VEDIC MATHEMATICS : Arithmetic Operations

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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

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### VEDIC MATHEMATICS : Arithmetic Operations

Arithmetic Operations

Positional Number System

43210

=

4 * 10,000

+ 3 * 1,000

+ 2 * 100

+ 1 * 10

+ 0

Arithmetic Operations

• Unary System

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

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

• Roman System

III + VIII

= XI

Too laborious

Not uniform,

Not incremental!

Arithmetic Operations

Arabic System

45 + 31

= 4 * 10 + 5

+ 3 * 10 + 1

= 7 * 10 + 6

= 76

Arithmetic Operations

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
• Unary System

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

= !!!!!!!

• Roman System

XI - IV

= VII

Arithmetic Operations

Arabic System

45 – 31

= 4 * 10 + 5

– [3 * 10 + 1]

= [4 – 3] *10 + [5 – 1]

= 1 * 10 + 4

= 14

Arithmetic Operations

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

43210

=

4 * 10,000

+ 3 * 1,000

+ 2 * 100

+ 1 * 10

+ 0

Arithmetic Operations

Prerequisite / Background
• 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
• 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

Method : Vertically and Crosswise Sutra

Correctness and Applicability

Arithmetic Operations

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
• 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

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
• 8 * 9 =
• 8 2
• 9 1
• 7 2
• 8 * 9 = 72

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
• (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

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 Example
• 5 * 8

5 5

8 2

3 10

4 0

Arithmetic Operations

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