Irrationality by Infinite Descent

The traditional proof that the is irrational (attributed to Pythagoras) depends on understanding facts about the divisibility of the integers. (It is often covered in courses and begins by assuming Sqrt[2]=x/y where x/y is in smallest terms, then concludes that both x and y are even, a contradiction. See the Hardy and Wright reference.)

But the proof we’re about to see (from the Landau reference) requires only an understanding of the ordering of the real numbers!

Proof. Suppose Sqrt[2] were rational. Then Sqrt[2]=x/y where x and y are integers and y > 0. We will show that it is also equal to another fraction x1/y1, where x1 and y1 are integers, y1 > 0 and y1 < y. If this were true, then this procedure could be applied over and over to each resulting fraction. Then the denominators would yield an infinite decreasing sequence of positive integers y > y1 > …, which is impossible.

So, suppose Sqrt[2]=x/y, that is, x2 = 2y2; then we show x1 = 2y – x, y1 = x – y works. By cross-multiplication, it is to check that

x/y = (2y – x) / (x – y).

So x1/y1 yields the same fraction as x/y.

Secondly, it must be the case that 0 < y1 < y, because this is the same as y < x < 2y, which is equivalent to 1 < (x/y) < 2. But this is equivalent to 1 < (x/y)2 < 4, and the last statement can be verified because (x/y)2 = 2 by hypothesis.

Thus we have found an equivalent fraction with smaller denominator, giving the desired contradiction. Therefore Sqrt[2] must have been irrational, after all. QED.

Presentation Suggestions:
After presenting this proof, ask students as homework to prove that Sqrt[N] is if N is a positive integer and not a perfect square.

Caution them not to prove “too much”: their proof must fail when N is a perfect square! You may give them a hint to use the analogous equation (where Sqrt[N] = x/y and k is an integer):

(x/y) = (Ny – kx) / (x – ky)

Sub-hint: the k to use is k = Floor[Sqrt[N]].

The Math Behind the Fact:
The reasoning about an infinite sequence of decreasing positive integers is another form of mathematical induction (both depend on the fact that any non-empty subset of the positive integers has a least element). This form of reasoning was invented by Fermat and is called the method of infinite descent.

How to Cite this Page: 
Su, Francis E., et al. “Irrationality by Infinite Descent.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Edmund Landau, Foundations of , Chelsea, 1966, Theorem 162.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford, 1979, Theorem 43.

Fun Fact suggested by:
Allen Stenger

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