Monday, January 17, 2011

square of a natural number is the sum of two definite natural numbers.

Square of a natural number can be found with the help of the following equation

(xy)^2 = x(x+1)y^2 + 20x(y−5).

Where y = 0 to 9; x = 0 to ∞.

Explanation:- value of first number is obtained by putting the value of x(x+1) next to the value of y^2.

Example:- square of 37 can be obtained by putting the value of 3(3+1) next to the value of 7^2 and add into them 20 times the value of 3(7−5).

Now 3(3+1) = 3×4 = 12 and 7^2 =49, put them next to each other. We get 1249.
In 1249 add 20 times the value of 3×(7−5).

Now 20×3(7−5)= 20×3×2 = 120

So 1249+120=1369. This is the required square of 37.

Another example
596^2 = 59(59+1)6^2 +20×59(6−5).
=(3540)36 + 1180
=354036+1180
=355216. This is the required square of 596. Here value of y is 6 and value of x is 59.

Another example
594^2 = 59(59+1)4^2 + 20×59(4−5)
= (59×60)4^2 + 20×59×(−1)
= (3540)16 − 1180
=354016 – 1180
=352836. This is the required square of 594. Here value of y is 4 and value of x is 59.

Note:- Numbers has its mirror image.