Why is it that it is impossible to work out accurately the area of a circle and Pythagoras' theorem?

Answer 1

First, let's clarify that given the measurements, you can get the calculated measurements as accurately as you want. If, in Pythagoras's Theorem, you have base sides of 1 and 1, the diagonal will be the square root of 2 - and that is the exact value. What is not possible is to express this as a fraction, with integers in the numerator and the denominator. Also, you can't write it exactly as a decimal. However, you can write lots of decimal digits and get it as accurate as you want - that is, make the error smaller than any given positive number. Similarly, if you express the area of a circle, with radius 3, as 9 times pi, that is exact; you can't express this exactly as a fraction of integers or as a decimal; but you can write it in such a way that any error becomes as small as you want.

School algebra books contain a basic proof that the square root of 2 (and, by a similar proof, the square root of any positive integer, except for perfect squares) is irrational - i.e., that it can't be written exactly with fractions of integers. I believe that the proof that pi is irrational is more complicated.

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Furthermore, there are some instances where the exact value for Pythagoras' theorem can be represented as ratios. For example, for a triangle with legs in the ratio of 3:4 units, the hypotenuse will be 5 units. That is perfectly exact. There are infinitely many such Pythagorean triplets.