Professor Stewart's Incredible Numbers by Ian Stewart

Author:Ian Stewart [Stewart, Ian]
Language: eng
Format: epub, pdf
Publisher: Profile Books
Published: 2015-03-18T22:00:00+00:00

Fig 95 Make the shaded rectangle the same shape as the original one.

The Greeks constructed such a rectangle geometrically, but we can use algebra. Assume that the sides are a and 1. The required condition is then

So a2 – 2a = 1, whence (a – 1)2 = 2, so a = 1 + . To sum up: Euclid’s algorithm implies that lengths 1 + and 1 are incommensurable, so 1 + is irrational.

Therefore is also irrational. To see why, assume that is rational, equal to . Then which again is rational. But it’s not, so we have obtained a contradiction, and our assumption is false.

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