Easily find the square of any number

Question

Getting the square of a number ending with 5 is very easy.

If the last digit of the number is 5, then the second part of the result will be 25 and for the first part of the result, multiply the number(leaving 5 i.e the last digit) with the next integer.

Ex: 35*35= 1225 { First part : 3*4=12 , Second part: 25 }
 245*245=60025 { First part : 24*25=600 , Second part: 25 }

Ok now it is simple to calculate squares of a number ending with 5.

How to calculate square of any number??

26*26=???

The following is a Very important property of squares:

1*1= 1 = 1
2*2= 4 = 1+3
3*3= 9 = 1+3+5
4*4= 16 = 1+3+5+7
an so on...

i.e. Square of a number "n" is equal to sum of first "n "odd numbers.

"n" th odd number for the above sequence is given by  2n-1

26*26=???

For calculating this you must use the result 25*25(which we have already learned)

26*26=25*25+26th odd number
 = 625+2(26)-1
 = 676

27*27= 25*25+ 26th odd no + 27th odd no
 but sum of 26th +27th odd numbers= 26(4) {similarly in other cases}
 = 625+26*4
 = 729

Traditional multiplication would seem better for the above mentioned examples but this would really fetch you for 3digit numbers. When you apply this concept with 3 digit numbers you will understand the beauty of the above mentioned technique.

Ex: 214*214=??

214*214= 215*215-215th odd number
 = 46225-  
 = 46625-429
 = 45796

Thank You
Rama Krishna Nagoju

duriangris · Answer

Quite interesting (though I'm too lazy to verify it).
I don't think it's faster than an old-fashioned multiplication but that can be useful to double check the result.

Thanks

Easily find the square of any number

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