Topics Mathematics

Question In a right angled isosceles triangle with equal sides of 1 unit and 1 unit, the third side will be sqroot(2) according to Pythagoras theorem. But sqroot(2)= 1.414213562373095... It is never ending. So theoretically we cannot determine its exact length. But physically it should have a definite length! The side is touching the other two sides of the triangle, so how can the length be theoretically indeterminate but physically determinate ? Does this mean the human understanding is limited and we cannot fully understand the mind of god ? Can you resolve this dilemma ?

Accepted:
April 20, 2011

Comments

Suppose someone had made the analogous argument about dividing a line of 1 unit into three equal parts. She tells us that "the length of each of these parts is 1/3 which is 0.333333333333 .... It is never ending. So theoretically we cannot determine the exact length of these parts."

I think this would be a bit overblown. We know that the length of each of these three parts is exactly 1/3, and we also know that, while this leads to an infinitely long expression in the decimal system, it would not do so in the duodecimal system (which is based on the number 12 rather than the number 10).

I want to suggest that you consider a similar response to your question. Yes, there is a notation in which we cannot express the length of the hypotenuse you have in mind with a finite number of signs. But there are other notations in which this is possible -- we can just call it "sqroot(2)". So, contrary to what you are saying, we can determine the exact length of that hypotenuse.

You can refresh your problem by pointing to some physical object and then asserting that we cannot determine its precise length. No matter how many digits we may manage to add (through clever measurement) behind the decimal point, there will still remain many further such digits unknown. Leaving modern physics aside, I agree with this but still see no dilemma, no tension with you exclamation that "physically it should have a definite length". Yes, it should, and it can have a definite length even if we cannot possibly ascertain what this length is with perfect precision.

Our human understanding is limited here, but fortunately our human curiosity is limited as well: beyond a few dozen digits after the decimal point, even the nanotechies lose interest in greater precision.