Simple question about rational and irrational numbers
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From: Craig Feinstein on 1 Jun 2010 13:09 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? Craig
From: Pubkeybreaker on 1 Jun 2010 13:15 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.
From: Arturo Magidin on 1 Jun 2010 13:23 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
From: Craig Feinstein on 1 Jun 2010 13:39 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. What about conditions in which both a, b are algebraic and irrational, c is rational, and ab=c^2 is impossible? Craig
From: Pubkeybreaker on 1 Jun 2010 13:44
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. > > What about conditions in which both a, b are algebraic and irrational, > c is rational, and ab=c^2 is impossible? |