{"id":489,"date":"2019-06-29T22:00:34","date_gmt":"2019-06-29T22:00:34","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=489"},"modified":"2020-01-03T19:34:45","modified_gmt":"2020-01-03T19:34:45","slug":"square-root-of-two-is-irrational","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/square-root-of-two-is-irrational\/","title":{"rendered":"Square Root of Two is Irrational"},"content":{"rendered":"\n

An\u00a0irrational<\/em>\u00a0number is a number that cannot be expressed as a fraction. But are there any\u00a0irrational numbers?<\/p>\n\n\n\n

It was known to the ancient Greeks that there were lengths that could not be expressed as a fraction. For instance, they could show that a right triangle whose side lengths (adjacent to the right angle) are both 1 has a hypotenuse whose length is not a fraction. By the\u00a0Pythagorean theorem\u00a0this length is Sqrt[2] (the square root of 2). We shall show Sqrt[2] is irrational.<\/p>\n\n\n\n

Suppose, to the contrary, that Sqrt[2] were rational. Then Sqrt[2]=m\/n for some integers m, n in lowest terms<\/em>, i.e., m and n have no common factors. Then 2=m2<\/sup>\/n2<\/sup>, which implies that m2<\/sup>=2n2<\/sup>. Hence m2<\/sup> is even, which implies that m is even. Then m=2k for some integer k.<\/p>\n\n\n\n

So 2=(2k)2<\/sup>\/n2<\/sup>, but then 2n2<\/sup> = 4k2<\/sup>, or n2<\/sup> = 2k2<\/sup>. So n2<\/sup> is even. But this means that n must be even, because the square of an odd number cannot be even.<\/p>\n\n\n\n

We have just showed that both m and n are even, which contradicts the fact that m, n are in lowest terms. Thus our original assumption (that Sqrt[2] is rational) is false, so the Sqrt[2] must be irrational.<\/p>\n\n\n\n

Presentation\u00a0Suggestions:<\/strong>
This is a classic\u00a0proof by contradiction.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
You may wish to try to prove that Sqrt[3] is irrational using a similar technique. It is also instructive to see why this proof fails for Sqrt[4] (which is clearly rational). The above proof fails for Sqrt[2] because at the point in the proof where we deduce that m2<\/sup> is divisible by 4, we cannot conclude that m is divisible by 4.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Square Root of Two is Irrational.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"

An\u00a0irrational\u00a0number is a number that cannot be expressed as a fraction. But are there any\u00a0irrational numbers? It was known to...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[37,4,10,167],"class_list":["post-489","page","type-page","status-publish","hentry","tag-irrational","tag-medium","tag-numtheory","tag-square-root-of-2"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/489","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=489"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/489\/revisions"}],"predecessor-version":[{"id":1651,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/489\/revisions\/1651"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=489"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=489"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}