Simple method to Square numbers near to the base

As children we have always had a difficulty dealing with the 9 tables. Atleast for me it was a real time task to multiply large numbers involving 9s like 998*998 etc... But I had this difficulty till I realised the importance of number nine and the ease with which it can be dealt with. Speaking truly numbers involving 9 are the easiest to multiply. But not by the methods we have studied in our school curriculum.

Before I tell you how to multiply numbers involving 9 let me just brief you about the importance of number 9. Every number has its own importance. Number 9 is of special significance among all the basic numbers i.e. 1,2...9. The question of why is it so sinificant can be answered with some few simple properties of 9.

1. Sum of the digits in a number(called basic number) divisible by 9 always turns to be 9. for example 27 = 2 + 7 = 9, 4437 = 4 + 4 + 3 + 7 = 18 = 1 + 8 = 9. Hence to find the remainder of a number after dividing by 9 just add the digits in that number and reduce it to a basic number which will be the remainder. For example the remainder of 5467 after dividing by 9 is 5 + 4 + 6 + 7 = 32 = 3 + 2 = 5(remainder).
2. To find the result of addition of 9 to any number just subtract 1 from the numbers unit place and add 1 to its tens place. for example 2343 + 9 = 2352. How to do this?? Consider the number 2343. To add 9 to this just subtract one from its units place i.e 3 - 1 = 2 and add one to its tens place i.e 4 + 1 = 5. Hence the result is now 2352.
   

There several such properties of number 9. I will try to get you a list of them in my future posts...

Now we know how simple it is to deal with number nine for finding the remainder and for adding it to other number... But these tasks can be done with no difficulty in traditional methods also... But what about multiplying numbers like 999 * 999 or 999998 * 999998 etc... Can you give the answer without a piece of paper and a pencil??? Yes you can... Follow the method I explain below...

Let me take an example to explain this. Consider finding answer for 98 * 98...

• Here number 98 is close to base 100 with a difference of 2. i.e 100 - 2 = 98.
• Since 98 is 2 less than 100 just subtract it from 98. we get 96... Keep this 96 which forms one part of the answer.
• Now since we subtracted 2, find the square of 2 i.e 2*2=4. And since we have choosen the base to be 100 initially we have to pefix 0 to number 4 i.e 04. This forms the other part of the answer.
  
• Now combine the result of above two steps to get the final result as 9604 :-)

This is the simplest method to square a number which is near to the base like 99, 98, 9999999999999996 etc... Take any large number near to the base and this method works even if your calculator dint work :-)

For example consider another number 999996 * 999996. So this is near to 1000000 and is less by 4. So result is 999992000016...

And this method also works for the numbers like 10004, 1008 etc. which are above the base but with a slight modification. For example to find the square of 1008 which is near to 1000 and more than 1000 by 8 we do as follows... Add the number 8 to 1008 to get 1016 and also find the square of the number 8 to be 64 and append it to 1016 to get 10160064.

Now if you are wondering how this method yiels a correct result for any number then here is a simple mathematical proof...

We all know a simple algebraic identity which we have used merely to solve the equalities in our high schools but never realised that it can be used in real life. i.e the simplest and our favourite a*a - b*b = (a+b)(a-b). The method i explained above is derived from this identity. As we know now our aim is to find square of a number say a*a. so we now have...
a*a = b*b + (a+b)(a-b)
say here a=98, our first example. We can choose a random number b. I will choose b=2 because its eases my multiplication task as the term a+b will now be 98+2=100. Now just substitute for a=98 ans b=2 in above identity...
98*98 = 2*2 + (98+2)(98-2)
= 4 + 100*96
= 4 + 9600
= 9604

Hope this post added some information to your database :-). Ill explain few more methods in subsequent blogs... Thank you...