Base Method for squaring Numbers



Base Method for Squaring numbers is a very interesting technique for obtaining the square of a number. Base Method has its origin in Vedic Mathematics. We have already discussed Base Method here and this technique for squaring numbers is an extension of Base Method technique for Multiplication.

This shortcut is very useful when you need to obtain the square of a number which is closer to powers of 10 like 10, 100, 1000 etc.

Let me explain the Base Method with a simple example. Suppose you want to obtain the square of 105 i.e., 1052


The entire steps can be illustrated using the following figure.

Base-Square-1

 

Step 1 : Select a nearest power of 10 as the base. So in this case, choose a base 100.

Step 2 : The number 105 is 5 more than the base 100. Here 5 is referred to as offset or surplus.

Step 3 : Since the given number is more than the base 100, we add the offset to the given number to get 110. This forms the LHS part of the answer.

Step 4 : Obtain the square of the surplus 5 to get 25. This forms the RHS part of the answer.

So, the final answer is 11025.

 

Example 2 : Let us consider another example so that it is clear. Let us say we need to obtain the square of 112 i.e., 1122.

The entire steps can be illustrated using the following figure.

 

Base-Square-2

 

Notice the carry forward from the RHS. So, we need to keep as many digits in the RHS as the number of zeros in the base. So the chose base 100 has 2 zeros. So, the RHS need to have 2 digits and the rest of the digits are carry forward. So the final answer is 12544.

 

Example 3 : Can you obtain the square of 1002 i.e., 10022

 

Base-Square-3


Notice, how 00 is appended to the RHS part of the answer. This depends on the base. In this example, the base chosen is 1000 which has 3 zeros. So, the RHS part of the answer need to have 3 digits. If we get less than 3 digits, 0 is appended and if we get more than 3 digits in RHS, rest of the digits are carry forward.

I hope, you find this shortcut very useful. Notice from the above examples, we have always chosen a number which is closer to base 10. As we need to obtain the square of the surplus, this shortcut can be very useful when the number is +30 or -30 to the base as we already generally the squares of the number till 30. If we do not know, please practice it. It comes real handy in the competitive examinations.

In order to obtain the square of the number using this shortcut, the number does not have to be always bigger than the chosen base. Let us consider an example where the number is smaller than the chosen base.

Example 4 : Can you obtain the square of 96 i.e., 962

Base-Square-4


Step 1 :
Select a nearest power of 10 as the base. So in this case, choose a base 100.

Step 2 : The number 96 is 4 less than the base 100. Here 4 is referred to as deficiency.

Step 3 : Since the given number is less than the base 100, we subtract the offset to the given number to get 92. This forms the LHS part of the answer.

Step 4 : Obtain the square of the deficiency 4 to get 16. This forms the RHS part of the answer.

So, the final answer is 9216.

Very easy right 🙂


The rule of appending zeros in the RHS or carry forward that we saw in Example 2 and Example 3 is applicable here as well.

Using this shortcut, can you obtain the square of the following numbers :

  1. 115
  2. 970
  3. 75
  4. 992


If you have any doubts using this shortcut, please leave a comment or contact me. I would be glad to help.




Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

2 Comments

  1. can u please tell me how to obtain the square of 159 as a example,can we obtain the ans with the same way you tell above if yes than how…

    • I would not use Base method for it as it is little tricky to use this method. There is an article where I discuss about squaring a 3-digit number. Please go through it.

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