Irrational Square Roots
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From: Gerry Myerson on 5 Aug 2010 21:27 In article <04205ded-ffaf-4243-93c8-36c80deb442a(a)v35g2000prn.googlegroups.com>, bill <b92057(a)yahoo.com> wrote: > On Aug 4, 4:56�pm, "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net> > wrote: > > On 4 Aug, 22:17, 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. > > > > 4.0 is a real number with sqrt = 2.0 > > 2.0 is a finite decimal > > 2.0x2.0 = 4.0, an integer > > To say that 2.0 is a finite decimal is to take > advantage of a deficiency in the definition of > a finite decimal. Which is why it's so important to get the definitions right. Which is why I've been so insistent that you define your terms. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Gerry Myerson on 5 Aug 2010 22:01 In article <l2ceiectuto.fsf(a)shaggy.csail.mit.edu>, Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: > Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: > > Butch Malahide <fred.galvin(a)gmail.com> wrote: > >> erry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>> wrote: > >>> �bill <b92...(a)yahoo.com> wrote: > >>>> > >>>> Conjectrure: �If N is an integer and �Sqrt(N) is a > >>>> decimal number; then Sqrt(N) is irrational. > >>> > >>> If you manage to give a precise definition of decimal number > >>> and do it in such a way that integers don't count as decimal > >>> numbers, you will find that your "conjectrure [sic]" was proved > >>> 2000 years ago. > >> > >> Are you sure? 2000 years ago seems early for such a general result. > >> And yet, I recall reading that Theodorus is supposed to have proved > >> the irrationality of the square roots of nonsquare integers up to and > >> including 17, sometime--let me look it up--in the 5th century BC. > >> Yeah, I guess another 4 or 5 centuries would be enough time for > >> somebody to work out the generalization. Do you know who it was? > > > > 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. > > Hopefully the above remark was not meant to be taken seriously. > > Euclid did in fact prove some results about geometric sequences that, > to us, easily implies such results about the irrationality of sqrts. > But no one appears to have noticed such applications in Euclid's time. > Indeed, only rarely are such applications noticed even nowadays. Perhaps Bill is referring to Proposition 22 of Book VII of the Elements. Here is a paragraph from Roger Cooke's essay, Life on the mathematical frontier, Notices of the AMS 57 (April 2010) page 473: It thus appears that in Plato�s time a general proof that the square root of a nonsquare integer is irrational had not been found. However, the Greeks did eventually find a proof. Proposition 22 of Book VII of the Elements asserts that if three integers are in proportion and the first is a square, then the third is, also. In our language, if a, b, and c are integers, a/b = b/c, and a is the square of an integer, then c is the square of an integer. (By writing this paragraph, I have deliberately produced a pseudo-syllogism of type 2. Euclid would not have recognized the proposition in the first sentence, since the phrase irrational number would have been an oxymoron to him.) -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Arturo Magidin on 5 Aug 2010 23:19 On Aug 5, 6:32 pm, bill <b92...(a)yahoo.com> wrote: > On Aug 4, 4:56 pm, "sttscitr...(a)tesco.net" <sttscitr...(a)tesco.net> > wrote: > > > On 4 Aug, 22:17, 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. > > > 4.0 is a real number with sqrt = 2.0 > > 2.0 is a finite decimal > > 2.0x2.0 = 4.0, an integer > > To say that 2.0 is a finite decimal is to take > advantage of a deficiency in the definition of > a finite decimal. Which suggests that one should not bandy about terms of art like "finite decimal" unless one has a very precise and very explicit definition, to avoid such alleged "deficiencies". Or perhaps that one should acknowledge that one did not express him or herself correctly (due to an honest mistake, ignorance, carelessness, or some other reason) and seek knowledge on how to express oneself correctly and precisely (a must in mathematics). -- Arturo Magidin
From: Bill Dubuque on 6 Aug 2010 00:16 Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: > Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >> Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> wrote: >>> Butch Malahide <fred.galvin(a)gmail.com> wrote: >>>> erry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>> wrote: >>>>> bill <b92...(a)yahoo.com> wrote: >>>>>> >>>>>> Conjectrure: If N is an integer and Sqrt(N) is a >>>>>> decimal number; then Sqrt(N) is irrational. >>>>> >>>>> If you manage to give a precise definition of decimal number >>>>> and do it in such a way that integers don't count as decimal >>>>> numbers, you will find that your "conjectrure [sic]" was proved >>>>> 2000 years ago. >>>> >>>> Are you sure? 2000 years ago seems early for such a general result. >>>> And yet, I recall reading that Theodorus is supposed to have proved >>>> the irrationality of the square roots of nonsquare integers up to and >>>> including 17, sometime--let me look it up--in the 5th century BC. >>>> Yeah, I guess another 4 or 5 centuries would be enough time for >>>> somebody to work out the generalization. Do you know who it was? >>> >>> 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. >> >> Hopefully the above remark was not meant to be taken seriously. >> >> Euclid did in fact prove some results about geometric sequences that, >> to us, easily implies such results about the irrationality of sqrts. >> But no one appears to have noticed such applications in Euclid's time. >> Indeed, only rarely are such applications noticed even nowadays. > > Perhaps Bill is referring to Proposition 22 of Book VII of the Elements. > Here is a paragraph from Roger Cooke's essay, Life on the mathematical > frontier, Notices of the AMS 57 (April 2010) page 473: > > It thus appears that in Plato¹s time a general proof that the square > root of a nonsquare integer is irrational had not been found. However, > the Greeks did eventually find a proof. Proposition 22 of Book VII of > the Elements asserts that if three integers are in proportion and the > first is a square, then the third is, also. In our language, if a, b, > and c are integers, a/b = b/c, and a is the square of an integer, then > c is the square of an integer. (By writing this paragraph, I have > deliberately produced a pseudo-syllogism of type 2. Euclid would not > have recognized the proposition in the first sentence, since the phrase > irrational number would have been an oxymoron to him.) One has to be very, very careful when interpreting ancient mathematics. In particular I would not trust any such claim unless it was made by someone who was an expert in the mathematics of that period. No such expert has ever made such a claim - to the best of my knowledge. --Bill Dubuque
From: Pubkeybreaker on 6 Aug 2010 14:15
On Aug 5, 5:55 pm, bill <b92...(a)yahoo.com> wrote: > > "A regular number, also called a finite decimal (Havil 2003, p. 25), > is a positive number that has a finite decimal expansion." > > I don't know what "....." means. If it means > "expand infinitely", then 3.99999..... is not > a finite decimal. Sigh. From your quote above, I extract the words: "has a finite decimal expansion" Note the quantifier, "a". The integer 4 has a finite decimal expansion. It fact, it has infinitely many different finite decimal expansions. 4.0, 4.00, 4.000, ..... It also has an infinite decimal expansion: 3.999999999........ Note that the text you quoted did NOT say "a UNIQUE finite decimal expansion". |