Mental Multiplication using Vedic Math - Part 3

Use Indian Vedic Techniques to excel in Math

Introduction

This is the part 3 post of the series on “Mental Multiplication using Vedic Math”. Here we will find out on “Multiply 2 numbers that are near a common 10^x”. Example: 98 x 109.

As you can see, one number is greater than 10^x and the other is lesser than 10^x.

If you need to refer basics of Vedic Math that is required to understand this post, pls click here for Post 1.

We will still use the same rule “Nikhilam Navatashcaramam Dashatah” meaning “All from 9 and the last from 10”.

Can you do this below multiplication mentally in few seconds ?

If this interests you, then this series of post is for you. Read on to find out how we can do this in few seconds.

Power of 10 Base

Just a recap - For Nikhilam multiplication, we need to associate numbers close to their corresponding 10^x base. If you need to under more on how to identify the base, pls read the previous post here

Examples of Nearest Base as 10^x:

and so on…

So in the Nikhilam Multiplication that we will see soon, we would be referring to these actual numbers and as well as the difference that we calculated from their correponding bases for calculations. Also note the sign for the differences as that is going to play a very important role in multiplication!

What is Nikhilam Sutra ?

Pls read the previous post here to understand more on this.

How Nikhilam Method Works?

Let us see, “How to Multiply 2 numbers that are near a common 10^x”. Example: Find the value of 98 x 109 ?

Nikhilam Method Working:

So, 98 x 109 = 10682

Steps to perform the Nikhilam Method:

1. Lets first get their corresponding 10^x & the difference from 10^x.

Actual Numbers to Multiply	Corresponding Difference from 10x
98	-02
109	+09

Note: Since these numbers are greater or lesser than 100, the difference is either POSITIVE or NEGATIVE.
2. Write the numbers that you want to multiply one below the other
3. Write the difference between the number and the closest base 10s number on the right side of the number.
4. Since these numbers are greater or lesser than 100, the difference is either POSITIVE or NEGATIVE
5. Now multiply the differences. The multiplication will be a negative number. Based on the base that you have chosen (here we have chosen 100), product of these two differences should be written in as many digits as zeros' in the 10^x. Like here product of the difference is (-02) x (+09) = (-18). Since base 100 has 2 zeroes, the answer should be noted in two digits. (-18) is a 2 digit number so no issues otherwise we would have to add a leading 0 zeroes to fill that gap. Write the product down.
6. On the left side, which way you cross add determines if you will add or subtract the difference. Both are correct. Hence you can take any of the difference with SIGN from right side and add it crosswise from the other number on the left. That is either 98 + 09 or 109 + (-02). Both will yield 107. Write this number down.
7. There is one additional step since we have a negative number in the right side. This number is called as BAR NUMBER in vedic terms. All you need to do is borrow 1 from left side and subtract this bar number from it. Based on the base you have chosen the 1 that you are borrowing from left side will turn out to be 1 x 10^x. So here it is 100. 100 - 18 = 82. Thats the right side.
8. There you go, the answer is 10682

Algebric Proof for Nikhilam Method

Are we sure that this method will work correctly? Or is it Vedic Math steps a mere coincidence?

Though the steps might seem so absurdly simple, internally Vedic Math follows every single mathematical principle and there is absolutely no deviation in that.

Like previous post, lets see the Algebric Proof for this as well.

Since the numbers to be multiplied can be represented based on (10^x) lets define the numbers A & B as follows.

Let
A = (10x + a)
B = (10x - b)

So A x B = (10x + a)(10x - b)
         = (10x)2 + 10xa -10xb - ab
         = (10x)(10x + a) - (10x)b - ab
         = (10x)(A) - (10x)b - ab  
         = (10x)(A - b) - ab
         = (10x)(A + (-b)) - ab

We can also say this as
So A x B = (10x + a)(10x - b)
         = (10x)2 -10xb + 10xa  - ab
         = (10x)(10x - b) + (10x)a - ab
         = (10x)(B) + (10x)a - ab  
         = (10x)(B + a) - ab

This is what we are doing in Nikhilam.
Cross add for first portion of the answer = (A + (-b)) or (B + a)
Vertically multiply for the last portion of the answer = -ab

Isn’t it cool!

Now you will be able to do this math right in your head and you wouldn’t need a calculator or a piece of paper. This can be extended to any higher order base 10s as well. And it will just work very fine.

See below for additional examples:

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  1502    + 502
x  998    - 002
-----------------
  1500     -1004
------------------
= 1499 + (1000 - 1004)
= 1499 + (-004)
= 1498 + (1000 - 004)
= 1498996

Important: We already said that the right side should have digits equivalent to as many zeros in your base 10^x. But here we have 4 digits on the right side whereas the base 10^x (1000) has only 3 zeroes. Moreover they are negative numbers. Hence we start borrowing numbers from Left side until the negative value is converted to positive. (1500 concat -1004 = 1499 concat -004 = 1498 concat 996.

For explanation sake we have shown couple of additional steps. Once you practice, we simply subtract them in mind and write the answer.

So answer is 1498996.

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  1001    + 001
x  997    - 003
-----------------
   998     -003
------------------
= 997 concat (1000 -003)
= 997997

Important: Since 1000 is the (10^x) we have chosen, the mulitplication of the differences should be represented in 3 digits (as many zeros in (10^x) chosen). Hence leading zeroes for 3! Again remember how we subtract the bar numbers from left side. Do that in your mind for practice. Refer above example if you get struck.

So the answer is 997997.

Conclusion

So lets conclude the post by finding the answer for the question.

If you were to do this in regular method, it would have been very very laborious. See below.

We will look into the other variations in the subsequent posts. Until then, happy mental math!