How to Square Numbers Ending in 1 ?

 

This shortcut can be used to obtain the squares of any number which ends  in a digit 1. You can also use the shortcut How to obtain the square of any number.

However, this shortcut will be much faster as the number will be closer to multiples of 10. You can use this shortcut to obtain the squares of the number such as 21, 31, 121 etc

Let us understand the algebraic proof behind this shortcut.

a2 = (a – 1 )2 + 2a – 1

a2 = (a +1 )2 + (a + a – 1)

 Let us understand the shortcut with an example. Let’s say we want to obtain the square of 21.

 

The entire steps is illustrated in the following figure.

Square_numbers_ending_11

 

Step 1: First step is to write 31 as 31+1. Obtain 302 = 900.

Step 2: Add 30 to the given number 31. You will get 61.

Step 3: Add 119  to 900 to get the final answer 961

 

Let us consider another example. Let’s say we want to obtain the 1212.

 

Square_numbers_ending_12

Very easy right 🙂

On the similar lines, try to calculate the squares of :

  • 612
  • 712
  • 812
  • 1412

 

There is another shortcut however, I hope you find it useful. Let’s see how does the square of  number which ends in 1 look like.

Let us consider a  number which ends with 1. The number would look something like a1, where a can be any number.  Let’s see how the square of this number looks like:

Using the modified formula mentioned in Mathematical Shortcut for squaring numbers,

 

Square_numbers_ending_1

 

The squares of any 2-digit number ending in 1 would be of the form a2/2a/1.

So, let us consider an example. Can you mentally calculate 1212 ?

 

Square_numbers_ending_2

 

So, 1212 = 14641.

Very easy right :).

On the similar lines, try to calculate the squares of :

212 = 4/4/1 =441

812 = 64/16/1 =7061

512 = 25/10/1 = 2601

 

Exercise: Try to obtain the squares of the following numbers using this technique:

  • 61
  • 81
  • 91
  • 141
  • 261

If you have any queries regarding this tutorial, please leave a comment. 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.

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