Squaring Numbers

Numbers that end in '5'

Choose any number that ends in 5. Remove '5' from the number. Multiply the left number to its predecessor (next number). Then write '25'.

Example:

(45)^2 = Remove 5 from the number. The number left is '4'. Multiply 4 with its next number i.e. 5. and write 25

1. (45)^2

      = 4(4+1)|25

  = 4(5)|25 

= 20|25 

= 2025  

2. (65)^2

       = 6(6+1)|25

   = 6(7)|25

= 42|25

 = 4225  

Proof

(10n+5)^2

                     = (10n)^2+2*10n*5+(5)^2

          = 100n^2+100n+25

      = (n^2+n)100+25

       = [n(n+1)]100+25

 = n(n+1)|25    

Numbers near to '50' (25-75)

To help you I recommend you to memorize some squares :-

1,2,3,4,5,............,23,24,25,50,100

Choose any number between 25-75. Find out how much greater or less than 50 is the number.

Case 1: Let the number is greater than 50 by 'a'. Add 'a' with 25. Than find a^2. If a^2 is a two digit number then write it after a+25, if not then add the hundredth digit with a+25

Example:

(58)^2 = Subtract 50 from 58, the result is 8. Add 8 with 25, the result is 33. Write 8^2 after it.

(58)^2

  = 8+25|8^2

= 33|64    

= 3364      

Proof

(50+n)^2

                = (50)^2+2*50*n+n^2

          = 2500+100n+n^2

        = (25+n)100+n^2

= 25+n|n^2 

Case 2 : Let the number is less than 50 by 'a'. Subtract 'a' from 25. Than find a^2. If a^2 is a two digit number then write it after 25-a, if not then add the hundredth digit with 25-a.

Example:

(46)^2 = Subtract 46 from 50, the result lis 4. subtract 4 from 25, the result is 21.  write 4^2 after 21.

(46)^2

 = 25-4|4^2

= 21|16    

= 2116      

Proof

(50-n)^2

                = (50)^2-2*50*n+n^2

         = 2500-100n+n^2

       = (25-n)100+n^2

 = 25-n|n^2   

Numbers near to '100' (75-125)

Choose any number between 75-125. Find out how much greater or less than 100 is the number.

Case 1: Let the number is greater than 100 by 'a'. Add 'a' with the original number (100+a)+a. Than find a^2. If a^2 is a two digit number then write it after (100+a)+a, if not then add the hundredth digit with (100+a)+a.

Example:

(106)^2 = Subtract 100 from 106, the result is 6. Add '6' with 106, the result is 112. Write 6^2 after 112.

(106)^2

  = 106+6|6^2

= 112|36    

= 11236      

Proof

(100+n)^2

                  = (100)^2+2*100*n+n^2

          = 10000+200n+n^2

          = (100+2n)100+n^2

             = (100+n+n)100+n^2

                = [(100+n)+n]100+n^2

       = (100+n)+n|n^2

Case 2 : Let the number is less than 100 by 'a'. Subtract 'a' from the original number (100-a)-a. Than find a^2. If a^2 is a two digit number then write it after (100-a)-a, if not then add the hundredth digit with (100-a)-a.

Example:

(96)^2 = Subtract 96 from 100, the result lis 4. subtract 4 from 96, the result is 92.  write 4^2 after 92.

(96)^2

 = 96-4|4^2

= 92|16     

 = 921616    

Proof

(100-n)^2

                  = (100)^2-2*100*n+n^2

         = 10000-200n+n^2

         = (100-2n)100+n^2

           = (100-n-n)100+n^2

              = [(100-n)-n]100+n^2

    =(100-n)-n|n^2

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