Vedic Maths #02: Multiplication by 999.... [with same number of digits]

Following method will give you easy way to multiply a number with 9's.  After learning this method you can easily do

1. 12 x 99
2. 234 x 999
3. 6316 x 9999 and so on...

Note this method is applied to numbers with same number of digits in them. Other variation are covered later.

Let's take an example of 234 * 999. Do try it on piece of paper using the usual method of multiplication.

Vedic Method is as follows. Answer is calculated in two parts and later combined together to get the final answer.

To know what is 10's complement you can refer (Vedic Maths #01: Complement method - Subtraction from 100/1000/10000). Here is mathematical analysis of the same.

Lets take a generic number xyz, and we need to do
xyz * 999

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

= (100x + 10y + z) * 999
= (100x + 10y + z) * (1000 - 1)
= 100000x + 10000y + 1000z - 100x - 10y - z
= 100000x + 10000y + 1000(z - 1) + 1000 - 100x - 10y - z
= 100000x + 10000y + 1000(z - 1) + (1000 - xyz)
= 100000x + 10000y + 1000(z - 1) + (1000 - xyz)
= 1000(100x + 10y + (z - 1)) + (1000 - xyz)
= 1000(xy[z-1]) + 10's complement of xyz
= 1st part of answer + 2nd part of answer

This applies to number as big as 3450534 * 9999999.

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