Simple question about rational and irrational numbers
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From: Craig Feinstein on 1 Jun 2010 13:56 On Jun 1, 1:44 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > On Jun 1, 1:39 pm, Craig Feinstein <cafei...(a)msn.com> wrote: > > > > > > > On Jun 1, 1:15 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > > > > On Jun 1, 1:09 pm, Craig Feinstein <cafei...(a)msn.com> wrote: > > > > > On Jun 1, 12:45 pm, "Mike Terry" > > > > > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > > > > "Craig Feinstein" <cafei...(a)msn.com> wrote in message > > > > > >news:bb2d0bb1-139b-4ada-b42a-f9d3077247b6(a)s41g2000vba.googlegroups..com... > > > > > > > On Jun 1, 12:24 pm, José Carlos Santos <jcsan...(a)fc.up.pt> wrote: > > > > > > > On 01-06-2010 17:19, Craig Feinstein wrote: > > > > > > > > > Is the following possible? > > > > > > > > > a,b are irrational. c is rational. ab=c^2. > > > > > > > > Sure. a = sqrt(2), b = 1/sqrt(2) and c = 1. > > > > > > > > Best regards, > > > > > > > > Jose Carlos Santos > > > > > > > Thank you, that was too simple. I really meant to ask is the following > > > > > > possible? > > > > > > > a^2 and b^2 are irrational. c is rational. ab=c^2. > > > > > > Well, there was nothing very special about the number 2 in José's example - > > > > > e.g. replace the number 2 with an irrational number like Pi... > > > > > > Mike.- Hide quoted text - > > > > > > - Show quoted text - > > > > > Thank you. You are correct. My real goal here is to come up with > > > > general conditions where a, b are irrational, c is rational, and > > > > ab=c^2 are impossible. Anyone aware of any such conditions? > > > > Define what you mean by "general conditions". It is not well defined. > > > > And yes, I know of such conditions. Consider, e.g. one of a, b is > > > algebraic, > > > and the other is transcendental. Their product is transcendental, > > > hence > > > c can not be rational.- Hide quoted text - > > > > - Show quoted text - > > > By general conditions, I mean the largest set of nontrivial conditions > > possible. > > This is still undefined. What is a 'trivial condition' in this > context? > > The most general condition is that ab = c^2 where c is rational is > impossible for all a,b, such that a*b is irrational. > > That is trivial. I'm looking for nontrivial conditions. For instance, what you said above about a transcendental number and an algebraic number is what I'm looking for, except I don't want transcendental numbers. > > > > > What about conditions in which both a, b are algebraic and irrational, > > c is rational, and ab=c^2 is impossible?- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
From: Pubkeybreaker on 1 Jun 2010 14:17 On Jun 1, 1:56 pm, Craig Feinstein <cafei...(a)msn.com> wrote: > On Jun 1, 1:44 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > > > > > > By general conditions, I mean the largest set of nontrivial conditions > > > possible. > > > This is still undefined. What is a 'trivial condition' in this > > context? > > > The most general condition is that ab = c^2 where c is rational is > > impossible for all a,b, such that a*b is irrational. > > That is trivial. Then DEFINE what *you* mean by 'trivial' and 'non-trivial'. >I'm looking for nontrivial conditions. For instance, > what you said above about a transcendental number and an algebraic > number is what I'm looking for, except I don't want transcendental > numbers. Then for &*(#&*#! sake, give us a RIGOROUS DEFINITION of the conditions that you do want. You can't do it, can you? This makes the question meaningless. Would you settle for the following: ab = c^2 is impossible with a,b, irrational and c rational if the extension degree of Q[a,b] is greater than 1???
From: Craig Feinstein on 1 Jun 2010 14:21 On Jun 1, 1:23 pm, Arturo Magidin <magi...(a)member.ams.org> wrote: > On Jun 1, 12:09 pm, Craig Feinstein <cafei...(a)msn.com> wrote: > > > > > > > On Jun 1, 12:45 pm, "Mike Terry" > > > <news.dead.person.sto...(a)darjeeling.plus.com> wrote: > > > "Craig Feinstein" <cafei...(a)msn.com> wrote in message > > > >news:bb2d0bb1-139b-4ada-b42a-f9d3077247b6(a)s41g2000vba.googlegroups.com.... > > > > > On Jun 1, 12:24 pm, José Carlos Santos <jcsan...(a)fc.up.pt> wrote: > > > > > On 01-06-2010 17:19, Craig Feinstein wrote: > > > > > > > Is the following possible? > > > > > > > a,b are irrational. c is rational. ab=c^2. > > > > > > Sure. a = sqrt(2), b = 1/sqrt(2) and c = 1. > > > > > > Best regards, > > > > > > Jose Carlos Santos > > > > > Thank you, that was too simple. I really meant to ask is the following > > > > possible? > > > > > a^2 and b^2 are irrational. c is rational. ab=c^2. > > > > Well, there was nothing very special about the number 2 in José's example - > > > e.g. replace the number 2 with an irrational number like Pi... > > > > Mike.- Hide quoted text - > > > > - Show quoted text - > > > Thank you. You are correct. My real goal > > So, the third try is your "real" goal? When this fails, will you come > up with some new "what you really meant"? > > > here is to come up with > > general conditions where a, b are irrational, c is rational, and > > ab=c^2 are impossible. Anyone aware of any such conditions? > > Yes: ab=c^2 with a,b irrational and c rational is impossible if and > only if c=0. The "if" clause if hopefully clear. > > Simply pick your favorite irrational number k; then let a = c^2*k, > which is of course irrational whenever c is rational and nonzero, and > let b=1/k, which is of course irrational. > > -- > Arturo Magidin- Hide quoted text - > > - Show quoted text - I think this answers my question. It's not the answer I wanted, but it answers the question. Thank you. Craig
From: Craig Feinstein on 1 Jun 2010 14:27 On Jun 1, 2:17 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > On Jun 1, 1:56 pm, Craig Feinstein <cafei...(a)msn.com> wrote: > > > On Jun 1, 1:44 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > > > > > By general conditions, I mean the largest set of nontrivial conditions > > > > possible. > > > > This is still undefined. What is a 'trivial condition' in this > > > context? > > > > The most general condition is that ab = c^2 where c is rational is > > > impossible for all a,b, such that a*b is irrational. > > > That is trivial. > > Then DEFINE what *you* mean by 'trivial' and 'non-trivial'. > > >I'm looking for nontrivial conditions. For instance, > > what you said above about a transcendental number and an algebraic > > number is what I'm looking for, except I don't want transcendental > > numbers. > > Then for &*(#&*#! sake, give us a RIGOROUS DEFINITION of the > conditions > that you do want. > > You can't do it, can you? This makes the question meaningless. > > Would you settle for the following: ab = c^2 is impossible with a,b, > irrational > and c rational if the extension degree of Q[a,b] is greater than 1??? Yes! That's the answer I was looking for. It appears that I don't need to give you a rigorous definition, since you seem to be able to read my mind. Can you prove that ab=c^2 is impossible with a,b, irrational and c rational if the extension degree of Q[a,b] is greater than 1?
From: Jim Burns on 1 Jun 2010 14:35
Craig Feinstein wrote: > On Jun 1, 1:44 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: >> On Jun 1, 1:39 pm, Craig Feinstein <cafei...(a)msn.com> wrote: >> The most general condition is that ab = c^2 >> where c is rational is impossible for all a,b, >> such that a*b is irrational. > > That is trivial. I'm looking for nontrivial conditions. > For instance, what you said above about a transcendental > number and an algebraic number is what I'm looking for, > except I don't want transcendental numbers. Perhaps we would have a better idea what you consider "trivial" if you were to give us some more background. Why do you find yourself asking this question? What larger question is it a part of? Jim Burns |