निखिलम्‌ continued ...

Here is another of the special cases, re-iterating the rule to thumb explained in last post

998 x 996

In this case, the closest base is 1000
Thus like in previous example we write,

998:2
996:4
-------
/

For calculating the left part we can do, 998 + 4 = 994 or 996 + 2 = 994
For calculating the right part we do 4 x 2 = 8

998:2
996:4
-------
994/8

The answer is clearly not correct. Applying the rule again -
The number of digits on the right hand side of the slash should be equal to the number of zeroes in the base. Rest of the digits should be carried over and added to the digits on the left hand side.

Hence as the base is 1000, we need to have three digits on the right. So the correct answer is,

998:2
996:4
-------
994/008

998 x 996 = 994008

So far the base has been greater than the multiplier and multiplicand both. However, same method can be applied even if the base is less than both or either of them. Here are a couple of examples illustrating that -

(1) 1903 x 1004
Let's take base as 1000. So we can write,

1903:903
1004: 4
-------
/

For calculating the left part we can do, 1903 + 4 = 1907 ( or 1004 + 903 = 1907, which is too much of a work :))
For calculating the right part we do 903 x 4 = 3612
Thus we can write,

1903:903
1004: 4
------------
1907/₃612

According to the rule, we can't have more than 3 digits on the right side of the slash as the base is 1000. So we carry the 3 over to the left side, giving us -

1903:903
1004: 4
------------
1910/612

1903 x 1004 = 1910612

(2) 18 x 7
Let's take base as 10. So we get -

18:8
7: 3
-------
/

For calculating the left part we can do, 7 + 8 = 15 or 18 + 3 = 15
For calculating the right part we do 8 x 3 = 24
Thus we can write,

18:8
7: 3
-------
15/ ₂4

Here we need to carry over the 2 to the left to get -

18:8
7: 3
-------
13/4

But again the final answer is not in the form we want.
If we refer to last post, we have said that the value of the number to the left hand side of the slash is actually that number multiplied by the base. Thus

13/4 = (13 x 10) + 4 = 126

Hence,

18 x 7 = 126

निखिलम्‌ continued

Let's see another example,

8 x 4

In this case, we can again take 10 as base.
Thus like in previous example we write,

8:2
4:6
----

For calculating the left part we can do, 8 + 6 = 2 or 4 + 2 = 2
For calculating the right part we do 2 x 6 = 12

8:2
4:6
----
2/12

However, the answer doesn't come out right in the form we want.
Here the answer 2/12 really means -

2/12 = (2 x base) + 12

as base is 10,

2/12 = 20 + 12 = 32

Simple rule of thumb is, The number of digits on the right hand side of the slash should be equal to the number of zeroes in the base. Rest of the digits should be carried over and added to the digits on the left hand side.
Another example -

98 x 86

Here closest power of 10 to the two numbers given is 100. So base is 100.
Therefore, we can solve this as -

98:2
86:14
-------

For calculating the left part we can do, 86 + 2 = 84
For calculating the right part we do 2 x 14 = 28

98:2
86:14
-------
84/28

Base is 100, so i have to keep two digits on the right side of the slash.
So answer is,

98 x 86 = 8428

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

------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------

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