वैदिक गणित Vaidik Mathematics

I came across a set of books written called "समजेपर्यंत जादू वाटणारे - वैदिक गणित" few years ago. It is a translation of the original book called "Vedic Mathematics" written by शंकराचार्य स्वामी भारती कृष्ण तीर्थ (Shankaracharya Swami Bharati Krishna Tirtha). It explains total 16 sutras developed by him which can be used to solve any difficult problem from any branch in mathematics. I found the books very interesting and thought of carrying them with me while entering the blog-world for the first time. I have tried to explain it the way I understood it. Correct me if there's any mistake. So here it is ...

सूत्र १ (sutra 1)
’ निखिलं नवतः चरमं दशतः ’

It's literal meaning is - all from nine and final from ten (It sounds weird but basically it's a method to do multiplication by using 10's complement.). Let's call it just "Nikhilam".
This is how it works.

Example 1:

7 x 8

Choose a power of 10 which is nearest to both the multiplier and the multiplicand.
In this case 10¹ = 10 is the nearest to both. Thus, base = 10.
Write the 10's complement of both the multiplier and the multiplicand to the right side of the colon as shown.

7 : 3
8 : 2
------------
_ / _

The number to the left hand side of the slash can be calculated by any of the four ways given below:

1. 7 + 2 = 5
2. 8 + 3 = 5
3. 3 + 2 = 5; since 5 is a 10's complement of 5 we write it as 5
4. 7 + 8 = 15; and as the answer is 5 more than the base we write 15 - 10 =5

Method no. 3 and 4 can be ignored for now. They will be clear from the explanation of this method at the bottom. I don't really use them while using Nikhilam. All we need to keep in mind is that we do the addition of either of the 2 numbers with the 10's complement of the other. So we write 5 at the left hand side.

7 : 3
8 : 2
------------
5/

The number on the right hand side of the slash is calculated by signed multiplication of the complements. So we get,

3 x 2 = 6

7 : 3
8 : 2
------------
5/6

And what we get is the final answer which is,

7 x 8 = (number to the left x base) + (number to the right) i.e.
7 x 8= 56

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Explanation:

7 = 10 -3
8 = 10 -2
Hence,
7 x 8
= (10 - 3) x (10 -2)
= 100 - 30 - 20 + 6
= 10(10 - 3 - 2) + 6

10-3-2 can be written in four different ways.
=(10 + [-3-2]) = (10 - 5) = 5 or
=([10-3] - 2) = ( 7 - 2) = 5 or
=([10-2] - 3) = ( 8 - 3) = 5 or
=([10 -3]+[10-2]-10) = (7+8-10) = 5

Thus,
7x8=10(5) + 6
7x8=56