Irrationality by Infinite Descent

The traditional proof that the is (attributed to ) 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!


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.

22 = 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 irrational 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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