Monday, 7 October 2013

Proof of 4=5

Proof:
                          -20 = -20
                    16 - 36 = 25 - 45
                4^2 - 9*4 = 5^2 - 9*5
4^2 - 9*4 + (9/2)^2 = 5^2 - 9*5 + (9/2)^2

          Since (a-b)^2 = a^2 + b^2 - 2ab

              (4 - 9/2)^2 = (5 - 9/2)^2

Taking square root on both sides, it becomes

                     4 - 9/2 = 5 - 9/2

-9/2 on both sides get cancelled.

                            4 = 5

i know it is not really true......
but where is the mistake.......?

Mistake:
Two different numbers can have the same square.

For example, (-3)^2 = 9 = 3^2, but -3 is not 3.

In fact, if you simplify the numbers being squared, we have
4 - 9/2 = 8/2 - 9/2 = -1/2.
5 - 9/2 = 10/2 - 9/2 = 1/2.

Like my example of 3 and -3, these two numbers are opposites of each other. When squared, their "oppositeness" goes away.

My point is that, just because their squares are equal is no good reason to believe the numbers themselves are equal.

(This is why, when taking the square root of both sides of an equation, one must include a "plus or minus" option.)