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VEDIC MATHEMATICS : Arithmetic Operations. T. K. Prasad http://www.cs.wright.edu/~tkprasad. Positional Number System. 4 3 2 1 0 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 + 0. Adding Numbers. Unary System !!! + !!!!!!!! = !!!!!!!!!!! Roman System

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vedic mathematics arithmetic operations

VEDIC MATHEMATICS : Arithmetic Operations

T. K. Prasad

http://www.cs.wright.edu/~tkprasad

Arithmetic Operations

positional number system
Positional Number System

43210

=

4 * 10,000

+ 3 * 1,000

+ 2 * 100

+ 1 * 10

+ 0

Arithmetic Operations

adding numbers
Adding Numbers
  • Unary System

!!! + !!!!!!!!

= !!!!!!!!!!!

  • Roman System

III + VIII

= XI

Too laborious

and unreadable!

Not uniform,

Not incremental!

Arithmetic Operations

slide4
Arabic System

45 + 31

= 4 * 10 + 5

+ 3 * 10 + 1

= 7 * 10 + 6

= 76

Arithmetic Operations

slide5
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

subtracting numbers
Subtracting Numbers
  • Unary System

!!!!!!!!!!! - !!!!

= !!!!!!!

  • Roman System

XI - IV

= VII

Arithmetic Operations

slide7
Arabic System

45 – 31

= 4 * 10 + 5

– [3 * 10 + 1]

= [4 – 3] *10 + [5 – 1]

= 1 * 10 + 4

= 14

Arithmetic Operations

slide8
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

recap positional number system
Recap: Positional Number System

43210

=

4 * 10,000

+ 3 * 1,000

+ 2 * 100

+ 1 * 10

+ 0

Arithmetic Operations

prerequisite background
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

multiplying single digit numbers
Multiplying Single Digit Numbers
  • 1 * 2 = 2
  • 1 x 2 = 2
  • 3 * 4 =
  • 4 * 3 =

12

  • 5 * 5 =

25

Arithmetic Operations

single digit multiplication of large digits in terms of small digits using vedic approach

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

Method : Vertically and Crosswise Sutra

Correctness and Applicability

Arithmetic Operations

10 s complement
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

method multiply 7 8
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

slide15
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

another example
Another Example
  • 8 * 9 =
  • 8 2
  • 9 1
  • 7 2
  • 8 * 9 = 72

Arithmetic Operations

questions
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

proof sketch
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

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

Correctness Argument:Two possibilities

Right digit

[Vertical

Product]

Right digit

[Vertical

Product]

Left digit

[Crosswise

Subtraction]

Left digit

[Crosswise

Subtraction]

Arithmetic Operations

another example1
Another Example
  • 5 * 8

5 5

8 2

3 10

4 0

Arithmetic Operations

yet another example
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

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