tag:blogger.com,1999:blog-1918995924244969903.post6372893274279104593..comments2017-08-17T09:53:01.951+12:00Comments on Kiwi Hellenist: Were the Greeks scared of irrational numbers?Peter Gainsfordhttps://plus.google.com/112181800578852902474noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-1918995924244969903.post-72901097245239464882017-08-03T10:54:28.801+12:002017-08-03T10:54:28.801+12:00The tantalising thing is: what did Theaetetus'...The tantalising thing is: what did Theaetetus' proof look like? One wonders. Presumably it would have been geometrical at heart, like most Greek mathematics of the time.<br /><br />There's speculation that Theaetetus is the actual author of book 10 of the Elements (or an earlier version of it), which deals with rationality and irrationality. I'm not overly familiar with the Elements, but on a glance through I can't see anything dealing with higher than 2 dimensions. So no help there.Peter Gainsfordhttps://www.blogger.com/profile/17448862214081111386noreply@blogger.comtag:blogger.com,1999:blog-1918995924244969903.post-2851292583902443362017-08-03T01:38:24.276+12:002017-08-03T01:38:24.276+12:00Here's a proof that the higher roots of 2 are ...Here's a proof that the higher roots of 2 are also irrational.<br /><br />Let ⁿ√2 = a/b, where n ≥ 3.<br /><br />Then 2 = aⁿ/bⁿ.<br /><br />2bⁿ = aⁿ.<br /><br />This has no solutions, by Fermat-Wiles.<br /><br />Hence ⁿ√2 is irrational.<br /><br />(This proof is not only topsy-turvy, proving an ancient result by a thoroughly modern proof, it is also circular, because Wiles's proof depends on the irrationality of the roots of 2. But it is correct.)John Cowanhttps://www.blogger.com/profile/11452247999156925669noreply@blogger.com