Download - PrasadArithmetic Operations1 VEDIC MATHEMATICS : Arithmetic Operations T. K. Prasad tkprasad
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Prasad Arithmetic Operations 1
VEDIC MATHEMATICS : Arithmetic Operations
T. K. Prasadhttp://www.cs.wright.edu/~tkprasad
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Prasad Arithmetic Operations 2
Positional Number SystemTEN-
THOUSANDSTHOUSANDS HUNDREDS TENS UNITS
43210=
4 * 10,000+ 3 * 1,000+ 2 * 100+ 1 * 10
+ 0
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Prasad Arithmetic Operations 3
Adding Numbers
• Unary System
!!! + !!!!!!!!
= !!!!!!!!!!!
• Roman System
III + VIII
= XI
Too laboriousand unreadable!
Not uniform,Not incremental!
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Prasad Arithmetic Operations 4
• Arabic System
45 + 31
= 4 * 10 + 5
+ 3 * 10 + 1
= 7 * 10 + 6
= 76
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Prasad 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
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Prasad Arithmetic Operations 6
Subtracting Numbers
• Unary System
!!!!!!!!!!! - !!!!
= !!!!!!!
• Roman System
XI - IV
= VII
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Prasad Arithmetic Operations 7
• Arabic System
45 – 31
= 4 * 10 + 5
– [3 * 10 + 1]
= [4 – 3] *10 + [5 – 1]
= 1 * 10 + 4
= 14
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Prasad 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
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Prasad Arithmetic Operations 9
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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Prasad 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
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Prasad Arithmetic Operations 11
Multiplying Single Digit Numbers
• 1 * 2 = 2
• 1 x 2 = 2
• 3 * 4 =
• 4 * 3 =
12
• 5 * 5 =
25
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Prasad Arithmetic Operations 12
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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Prasad Arithmetic Operations 13
10’s Complement
• TC(1) =
9• TC(3) =
7• TC(4) =
6
• TC(6) =
4• TC(8) =
2• TC(9) =
1
10’s complement of a digit d is (10 – d).
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Prasad 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
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Prasad Arithmetic Operations 15
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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Prasad Arithmetic Operations 16
Another Example
• 8 * 9 =
• 8 2
• 9 1
• 7 2
• 8 * 9 = 72
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Prasad 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?
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Prasad 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
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Prasad Arithmetic Operations 19
Left digit[Crosswise
Subtraction]
Left digit[Crosswise
Subtraction]
Correctness Argument:Two 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]
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Prasad Arithmetic Operations 20
• 5 * 8
5 5
8 2
3 10
4 0
Another Example
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Prasad Arithmetic Operations 21
• 3 * 2
3 7
2 8
– 5 56
– 5+5 6
0 6Pointless in practicePointless in practice but need proof to feel comfortable!
Yet Another Example