Talk:Square root of 2

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[edit] Historical evidence

Certainly this number seems to be widely believed by mathematicians to be the first known irrational number. But what is the historical evidence? Michael Hardy 23:25, 15 Apr 2005 (UTC)

I suppose you mean "the first number known to be irrational". I guess it's hard to document that something is really a "first" like that. I've seen quasi-serious speculations suggesting the golden ratio was the first number known to be irrational. Both numbers were known, in geometrical form, to the Pythagoreans, who were fond of the pentagram (full of golden ratios). I suppose the squareroot of 2 is just the most likely candidate. Anyway, reliable historical evidence (sources) seems to be a problem with most things involving the Pythagoreans.--Niels Ø 17:19, 29 October 2006 (UTC)

[edit] Usage?

As I see it,

Every number except 1

means something different from

Every number except one

Accordingly, I think this page should be called square root of 2.

Michael Hardy 23:25, 15 Apr 2005 (UTC)

Agreed. Fredrik | talk 23:53, 15 Apr 2005 (UTC)

[edit] Image title

As from the above disussion, normally any number which is written less than (<) 10 is written in its letter form; and anything that is written greater than (>) 10 is written as their actual number. So what made you change it to "Square root of 2". I find it misleading. --Kilo-Lima 17:08, 12 November 2005 (UTC)

That may be the convention in journalism, but not in mathematics. 84.70.26.165 11:46, 29 October 2006 (UTC)
Indeed. In fact, I think even most journalistic style guides prescribe the use of the numeral when referring to the number itself instead of to a quantity. —Caesura(t) 01:43, 31 October 2006 (UTC)

[edit] Redirect Link

Down at the bottom where I added something about silver means, the redirect link from silver means goes to the Plastic Number article, I think that it would be much more useful to have it go to the article about the Silver Ratio --Carifio24 15:58, 7 July 2006 (UTC)

[edit] Factual accuracy

Article :« The first approximation of this number was given in ancient Indian mathematical texts, the Sulbasutras (800 B.C. to 200 B.C.) as follows: Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth. »

The Babylonian clay tablet YBC 7289 (1700 ± 100 BCE) displays an approximation of √2 with an accuracy of 6 × 10-7 (1.24 51 10 in sexagesimal base). See for instance Square root approximations in Old Babylonian mathematics : YBC 7289 in context. The exact date of the Salbasutra is too imprecise to be the first approximation ever, even compared to 3/2 in Meno :-). Lachaume 21:55, 29 August 2006 (UTC)

Yes you are quite right, I've updated the article accordingly. Thanks. Paul August ☎ 20:04, 30 August 2006 (UTC)

[edit] Approximations

The article currently mentions the approximation 99/70. For certain applications (e.g. line widths of diagonals in bitmap art), this is a bit unwieldy. Perhaps it would be worthwhile to have a section stating the following common approximations and their errors:

7/5, -1.005%
10/7, +1.015%
17/12, +0.173%
24/17, -0.173%
41/29, -0.030%
58/41, +0.030%
99/70, +0.005%
140/99, -0.005%
239/169, -0.001%
338/239, +0.001%
et cetera...

These approximations actually form two sequences, and so can be extended to find even more accurate approximations - the rule being that if your last term was N/D, your next will be (N+2D)/(N+D) (there's probably a proof of this somewhere online). The sequences relate to eachother in that you can flip each fraction and double it (because we're approximating root-2). It may be worth noting that the errors of each term in one sequence is NOT actually the negative of the corresponding term in the other sequence (they're just very much in the same ballpark).

So should there be a section on approximations (I find them useful and interesting, but then again I'm biased)? A435(m) 15:24, 30 December 2006 (UTC)

What are you waiting for? Go ahead and put it in. Sympleko (Συμπλεκω) 01:17, 26 June 2007 (UTC)
Two things. First, why bother with 24/17 when it's no better than 17/12 yet involves larger integers? Second these ratios (less the suboptimal ones, namely every second item in your list) are what you get by taking only the first n terms of the continued fraction expansion given in the article. So the logical thing to do is simply list the values of these truncations after the expansion itself, namely the odd numbered lines of your table (which should start out 1/1, 3/2, 7/5, 17/12, ...) --Vaughan Pratt (talk) 23:14, 16 March 2008 (UTC)

[edit] much simpler proof

the proof in the article seems overly complex, a much simpler proof would be

  1. Assume that √2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = √2.
  2. Then √2 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)2 = 2.
  3. Clearly √2 is not an integer so b > 1
  4. If a and b are coprime then a2 and b2 must also be coprime since squaring introduces no new prime factors
  5. therefore a2/b2 must be an irriducible fraction with a denominator greater than 1.
  6. therefore a2/b2 cannot be 2 which contridicts the initial assumption

comments? Plugwash 22:31, 10 July 2007 (UTC)

Does this argument presuppose uniqueness of prime factorization, rather than only the simpler fact about divisibility by 2? Michael Hardy (talk) 01:05, 17 March 2008 (UTC)
I think you mean contradicts
Irreducible
DarkestMoonlight (talk) 19:37, 20 March 2008 (UTC)

[edit] "Positive"

"The square root of 2... is the positive real number that, when multiplied by itself, gives the number 2"

No, I may have only taken up to intermediate algebra, but I'm pretty damn certain that there's a positive and negative square root of 2.

—Preceding unsigned comment added by Mqduck (talk • contribs)

True, but when we talk about "the" square root of two, as a real number, we mean the positive one. —David Eppstein 15:05, 7 October 2007 (UTC)

[edit] Generalized proof

From the article: This proof can be generalized to show that any root of any natural number is either a natural number or irrational. Where can I find such a proof? --Steerpike (talk) 19:06, 18 December 2007 (UTC)

Take the proof by infinite descent and plug in \sqrt{n} for any positive integer n in place of \sqrt{2}. --69.91.95.139 (talk) 02:15, 26 January 2008 (UTC)
The most obvious proof is to start from the other direction. The square of a non integer rational must be a non integer rational because squaring introduces no new prime factors to either the numerator or the denominator. Therefore the squre root of an integer must be either an integer or irrational (since for it to be a non integer rational would be a contradiction). Plugwash (talk) 02:29, 26 January 2008 (UTC)
Oh, interesting. Never thought about it that way. You learn something new every day. --69.91.95.139 (talk) 03:15, 6 February 2008 (UTC)

[edit] Suggested merge

The overlap with Irrational number is so great (strong emphasis there on sqrt(2)) that this article could be merged with it with negligible loss, leaving just a redirect at this article. The bit about continued fractions can be generalized to the observation that every positive algebraic number has a periodic branching continued fraction expansion (2006 observation of N.R. Zakirov), with the quadratic irrationals such as sqrt(2) being exactly the nonbranching periodic continued fractions (sqrt(2) as a simple example). --Vaughan Pratt (talk) 23:40, 16 March 2008 (UTC)

If there's some need to merge this with another article, wouldn't silver ratio be the more obvious choice? —David Eppstein (talk) 23:55, 16 March 2008 (UTC)