Talk:Irrational number

I moved this from the main page:

[I regret that I cannot carefully edit this paragraph at this time, but in good conscience I must question whether any responsible historian of mathematics ascribes the argument below to Pythagoras. An algebraic argument for the irrationality of the square root of 2 observes that if √2=m/n, then √2=(2n-m)/(m-n), so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction, completing the reductio ad absurdum. To me it is plausible that Pythagoras or someone of his school discovered a geometric argument showing that if n and m are respectively the leg and the hypotenuse of an isosceles right triangle, then m-n and 2n-m are respectively the leg and the hypotenuse of a smaller isosceles right triangle. An ancient Greek geometer would have constructed the smaller triangle from the larger one, rather than doing algebra, as we do today. I would recommend that any mathematician editing this page look at Thomas Heath's translations of the writings of ancient Greek geometers before ascribing anything to Pythagoras.]

and changed the paragraph about Pythagoras's discovery accordingly. AxelBoldt 03:30 Oct 23, 2002 (UTC)

I suspect the Greeks' argument might also have used Euclid's own version of Euclid's algorithm, involving repeated subtraction rather than the division used in today's optimised variant. PML.

About the Irrationality of the squareroot of 2. My math teacher said today that Pythagoras believed that sqrt(2) actually WAS a rational number and that that was a thought that his followers the Pythagoreans also thought. He also said that someone during the Middle Ages proved that sqrt(2) = irrational and that that guy subsequently was murdered. BL 22:58, 16 Sep 2003 (UTC)

That is ignorant nonsense; just look at Euclid's Elements and you will see that irrationality was know to the ancient Greeks. Michael Hardy 22:12, 17 Aug 2004 (UTC)

Euclid was born about a hundred years after pythagoris' death. The proof of the irrationality was discovered by one of pythagoris' followers, but if I remember my reading correctly he was banished, not killed. --Starx 01:09, 18 Aug 2004 (UTC)

In other words, as I said, it is ignorant nonsense to say it was not done until the middle ages. Michael Hardy 01:52, 18 Aug 2004 (UTC)

If Euclid was around after pythagoras, then the fact that he knew of the irrationality of the square root of 2 is not surprising, considering it was during pythagoras' time that it was first proven. Do you have any sources? Cause there are plenty documented sources saying it was, in fact, a follower of pythagoras. --Starx

So you and I both agree that it was known to the Pythagoreans and therefore to Euclid, who came later. And we both agree therefore that it is ignorant nonsense to say that it was not done until the middle ages. Right? As for sources, I've read some of Thomas Heath's books, but it's been a while, so I cannot cite chapter and verse. On another matter, why do you keep deleting my assertion in the article that the conventional algebraic argument is not the one that the Pythagoreans used? Michael Hardy 21:40, 18 Aug 2004 (UTC)

Because everything I've read has said that that was the proof. The only one I can think of off the top of my head is the golden ratio by mario livio, which has quite a bit on the history of math. If you have a better source that says otherwise then I'll concede, but all you've done so far is claim that it's ignorant nonsense. If we both agree that it was known to the pythagoreans, and I'm saying it was the pythagoreans who first discovered it, where do you get the middle ages?? Who brought that up? --Starx 01:10, 19 Aug 2004 (UTC)

I did not "get middle ages"!! That is what I called "ignorant nonsense". I never said that it is "ignorant nonsense" to say that the relatively recent algebraic proof of irrationality is how the Pythagoreans did it. It is not how the Pythagoreans did it; it is how many mathematicians believe (and write) that the Pythagoreans did it; I never said that that error is "ignorant nonsense" -- only that it is an error. Michael Hardy 02:31, 19 Aug 2004 (UTC)

I'm not debating about anything that happened during the middle ages. I'm debating about whether or not the proof displayed on the page was done by one of pythagoras' followers. That's what our recent edits have concerned so I think it would be fairly obvious that that is what the discussion is about. I don't understand why you're still bringing up the comment another user made on the middle ages, that's not the subject of the debate and that's why I want to know where you're getting that from. I'm sorry if I was unclear. I'm asking what referances do you have pertaining to what proof pythagoras used to determine the irrationality of the square root of two. Because I have referances that say that what's displayed is the correct proof. I said this in my above post and I'll say it again: If you have a better source that says otherwise then I'll concede. --Starx 03:41, 19 Aug 2004 (UTC)

I will get the references.

What I called "ignorant nonsense" was the statement about the middle ages. Then you attacked me for calling your statements about the Pythagoreans and Euclid "ignorant nonsense". That's why I brought up the matter of the middle ages. Michael Hardy 18:35, 19 Aug 2004 (UTC)

Complicated proof?

Isn't the first proof for the irrationality of

{\sqrt {2}}

overly complicated? It basically states that when you transform

\left({\frac {p}{q}}\right)^{2}=2

to

p^{2}=2q^{2}

, the multiplicity of prime factor 2 is even on the left side, and odd on the right side -> contradiction.

Aragorn2 21:00, 17 Sep 2003 (UTC)

No, because the proof builds on other proofs that has to be explicitly stated. Like that the square of an even number also is even. As it is on the page is how my math teacher described it. BL 21:27, 26 Sep 2003 (UTC)

The recent posting on the history is directly taken from Article 3 of a 1906 book at www.gutenberg.net/etext05/hsmmt10p.pdf .

I'll leave it there for the present; but in any case it would need a thorough edit.

Charles Matthews 16:50, 29 Jan 2004 (UTC)

BL: a root of a natural number m (i.e. a positive/non-negative integer) is either a natural number or an irrational: Suppose we are looking at m^(1/n) and this was a/b (i.e. rational with a,b integers), so a^n=m*b^n. Then write m in terms of a product of powers of prime numbers (m=p^x * q^y * r^z * ...). Do the same with a and b, and then match exponents on each side.

If all of x,y,z,... are multiples of n, we will be able to take the n-th root of m and get a natural number. If any of them are not, then we will not even be able to get a rational number because the LHS of a^n=m*b^n will be a product of powers of primes where all the exponents are multiples of n while the RHS will not be, which based on the fundamental theorem of arithmetic leads to a contraction of the hypothesis that m^(1/n) is rational. --Henrygb 23:28, 13 Feb 2004 (UTC)

The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters a periodic pattern

I know that is true but there is no need to invoke decimal when describing irrational numbers. I have witnessed confusion when irrational numbers are defined thus. People think that the set of irrational numbers are different in base-2 than they are in base-10 because of definitions like that. Paul Beardsell 05:03, 20 Feb 2004 (UTC)

Grammar

From the article: (because none of its prime factors is 2) Factors is plural, so shouldn't it be are instead of is? --Starx 01:51, 20 Dec 2004 (UTC)

No. "Its factors" is the object of the preposition "of". If I wrote "Not even one of its factors is prime", obviously it would be grossly wrong to write "are". Similarly if I wrote "Just one of these factors is prime", would you say I should have written "are", when I'm writing about only one, on the grounds that "factors" is plural? Traditionally, "none" is singular. Of course, recently many people have used "none" as plural, but even so, there can hardly be a grammatical objection to using a singular "none". (And somehow the misspelling of "grammar" in the edit summary doesn't inspire confidence either.) Michael Hardy 23:24, 20 Dec 2004 (UTC)

... and also, when you say "because factors is plural", I almost fear that next you'll write something like "One of these are correct". I actually hear people say that from time to time; it's as if the fact that these is plural means that the phrase one of these is plural. Obviously the phrase one of these is singular and should be followed by is, not are. Michael Hardy 23:57, 20 Dec 2004 (UTC)

It's nice to see you take an honest question and be a dick about answering it. You sounded like a decent human being right up until the parenthetical remark in your first responce. But that one remark wasn't enough, you had to go back for a second responce. Kudos. --Starx 06:16, 21 Dec 2004 (UTC)

I am in fact a decent human being. And I stand by what I wrote above: misspelling grammar two times running doesn't inspire confidence; it may be useful for you to know that. If you disagree with that or any of the other points above, you could argue the point instead of engaging in name-calling. What, specifically, do you object to in the second response? Writing "One of these are correct" is in fact grammatically parallel to the usage you raised a question about. Michael Hardy 21:44, 21 Dec 2004 (UTC)

It may be grammatically parallel in a technical sense, but the original case is far more obscure. Your example states one of these factors, it's obviously singular. The snippet from the article isn't so clear. In either event you very nicely explained things to me and should have stopped there. But instead you chose to make remarks about my spelling not inspiring confidence and how you "fear" I'll do something even stupider. I stand by what I wrote above, you're being a dick for no reason I can see other then possibly a superiority complex. --Starx 22:44, 21 Dec 2004 (UTC)

In discussions of politics or scientific controversies a rhetorical device such as "Since you're advocating X's theory, next I expect you'll be saying the Big Bang didn't happen" is not generally construed literally; people aren't so touchy. But when the topic is grammar, it seems they are. I don't understand why the difference. Let me rephrase my comment that was found offensive. Originally I wrote:

... and also, when you say "because factors is plural", I almost fear that next you'll write something like "One of these are correct". I actually hear people say that from time to time; it's as if the fact that these is plural means that the phrase one of these is plural. Obviously the phrase one of these is singular and should be followed by is, not are.

Here is a rephrasing:

... and also, the phrase "because factors is plural", is syntactically parallel to, "One of these are correct". I actually hear people say that from time to time; it's as if the fact that these is plural means that the phrase one of these is plural. Obviously the phrase one of these is singular and should be followed by is, not are.

If I had not thought that was obviously what was meant, I would have phrased it in that literal way originally. Michael Hardy 23:22, 30 Dec 2004 (UTC)

<math>

Should this page be converted to use <math> tags rather than radical symbols? IMO, the radicals with no overline look really ugly. If nobody objects within a few days, I'll switch it over. --Simetrical 01:13, 30 Dec 2004 (UTC)

Please note that there is no consensus about using <math> tages, i.e. TeX, for inline symbols, rather than displayed formulae. It is generally preferred that inline MathML is left as such, until there is more agreement. How it appears may well be browser-dependent, so that changing it to suit one user may not have a good effect for another. It is often reported that inline TeX looks odd. Charles Matthews 07:14, 30 Dec 2004 (UTC)

Hmm. Well, I very much prefer it even for inline, but I can see how some might have problems with it. Is there a talk page for discussing the use of inline TeX? --Simetrical 00:16, 31 Dec 2004 (UTC)

I think there is; I don't know the URL at this moment. Formerly it failed to get centered and looked terrible. Now, when it does get centered, you have things like

2^{x}\,

getting centered that should not -- the bottom of the "2" should be at the same level as the bottoms of the letters, as in 2^x. Also, in

M+N\,

the variables appear much bigger than they should, whereas in M + N they look good. Various other problems like those, too. Michael Hardy 00:25, 31 Dec 2004 (UTC)

I just found the URL myself: Wikipedia:WikiProject_Mathematics. Anyway, maybe we should at least overline the roots—√2. Unless that shows up funny in some browsers, I don't see any reason not to. --Simetrical 00:38, 31 Dec 2004 (UTC)

Repitend thing

Michael Hardy wrote:

I think this should say either "the period is 3" or "the length of the repitend is 3", but NOT "the length of the period is 3".

You are right. My spell-checker gave me "repitend" as an option. I should have looked up a dictionary and confirm this is correct. I instead chose to replace it with "period" assuming it will be the same thing. I would actually appreciate a bit of clarification here, if it would not take too long. Oleg Alexandrov 02:51, 4 Apr 2005 (UTC)

Actually, I think I figured it out myself. All it took is reading what you wrote, and actually thinking about it. Thanks! Oleg Alexandrov 02:56, 4 Apr 2005 (UTC)

I would appreciate in general more feedback with my spelling. And sorry for "polluting" your watchlist, I believe quite a bit of my bot changes show up there. Oleg Alexandrov 03:04, 4 Apr 2005 (UTC)

I feel rather foolish for asking this...

Latest comment: 19 years ago5 comments4 people in discussion

...but I can't figure out the logic behind the statement, "if √2=m/n, then √2=(2n-m)/(m-n)." Can someone derive that, or point me to another site that has the derivation? --Jay (Histrion) 16:50, 26 October 2005 (UTC)Reply

One way is by algebra: If √2 = m/n, then n√2 = m, and then we have

{2n-m \over m-n}={2n-n{\sqrt {2}} \over n{\sqrt {2}}-n}={n{\sqrt {2}}({\sqrt {2}}-1) \over n({\sqrt {2}}-1)}={\sqrt {2}}.

Well, I feel silly now — a straightforward substitution. I was trying to derive the expression, when I could have just simplified it. As Strong Bad might say, "Holy crap!" --Jay (Histrion) 20:17, 26 October 2005 (UTC)Reply

Another way is by geometry: if m is the diagonal, and n the side, of an isosceles right triangle, then by a simpler ruler-and-compass construction one creates a smaller isosceles right triangle in which the the respective lengths of diagonal and side are 2n − m and m − n. Someone's added a diagram showing this to the article titled square root of 2. Michael Hardy 18:59, 26 October 2005 (UTC)Reply

Says we have a fixed point of the mapping x → (-x + 2)/(x - 1). Well, easy to check the fixed points are √2 and −√2. Charles Matthews 19:32, 26 October 2005 (UTC)Reply

Actually, that brings up a good question -- how much of the material in this entry is repeated in that √2 entry? Should we look at merging them -- or, alternately, moving some text from this entry to the other? -- written and unsigned at 13:21, 26 October 2005 by Histrion

I would oppose the merger. The main article for irrational numbers is this one. As for the square root of two, it is an important enough example of irrational numbers to keep its own article. Oleg Alexandrov (talk) 23:28, 26 October 2005 (UTC)Reply