Saturday, 29 December 2012

Calculations made easier...

If calculating squares was a tedious job, calculating the difference of squares can be an even tougher task especially if there are bigger numbers involved. 
However, there is a neat trick that we have found which can produce the desired answer without actually calculating the squares.

There are two approaches, one is the one developed by us and the other one is more conventional, but equally effective.

  • ·       The 1st method involves generalizing the equation of differences of squares:  Let ‘n’ be the greater number and ‘n-d’ be the smaller number where‘d’ is the difference between these two numbers. Then, (n) 2-(n-d) 2 = n2-(n2+d2-2dn) = 2dn-d2. Hence we can express the difference between two squares via this simple expression (2dn-d2). The best thing is that now we need not calculate the squares of the bigger numbers involved, we just have to calculate the square of the difference which would be much smaller. 

Eg. 212-122 (n=21 d=9) Using 2dn-d2
2(9) (21)-(9)2=378-81=297

In case there is a negative number involved, one need not consider the negative sign for the difference, because square of a negative number is a positive number.
(15)2-(7)2 = (15)2-(-7)2

  • ·       The 2nd method makes use of the identity [a2-b2= (a+b) (a-b)]. This method too saves the hassle of calculating the squares individually and is especially useful if there are small differences involved.

Eg. 802-782= (80+78) (80-78) =158x2=316

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