The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the divisibility of the integers. (It is often covered in calculus 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.)<\/p>\n\n\n\n
But the proof we’re about to see (from the Landau reference) requires only an understanding of the ordering of the real numbers!<\/p>\n\n\n\n
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.
<\/p>\n\n\n\n
So, suppose Sqrt[2]=x\/y, that is, x2<\/sup>\u00a0= 2y2<\/sup>; then we show x1 = 2y – x, y1 = x – y works. By cross-multiplication, it is easy to check that<\/p>\n\n\n\n x\/y = (2y – x) \/ (x – y).<\/p>\n\n\n\n So x1\/y1 yields the same fraction as x\/y.<\/p>\n\n\n\n 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<\/sup> < 4, and the last statement can be verified because (x\/y)2<\/sup> = 2 by hypothesis.<\/p>\n\n\n\n Thus we have found an equivalent fraction with smaller denominator, giving the desired contradiction. Therefore Sqrt[2] must have been irrational, after all. QED.<\/p>\n\n\n\n Presentation Suggestions:<\/strong> 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):<\/p>\n\n\n\n (x\/y) = (Ny – kx) \/ (x – ky)<\/p>\n\n\n\n Sub-hint: the k to use is k = Floor[Sqrt[N]].<\/p>\n\n\n\n The Math Behind the Fact:<\/strong> How to Cite this Page:<\/strong> References:<\/strong> Fun Fact suggested by: <\/strong> The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,7,3,37,10,167],"class_list":["post-575","page","type-page","status-publish","hentry","tag-advanced","tag-calculus","tag-easy","tag-irrational","tag-numtheory","tag-square-root-of-2"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/575","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=575"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/575\/revisions"}],"predecessor-version":[{"id":1491,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/575\/revisions\/1491"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=575"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=575"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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.<\/p>\n\n\n\n
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<\/em>.<\/p>\n\n\n\n
Su, Francis E., et al. “Irrationality by Infinite Descent.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Edmund Landau, Foundations of Analysis<\/em>, Chelsea, 1966, Theorem 162.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers<\/em>, 5th edition, Oxford, 1979, Theorem 43.<\/p>\n\n\n\n
Allen Stenger<\/p>\n","protected":false},"excerpt":{"rendered":"