Further thoughts on the above: Let [symbol]a[/symbol] be any irrational number. Consider rational approximations to [symbol]a[/symbol], written a/b. Since [symbol]a[/symbol] is sandwiched between two consecutive fractions a/b, it is obvious that we can approximate [symbol]a[/symbol] by a fraction a/b so that there is an error of less than 1/2b. How much better than this can we do? The above considerations work for every denominator b; if we are allowed to choose the denominator, we can do considerably better. In particular, consider approximations which satisfy inequalities of the form
|[symbol]a[/symbol] - a/b| < 1/Ab[sup]2[/sup] (*)
Then we have:
(i) For every irrational [symbol]a[/symbol], there are infinitely many rational numbers a/b which satisfy () when A = sqrt(5).
(ii) If A is any number greater than sqrt(5), there is an irrational number [symbol]a[/symbol] such that () has only finitely many rational solutions. The significance to this thread is that we can take [symbol]f[/symbol] as such an irrational.
The property of [symbol]f[/symbol] which is fundamental here is its continued fraction expansion. Any number which has the same expansion from some point on, i.e. any number whose continued fraction expansion is eventually all 1s, can be used instead of [symbol]f[/symbol] in (ii). However,
(iii) If [symbol]a[/symbol] is any irrational number whose continued fraction expansion is not eventually all 1s, then we can find infinitely many rational numbers a/b which satisfy (*) when A = sqrt(8).
I’m glad osmeone epxlained that hyruu stuff. I tried searchign for phi… but of course a 3 letter word is too short for a search. Soo I tried golden ratio but nothing good came up. So,what i gather is that its not important to any fundamental equations? I have heard from several places that phi has some importance in the shape of galaxies? Is that just another result of Fibonacci numbers? So, might it be possible that aliens taught the ancient Egyptians about phi so they could build the pryamids? Then the Egyptians could harnass love power and… no I’m just kidding (except for the galaxies part)… I have no idea who this hyruu character is… although I am looking out for mysterious private messages “sooo… they’ve gotten to you too?”
Just in case anyone fancies a look at one of Hiyruu’s threads on the subject:
(PLEASE DO NOT POST IN THISLINKED THREAD; IT IS LONG DEAD AND BETTER OFF THAT WAY): Has E=MC^2 been replaced by Gravity=Light*PHI^n?
Ok, you asked. Unfortunately, I am not very good at doing mathematics here. But the sense is the error compared to the denominator. Let x be a number. Fix a positive integer Q and ask the question, among all numbers q =< Q find the rational fraction p/q (by a rational fraction is just the ratio of two integers and here I assume that p and q are always integers) for which the error |p/q - x| is as small as possible. IIRC, it can always be made =< 1/(sqrt(5)Q^2) and, except for phi and its close relatives, it can always be made < that bound. So phi is the hardest to approximate in this sense. I could have this wrong, since I haven’t looked at a number theory book in years, but it is something like that.
Back when I was an undergrad, I used to make a pest of myself on this topic. It’s one of those things you think is really cool when you first find out about it. One researcher actually claimed that a study of women, measuring the ratios of their heights to the heights of their navels yielded the golden ratio.
There was, and apparently still is, a small journal devoted to Fibonacci numbers and related topics:
(Of course, since the golden ratio is related to things like log spirals and fibonacci sequences, there are any number of naturally occuring processes that cause the number to keep turning up in odd places.)
Ok, what I said above was not quite right. Here is TSD. I take this from W.J. LeVeque’s Topics in Number Theory and this is on page 187 of the 1956 edition. (I took the course in 1957, BTW.)
To begin, say of two numbers x and y that x ~ y if for integers a, b, c, and d for which |ac - bd| = 1, we have x = (ay + b)/(cy + d). One easily shows that this relation is an equivalence relation (which I won’t define since if you don’t understand that, you are likely not reading this anyway). Anyway, it means that x is just as hard to approximate as y. Then given any number x there are infinitely many pairs of integers p and q for which |x - p/q| < 1/(sqrt(5)q^2). Moreover, either x ~ phi or there are infinitely many pairs of integers p and q for which |x - p/q| < 1/(sqrt(8)q^2) and both of these inequalities are best possible. In particular, you cannot replace sqrt(5) by any larger number in the estimate for phi, so its rational approximations are worse than those of any number not equivalent to it by a factor of at least sqrt(8/5).
