In Plato's Theaetetus we learn that mathematician Theodorus has established irrationality of square roots for all non-square numbers up to 17 and then he stopped. This is definitely a curious question as to what happened, why he stopped at 17. There is no documentation that would point to a method of proof he used. We may only guess and speculate.
In a 1976 paper, McCabe offers this reasoning for a possible Pythagorean proof:
The Pythagorean proof of the irrationality of √2 is well known. If we assume that √d = a/b and that a/b is in lowest terms, then |
A slightly different argument is offered in the collection of other irrationality proofs of √2.
It is noteworthy that neither appeals to the general properties of prime numbers which were probably unknown at the time of Pythagoras. Instead of claiming that 2 is a prime and invoking the Fundamental Theorem of Arithmetic, the above argument plays on the juxtaposition of odd and even numbers. It directly applies to non-square even numbers, while the following might be considered the application of that idea to proving the irrationality of √3.
Assume √3 = a/b, in lowest terms. Then
|2 = 4(k² + k - 3(m² + m)),|
|1 = 2(k² + k - 3(m² + m)),|
which is impossible since the left-hand side is odd, whereas the right-hand side is even.
This argument adapts to other odd numbers with slight modifications. Easily to the numbers, 7, 11, 15, ... in the form
|4 = 4(k² + k - 5(m² + m)),|
and then to
|1 = k² + k - 5(m² + m),|
where we observe that the sum of a number with its square is always even, making even the right-hand side. The argument works for 13 but becomes cumbersome for proving the irrationality of √17.
McCabe suggests that a theorem by W. R. Knorr may have been the foundation for the Pythagorean reasoning:
If p is a positive integer which can be written in any one of the following forms,
Assuming, for example that √8n + 5 = a/b, in lowest terms, we eventually arrive at
|8nm² + 8nm + 2n + 5(m² + m) + 1 = k² + k.|
with an odd number on the left and even number on the right.
The only odd numbers not covered by the theorem are those in the form
Hardy and Wright up to the 1960 edition of their classical book tried to extend the odd/even argument in a different way. For example, for √5, they observed that every integer is in one of the following forms: 5n,
Below is another attempt along the same lines by a notable Russian historian of mathematics, to reverse engineer the Pythagorean thought process.
Assume more generally that we'd like to demonstrate irrationality of √d, for some d, a non-square whole number. If d is even we can proceed as in the case
|(1)||a2 = db2.|
As before, a and b must be both odd. Assuming
|8(k(k + 1)/2 - dm(m + 1)/2) = d - 1.|
The left hand side here is divisible by 8 and so the right hand side must be of the form 8t, so that we ought to have
- I. G. Bashmakova, A. I. Lapin, Pifagor, Kvant, no 1, 1986, p. 10 (in Russian)
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications, 1996.
- R. L. McCabe, Theodorus' Irrationality Proofs, Math Magazine, Vol. 49, No. 4. (Sep., 1976), pp. 201-203.
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