Quandaries and Queries


Name Madhumita
Who is asking: Student
Level:         Middle

From Pythagoras theorem we can draw square root 2 as a finite distance but it is irrational number which is endless. Explain how we can equate these two.



Hi Madhumita,

You are not the only person who has been bothered by this dilemma. Probably the first person who realized that root 2 is irrational was Hippasus, one of the students of Pythagoras. Legend has it that Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans[1].

Your are correct that the line segment which is the diagonal of a unit square is of length root 2, and when you try to express this length using decimal notation, you can't do it with a finite number of decimal places.

root 2 is 1.41421.....

or, using common fraction notation

root 2 is 1 + 4/101/1024/1032/1041/105 +.....

A proof of the fact that this sum of an infinite number of fractions has a finite value requires a knowledge of Calculus. It is this problem of having a finite value as the result of an infinite process that led to the development of Calculus.



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