Thursday, July 27, 2006

Vedic Arithmetic Part 2: Multiplication

1. Multiplier is 9, 99, 999, 9999, ….

17 X 9 = 153, 17 X 999 = 16983

Here, we treat 17 as m (multiplicand) and 9, 999,…. (multiplier) as n

1.1. no of digits of n >= no of digits of m

Answer : (m – 1) | (n - LHS)

eg. 17 X 999

Answer: (17 - 1) | (999 - LHS) >> 16|(999 - 16) >>> 16983

Sly 17 X 99 >> 16|(99-16) >> 1683

1.2. else

Therefore we split m into p|q where no of digits of q = no of 9’s in n

Answer : (m - 1 - p) | (n + 1 - q)

Eg. 177 X 99

Split >> 1 | 77

Answer: (177 – 1 - 1) | (99 + 1 - 77) >> 175|23 >> 17523

Sly, 243 X 9 >> 24 | 3 >> (243-1-24)|(9+1-3) >> 218|7 >>2187

2. Multiplier = ca, multiplicand = cb & a+b = 10, 100, 1000, ….

ie total of last digits is 10, 100, 1000, … & whose prev part is exactly same

Answer: c(c+1) | ab

Eg. 237 X 233

Here c = 23, a = 7, b = 3

Answer: 23 X 24 | 7 X 3 >> 552 | 21 >>55221

Sly, 54 X 56 >> 30|24>>3024

3. Multiplicands & multiplier are near base

Find 2 small multipliers($) with signs

3.1. Base is 10, 100, 1000, ….

Answer: (base+ sum of 2 small multipliers) | (product of 2 small multipliers : Note base)

For, 97 X 105,

Base is 100, and $ are -3, +5

Answer: (100 + (-3+5) ) | -15 >> 102 | -15

To remove –ve, take carry :: 101 | (100 -15) >> 101|85 >> 10185

Sly, 102 X 103 >> 105|06 (Since base is 100, therefore 2 digits after |)

3.2. Else (working base (wb))

Multiply or divide only LHS of answer

(if wb is greater,multiply else divide)

I always prefer to keep wb greater.

Consider 31 X 33

Here wb is 30 (& base is 10), $ are +1, +3

Therefore, (30+1+3) | 3 >> 34 | 3

Since wb greater, multiply by 30/10 = 3

Answer: 34X3 | 3 >> 102 | 3 >> 1023

4. General case : Cross Multiplication

4.1. 2 digit multiplication

ab X cd >>> ac | ad+bc | bd

Eg. 23 X 12 >> 2 | 4+3 | 6 >> 276

4.2. 3 digit multiplication

abc X def >>> ad | ae+bd | af+dc+be | bf+ce | cf

Eg. 103 X 143 >> 1 | 4 | 6 | 12 | 9 >> 14729 (1 carried forward)

1 comment:

rk said...

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