Sunday, October 9, 2011

Are sqrt(2) and the constant pi, Rational or Irrational Numbers?

Two of the concepts which at a time really confused me are sqrt(2) and the other related one is circumference of a circle.

If we have a triangle with a base and perpendicular of unit length, then it's hypotenuse would be sqrt(2) long. Obviously, such a length sandwiched between two other definite and measurable lengths should also be definite and measurable. Here, I use the words "definite" and "measurable from mathematics' point of view. If humans can't measure anything definitely or absolutely, it's their limitations, not mathematics'. As sqrt(2) is irrational number and has never-ending trail of digits after the decimal point, so as a result one might argue that if something never ends then how can it be used to represent something definite (in this case hypotenuse of the triangle)? In calculus we find limits for functions using mathematical methods and we say it's the limit which the function tries to reach without ever actually reaching there. Likewise, the length of the hypotenuse which equals the value of sqrt(2) is the 'pictorial' limit and in this case human methods are the reason which prevent us from really calculating the value of sqrt(2).

The same argument goes for circumference of a circle which is 2*pi*r. Pi is an irrational number just like sqrt(2) with unending trail of digits after the decimal point. We have the 'pictorial' limit before us but we don't have the instrument to reach it.

I understand my reasoning is, perhaps, convoluted and a little erroneous. But I'm sure you can see beyond what I wrote and can extract the correct bits of information, and can fill the gaps between them to make the reasoning understandable for you. What is your opinion on length sqrt(2) and circumference? Please let me know. Thank you.

Regards, BMS