Prasad Arithmetic Operations 1

VEDIC MATHEMATICS : Arithmetic Operations

T. K. Prasadhttp://www.cs.wright.edu/~tkprasad

Prasad Arithmetic Operations 2

Positional Number SystemTEN-

THOUSANDSTHOUSANDS HUNDREDS TENS UNITS

43210=

4 * 10,000+ 3 * 1,000+ 2 * 100+ 1 * 10

+ 0

Prasad Arithmetic Operations 3

Adding Numbers

• Unary System

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

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

• Roman System

III + VIII

= XI

Too laboriousand unreadable!

Not uniform,Not incremental!

Prasad Arithmetic Operations 4

• Arabic System

45 + 31

= 4 * 10 + 5

+ 3 * 10 + 1

= 7 * 10 + 6

= 76

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

Prasad Arithmetic Operations 6

Subtracting Numbers

• Unary System

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

= !!!!!!!

• Roman System

XI - IV

= VII

Prasad Arithmetic Operations 7

• Arabic System

45 – 31

= 4 * 10 + 5

– [3 * 10 + 1]

= [4 – 3] *10 + [5 – 1]

= 1 * 10 + 4

= 14

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

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

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

Prasad Arithmetic Operations 11

Multiplying Single Digit Numbers

• 1 * 2 = 2

• 1 x 2 = 2

• 3 * 4 =

• 4 * 3 =

12

• 5 * 5 =

25

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

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).

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

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

Prasad Arithmetic Operations 16

Another Example

• 8 * 9 =

• 8 2

• 9 1

• 7 2

• 8 * 9 = 72

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?

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

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]

Prasad Arithmetic Operations 20

• 5 * 8

5 5

8 2

3 10

4 0

Another Example

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