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PI - Pictorial Explanation

Go back to your school days…; Mathematics class…; Geometry lessons…

Pi or π is a mathematical constant whose value is 3.14159. This is the ratio of any circle’s circumference to its diameter.

We all know about the above facts and can give many pages of explanations for the value 3.1.4159.

But the picture given above is sufficient to replace all such explanations.

A picture is worth a thousand words.

Image Credit : Wiki

The number system what we use today (1,2,3,4,5…) is called Arabic Numbers System. We also have Roman Numbers System (I, II, III, IV, V, …) which is rarely being used.

Have you ever thought WHY One is 1, Two is 2, Three is 3 … and so on?

It is all because of Angles. Yes, it is the number of angles in each letter.

The following image has ancient Arabic numbers. All the angles that are formed in these letters are mentioned by the letter o.

Ancient Number System - Arabic Letters

The number of angles gives meaning to each letters.

Number 1 has one angle
Number 2 has two angles
Number 3 has three angles
….
and so on.

Observe the letter Zero. It has no angles.

Interesting. Isn’t?

False Proof is one of the concepts in Mathematics. This is a quite interesting area. One of my Mathematics professor started his first class with the following question.

Can you prove 1 = -1?

We all said nope.

Then he went to the blackboard and wrote the following proof.

Version 1:

Amazing. Isn’t? This is called False Proof.

There is one more way to prove 1 = -1.

Version 2 :

Now let us see, how is this possible?

Let us take the first proof (False Proof)

The rule  is generally valid only if at least one of the two numbers x or y is positive, which is not the case here. Alternatively, one can view the square root as a 2-valued function over the complex numbers; in this case both sides of the above equation evaluate to {1, -1}.

How about the second version? Is that true? The answer is false again. Below is the explanation.

The equation a ^bc = (a ^b )^c, when b and/or c are fractions, is generally valid only when a is positive, which is not the case here, leading to an invalid proof.