Irrational Square Roots
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From: bill on 11 Aug 2010 17:52 On Aug 9, 3:45 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote: > In article > <7d557612-615b-4a98-9f40-ca165762d...(a)s24g2000pri.googlegroups.com>, > > > > bill <b92...(a)yahoo.com> wrote: > > On Aug 4, 2:17 pm, bill <b92...(a)yahoo.com> wrote: > > > Conjectrure: If N is an integer and Sqrt(N) is a > > > decimal number; then Sqrt(N) is irrational. > > > > Let R be any real number with a square root. > > > If Sqrt(R) is a finite decimal, then [Sqrt(R)]^2 > > > cannot be an integer. Therefore, Sqrt(N) is an > > > infinite decimal. > > > > regards, Bill J > > > Some corrections: > > > Conjecture: If N is an integer and > > Sqrt(N) > floor[( Sqrt(N)], then Sqrt(N) is not rational. > > To you, it's a conjecture. "Conjecture" is word that use I use carelessly and occasionally, incorrectly. This could be one of those occassions. To anyone whose mathematical education > has gotten up to where Euclid was, I know that Euclid was up to prime numbers. I was not. I feel that E. knew something about number theory. I don't! it's a simple theorem, indeed, So is the Four Color Theorem. > it's just a minor rewriting of the theorem we've been talking about > since the beginning of this thread. I just thought that I could stimulate my brain by attempting to prove that theorem. I expected to enjoy additional stimulation by responding to the posts, which I suspected would be mostly critical. > > > Suppose that Sqrt(N) is rational. Then Sqrt(N) can > > be written as an integer + a fraction, say (I + F) > > Where, by "fraction", you mean, a rational number > strictly between zero and one. Specifically, I mean 0 < F < 1; but not 0 <= F <=1 > > > Or (N) = I^2 + 2*I*F + F^2. Can (2*I*F + F^2) > > be an integer? Only if F = 0.9... . > > True, but I'd like to see you prove it! > I'd like to try. But offline. > -- > Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email) BJ
From: Gerry Myerson on 11 Aug 2010 23:42 In article <baf9b975-a4e9-4eae-847c-71a300ce744c(a)q16g2000prf.googlegroups.com>, bill <b92057(a)yahoo.com> wrote: > On Aug 9, 3:45�pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> > wrote: > > In article > > <7d557612-615b-4a98-9f40-ca165762d...(a)s24g2000pri.googlegroups.com>, > > > > �bill <b92...(a)yahoo.com> wrote: > > > > > Conjecture: �If N is an integer and > > > Sqrt(N) > floor[( Sqrt(N)], then Sqrt(N) is not rational. > > > > To you, it's a conjecture. > > "Conjecture" is word that use I use carelessly and > occasionally, incorrectly. This could be one of > those occassions. > > > To anyone whose mathematical education > > has gotten up to where Euclid was, > > I know that Euclid was up to prime numbers. I was not. > I feel that E. knew something about number theory. > I don't! > > it's a simple theorem, indeed, > > So is the Four Color Theorem. It is simple to state, as is the 4CT. But it is also simple to prove, which 4CT is not. > > it's just a minor rewriting of the theorem we've been talking about > > since the beginning of this thread. > > I just thought that I could stimulate my brain by > attempting to prove that theorem. I expected to > enjoy additional stimulation by responding to the posts, which I > suspected would be mostly critical. Why not stimulate your brain by picking up an intro Number Theory textbook, and reading it, and trying to do the exercises? I think you'll learn more by doing that than by subjecting yourself to my sarcastic responses to your posts here. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Bart Goddard on 12 Aug 2010 00:49 Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote in news:gerry- C4F159.13422012082010(a)mx01.eternal-september.org: > Why not stimulate your brain by picking up > an intro Number Theory textbook, and reading it, > and trying to do the exercises? I think you'll learn > more by doing that than by subjecting yourself > to my sarcastic responses to your posts here. That gives me an idea for a new number theory text: Elementary Sarcastic Number Theory. Exercise 1. Stop breathing through your mouth, wipe the drool off your chin and try to find the inverse of 3 modulo 7. If you can manage that, try solving 3x = 4 (mod 7) without just trying every number. If your answer is 0.75, fill out the attached drop slip and hand it to your instructor. Exercise 2. Find the correct time and room of the final exam. Practice going there several time throughout the semester. Exercise 3. Hey, genius, have you bought a stapler yet? OK, sorry, I mean, "Did your mom buy you stapler yet?" Exercise 4. Appendix A consists of a page of tear out Band-Aids. Put one on your index finger. Next time, try to keep it out of the way of the staple. Exercise 5. Google "Fermat's Last Theorem", and then construct a proof of it consisting of 3 examples. Write an essay explaining how no one ever thought of _that_ before. Exercise 6. btw turn off ur ipod thx -- Cheerfully resisting change since 1959.
From: Gerry on 12 Aug 2010 08:45 On Aug 12, 2:49 pm, Bart Goddard <goddar...(a)netscape.net> wrote: > Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote in news:gerry- > C4F159.13422012082...(a)mx01.eternal-september.org: > > > Why not stimulate your brain by picking up > > an intro Number Theory textbook, and reading it, > > and trying to do the exercises? I think you'll learn > > more by doing that than by subjecting yourself > > to my sarcastic responses to your posts here. > > That gives me an idea for a new number theory > text: Elementary Sarcastic Number Theory. > > Exercise 1. Stop breathing through your mouth, > wipe the drool off your chin and try to find the > inverse of 3 modulo 7. If you can manage that, > try solving 3x = 4 (mod 7) without just trying > every number. If your answer is 0.75, fill > out the attached drop slip and hand it to your > instructor. > > Exercise 2. Find the correct time and room of > the final exam. Practice going there several > time throughout the semester. > > Exercise 3. Hey, genius, have you bought a > stapler yet? OK, sorry, I mean, "Did your > mom buy you stapler yet?" > > Exercise 4. Appendix A consists of a page of > tear out Band-Aids. Put one on your index > finger. Next time, try to keep it out of > the way of the staple. > > Exercise 5. Google "Fermat's Last Theorem", and > then construct a proof of it consisting of 3 > examples. Write an essay explaining how no one > ever thought of _that_ before. > > Exercise 6. btw turn off ur ipod thx There's hope for you yet. -- GM
From: Ludovicus on 13 Aug 2010 11:17
On Aug 4, 5:17 pm, bill <b92...(a)yahoo.com> wrote: > Conjecture: If N is an integer and Sqrt(N) is a > decimal number; then Sqrt(N) is irrational. > > regards, Bill J It is not a conjecture but a theorem. Proof: Given the equation N - x^2 = 0 ; N = integer > 0 If testing the integers 1,2,3...x util x^2 > N and not finding a solution, then sqrt(N) is irrational. Ludovicus |