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### VEDIC MATHEMATICS : Arithmetic Operations

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

T. K. Prasad

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

Arithmetic Operations

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 NumbersArithmetic Operations

= 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 NumbersArithmetic Operations

Method : Vertically and Crosswise Sutra

Correctness and Applicability

Arithmetic Operations

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 Complement100’s complement of a 2-digit number n is (100 – n).

Arithmetic Operations

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

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

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

- (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

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 possibilitiesRight digit

[Vertical

Product]

Right digit

[Vertical

Product]

Left digit

[Crosswise

Subtraction]

Left digit

[Crosswise

Subtraction]

Arithmetic Operations

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

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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