Irrational Square Roots
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From: Gerry on 7 Aug 2010 19:43 On Aug 8, 8:45 am, bill <b92...(a)yahoo.com> wrote: > What does the "|" in "M|N" mean? M | N means M divides N, which is to say, N / M is an integer. Please pick up an introductory Number Theory textbook. Pretty please. You'll feel so much better. -- GM
From: Gerry on 7 Aug 2010 19:49 On Aug 8, 8:34 am, bill <b92...(a)yahoo.com> wrote: > On Aug 6, 5:02 pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <38d15a0a-0045-419e-bb28-3c5bafb41...(a)o7g2000prg.googlegroups.com>, > > bill <b92...(a)yahoo.com> wrote: > > > Then, the prime factors of A are prime factors of B. > > > > Therefore, A is a factor of B. > > > Suppose A = 12 and B = 6. > > > The prime factors of 12 are all prime factors of 6 but 12 is not a > > factor of 6. > > > Oops. > > If B/A < 1, Then (B^2)/(A^2) << 1. Therefore > your example is invalid. It is invalid as a counterexample to the statement that if x is rational and x^2 is an integer then x is an integer - but of course, that statement is true, so there are no valid counterexamples. It is, however, valid as a counterexample to one of the steps in your non-proof of the true statement. -- GM
From: Gerry on 7 Aug 2010 19:54 On Aug 8, 2:17 am, Bill Dubuque <w...(a)nestle.csail.mit.edu> wrote: > I regret to report that - after perusing some of Roger Cooke's works > on historical matters - the above cautionary remark is not strong enough. > Mr. Cooke's writings not only contain historical inaccuracies but also > serious mathematical errors. For example, the first few places that I > looked in his book "Classical Algebra - It's Nature, Origins, and Uses" > had errors that would be obvious even to a beginning algebra student: Not to mention an error in the use of the apostrophe that should be obvious even to a beginning English student. Did he really put "It's" in the title of his book, when it should have been "Its"? -- GM
From: Bill Dubuque on 7 Aug 2010 21:00 Gerry <gerry(a)math.mq.edu.au> wrote: >Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: >> >> No. But with guys like Archimedes and Diophantus and Eudoxus >> and Euclid all hanging around, I'm confident that someone >> figured it out back then. > [...] > Not to mention an error in the use of the apostrophe > that should be obvious even to a beginning English student. > Did he really put "It's" in the title of his book, when > it should have been "Its"? Gerry: your prior two posts in this thread seem to indicate that you do not always take historical matters very seriously. Should that change I'll be happy to continue the discussion, just as we did last time around on closely related historical topics (Estermann's irrationality proof [1]). --Bill Dubuque [1] sci.math, 20 May 2009, Irrationality of sqrt (n^2 - 1) http://groups.google.com/group/sci.math/msg/498e27a63c196e79 http://google.com/groups?selm=y8zws8b8w1f.fsf%40nestle.csail.mit.edu
From: Counterclockwise on 7 Aug 2010 17:36
> Oh? Please explain this "deficiency" and why you > believe > that the standard definition is deficient. Also > explain > how you would correct the standard definition so that > (in your eyes) > it is NOT deficient. Every definition is deficient on some level. Even the most clear and context precise ones. In fact, there's an entire part of philosophy dedicated to arguing that philosophers've gone overboard in trying to define everything in a general sense, (Structure scepticism or something like that). Anyhow, the important bit is to be able to clarify for someone who doesn't understand you when they ask, not to give a definition that is clear in every case & context, since that's debatably impossible. In this case, perhaps a fitting definition would be, "A number that could be represented without the use of a decimal point", or more precisely again, "A number that only has 0's after the decimal". |