## Aug 31, 2015

### Vedic Maths #04: Multiplication by 999.... [with lesser number of digits]

This is 2nd extension case of already covered articles - Vedic Maths #02: Multiplication by 999.... [with same number of digits] and Vedic Maths #03: Multiplication by 999.... [with greater number of digits].
It would good be if reader is aware of both the article. After learning this method you can easily do
1. 1234 x 99
2. 2343 x 999
3. 63164 x 9999 and so on...

Note this method is applied to numbers in which 99... has lesser number of digits when compared to other number. Other variation are already covered before.

Vedic Method is as follows.
1. Two part method is used with slight variation.
2. First part calculation is as before, where 1 is subtracted from the number.
3. Only as many digits in 99.. is used to find the 10's complement.
4. Remaining most significant digits are carried directly.
5. Both parts are written in overlap fashion.
6. Overlap region has as many digit as the difference of digits in the number.
7. Overlap region is subtracted to get final answer

Example will explain the method in better way. Lets calculate 2343 x 999

Mathematical proof is as follows

xyz * 99

xyz can be value can be written as x * 100 + y * 10 + z * 1

= (100x + 10y + z) * 99
= (100x + 10y + z) * (100 - 1)
= 10000x + 1000y + 100z - 100x - 10y - z
= 10000x + 1000y + 100(z - x) - 10y - z
= 10000x + 1000y + 100(z - x - 1) + 100 - 10y - z
= 10000x + 1000y + 100(z - x - 1) + (100 - yz)
= 100(100x + 10y) + (z - x - 1) + (100 - yz)
= 1 st part + overlap + 10's complement of yz

This method applies to number as big as 12345678 * 999.