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.)

Monday, 18 February 2013

Building Height

Use some simple trigonometry

With a few simple measurements, it's possible to estimate heights with some accuracy. Take a look at the figure below. All you need to know is:

   1. your distance from the building
   2. your eye height
   3. the angle between the ground and the top of the building




Use this formula to calculate the height of the building:
Height = (tan(angle) x distance ) + eye height

Example: Given a building distance of 25 meters, an angle of 37 degrees, and an eye height of 1.75 meters, the formula would be:

                                             Height = (tan(37) x 25m) + 1.75m
                                                         = (0.75355 x 25m) + 1.75m
                                                         = 20.6m