• XI

Too laborious and unreadable!
Not uniform, Not incremental!
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• Arabic System
• 45 31
• 4 10 5
• 3 10 1
• 7 10 6
• 76

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

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Subtracting Numbers

• Unary System
• !!!!!!!!!!! - !!!!
• !!!!!!!
• Roman System
• XI - IV
• VII

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• Arabic System
• 45 31
• 4 10 5
• 3 10 1
• 4 3 10 5 1
• 1 10 4
• 14

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• 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

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Recap Positional Number System
TEN-THOUSANDS THOUSANDS HUNDREDS TENS UNITS
43210 4 10,000 3 1,000 2 100 1
10 0
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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

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Multiplying Single Digit Numbers

• 1 2 2
• 1 x 2 2
• 3 4
• 4 3
• 12
• 5 5
• 25

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Single Digit Multiplication (of Large Digits in
terms of Small Digits) using Vedic Approach

1. Method Vertically and Crosswise Sutra
2. Correctness and Applicability

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10s Complement
10s complement of a digit d is (10 d).

• TC(6)
• 4
• TC(8)
• 2
• TC(9)
• 1

• TC(1)
• 9
• TC(3)
• 7
• TC(4)
• 6

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Method Multiply 7 8

• Write the first digit to be multiplied and its
  10s complement in the first row, and the second
  digit to be multiplied and its 10s complement in
  the second row.
• 7 3
• 8 2

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• 7 3
• 8 2
• To determine the 2-digit product
• subtract crosswise 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

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Another Example

• 8 9
• 8 2
• 9 1
• 7 2
• 8 9 72

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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?

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Proof Sketch

• (8 3) (7 2) 5
• Why are they same?
• That is, the difference between first digit and
  10s complement of the second digit.
• (8 (10 7)) (8 7 10) (15 10) 5
• (7 (10 8)) (7 8 10) (15 10) 5

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Correctness ArgumentTwo possibilities

• 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

Right digit Vertical Product
Right digit Vertical Product
Left digit Crosswise Subtraction
Left digit Crosswise Subtraction
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Another Example

• 5 8
• 5 5
• 8 2
• 3 10
• 4 0

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Yet Another Example

• 3 2
• 3 7
• 2 8
• 5 56
• 55 6
• 0 6
• Pointless in practice but need proof to feel
  comfortable!