vol 4: Glossary
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Proof that there are incommensurable numbers
We learn from Pythagoras' Theorem that the length of the diagonal of a unit square is the square root of two. We can easily prove that no natural number a , or ratio of natural numbers a/b, can represent the square root of two.
Assume that it can: the square root of two is equal to a rational number, a/b, where a and b have no common divisor.
If the square root of two is a/b it follows that a2 = 2b2.
a2 (and a) must therefore be even. b must therefore be odd, otherwise a and b would both be even with the common factor 2 which would contradicts our assumption that a and b have no common divisor.
Since a is even, we can replace it by (say) 2x, and write 4x2 = 2b2, or 2x2 = which tells us that b2, and hence b, must be even. We have just proved above that the assumption that a/b equals the square roots of 2 implies that b is odd. Since this assumption leads to contradictory results, we assume that the assumption is wrong. In other words, the square root of two cannot be represented by any a/b where a and b are natural numbers.
This type of proof, where an assumption is proved false by showing that it leads to contradiction is called reductio ad absurdum or indirect proof.
We say that the square root of two and unity are incommensurable numbers.
Euclid published a more general version of this simple proof in his Elements, Book X
A modern result similar to this was obtained by Georg Cantor using his diagonal argument.
Books
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 3, X-XIII), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' Amazon back |
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