Golden irrationality
Brian Chandler
asserts:
"Number theorists have long known that the most irrational number is
the golden number. It is 'badly approximable' by rational numbers, and
if you quantify how badly it's the worst of them all."
He's talking about sunflowers, and the "golden number" is phi =
(root(5)-1)/2 = 0.6182...
OK, but he doesn't explain further: what is this "most irrational"?
John Savard
Chandler) wrote, in part:
The reason this is said is that the nicest approximations by fractions
to a number can be found by continued fractions; thus, pi becomes 3
1/7 and 3 16/113 from the expansion
1
3 + -----------
1
7 + ----
16 ...
As it happens, the golden number, 1.6182... is
1
1 + ---------
1
1 + ---
1 ...
so there is no case where you have 1/(something big) in the expansion,
such that the approximation, with a small numerator and denominator,
is better than one might expect for one with such small numbers making
it up.
Virgil
imagin...@despammed.com (Brian Chandler) wrote:
An irrational cannot, by definition, be given exactly by a quotient
of integers. But quotients of integers can come arbitrarily close to
any given irrational.
Given the set of rationals with denominators less than or equal to
some given integer, n, how close can you get to some given
irrational?
If f(n,r) = inf{|p/q - r| : |q| <= n} tends to be large for many
positive integers, n, then r is badly approximable, whareas if it is
small, then the number is easily approximable.
A "most irrational" number, r, would be one in which the values of
f(n,r) are collectively larger than for all other irrationals.
I am not sure that this characterization as "most "irational" has
been proved, but it is true that sqrt(5)-1)/2 is not easily
approximable in the sense above.
d...@cmtzone.net
(John Savard) wrote:
>On 27 Jan 2002 11:24:04 -0800, imagin...@despammed.com (Brian
>Chandler) wrote, in part:
>
>>"Number theorists have long known that the most irrational number is
>>the golden number. It is 'badly approximable' by rational numbers, and
>>if you quantify how badly it's the worst of them all."
>
>As it happens, the golden number, 1.6182... is
>
> 1
>1 + ---------
> 1
> 1 + ---
> 1 ...
>
>so there is no case where you have 1/(something big) in the expansion,
>such that the approximation, with a small numerator and denominator,
>is better than one might expect for one with such small numbers making
>it up.
Funny how the "most irrational" number is also supposedly the most
aesthetically pleasing to us! What does this say!?
More on the golden ratio, for those interested:
http://pauillac.inria.fr/algo/bsolve/constant/gold/gold.html
I -love- those pages on popular mathematical constants.
-don
Paul Pollack
Theorem (Hurwitz). Let alpha be an irrational number. Then for infinitely
many rational numbers p/q, we have the inequality
|p/q - alpha| < q^{-2}/sqrt(5).
This is best possible in the sense that there are irrational values of alpha
for which the constant sqrt(5) cannot be replaced by anything larger. One
such example is alpha=phi, which explains why phi is sometimes called the
most irrational number.
(For a proof, see Hardy & Wright or LeVeque's _Fundamentals of Number
Theory_ or Burger's _Exploring the Number Jungle_...)
For many irrationals the constant sqrt(5) _can_ be improved: for alpha
irrational, define M(alpha) to be the supremum of those numbers L for which
|p/q-alpha| < q^{-2}/L has infinitely many solutions. (So by Hurwitz's
theorem M(alpha) >= sqrt(5) for every alpha.) The larger the value of
M(alpha), the "better approximable" alpha is by rationals. Then we have the
following
Theorem: Say two irrational numbers are equivalent if their continued
fraction expansions agree from some points on. Then if alpha is equivalent
to alpha', M(alpha) = M(alpha'). For every irrational alpha, M(alpha) >=
sqrt(5) with equality iff alpha is equivalent to phi. For alpha not
equivalent to phi, M(alpha) >= sqrt(8) with equality iff alpha is equivalent
to sqrt(2). For alpha not equivalent to phi or sqrt(2), M(alpha) >= 17/6.
For more details as to what is going on here, look in Burger's book
mentioned above under Module 6 on the Markoff spectrum.
HTH,
Paul
"Brian Chandler" <imagin...@despammed.com> wrote in message
news:f2c35871.0201...@posting.google.com...
Michael Barr
I believe that he means that it is the number that is hardest to
approximate well with rationals. In comparing approximations, the
thing to look at is
|qx - p| where x is your irrational and p, q are integers. This is
the ratio of |x - p/q| to q and tells you how big a denominator you
need to get a good approximation. If you can get a good approximation
with a small denominator, then your irrational is well approximated.
Basically, the smaller the numbers (integers) in the continued
fraction expansion, the worse the approximation in that sense. Well,
the continued fraction expansion of the golden number is all 1's and
that is as bad as can be. For example, there are places in the CF
expansion of pi that are very large and that is where you get good
approximations to pi. The number (doubtless transcendental) whose CF
expansion is [1,2,6,...,n!,...] has very good rational approximations
in that sense. The theorem of Liouville says that any number with
very good approximations that is not actually rational is
transcendental. The full statement and a very easy proof is found in
Courant and Robbins. It depends only on the fact that a non-zero
positive integer is at least 1, as is much of transcendental number
theory.
JohnW
itself, or than any other square root, or other root (surd), for that
matter. Theorems on the irrationality or transcendalism of roots
(whether integral or not) of natural numbers (or non-integers) cannot
distinguish between values.
John W.
Brian Chandler wrote:
>
> On p. 141 of my copy of Ian Stewart's book "Nature's Numbers" he
> asserts:
>
> "Number theorists have long known that the most irrational number is
> the golden number. It is 'badly approximable' by rational numbers, and
> if you quantify how badly it's the worst of them all."
>
> He's talking about sunflowers, and the "golden number" is phi =
> (root(5)-1)/2 = 0.6180339887....
Oscar Lanzi III
expansion. Any truncation of the continued fraction expansion, if
simplified to a simple fraction, gives the closest approximation to the
number without using a higher denominator. If, for example, we take the
continued fraction for sqrt(2) and cut it off after three layers, we get
1+1/(2+1/2)), which is 7/5. Thus any rational approximation for sqrt(2)
that is more accurate than 7/5 has to have a denomiator greater than 5.
Now the higher your denominator, the better accuracy this guarantees for
your approximation. So in a continued fraction approximation you want
the denominator to go up as fast as possible. But for the golden ratio
the denominator goes up only slowly because the integers in the
continued fraction are all 1's. Thus for a given accuray, the golden
ratio requires more continued-fraction layers than any other irrational
number with a known continued-fraction expansion.
Not all irrationals have known continued fraction expansions. The
expansions have a simple structure only for quadratic surds and a few
odds and ends like e.
--OL
dnahlhytbhy
http://www.ams.org/new-in-math/cover/irrational2.html
Earle Jones
"dnahlhytbhy" <a@b.c> wrote:
> check this out:
>
> http://www.ams.org/new-in-math/cover/irrational2.html
*
Amazing! sqrt(2) is more irrational than pi.
earle
*
Brian Chandler
Thanks for the responses. (It's a relief to know that people have time
for other things in the midst of all this Harris dreck.)
I could have written the "Being *more irrational* is like being *more
pregnant*" response myself; nevertheless, I could see intuitively that
phi is "keeping away from simple fractions as much as possible" (as
Stewart explains w.r.t. the sunflower seeds), but not how to give this
a rigorous backing.
I'm still not clear on the claim that phi (= 1+ 1/(1+1/(1+... ) is
*the most* irrational. Surely, with a bit of hand waving and appeal to
symmetry, phi-1, or 2-phi is *equally* irrational, in that except for
the beginning the continued fraction is essentially the same? In
particular, phi-1 is just phi with "1+" missed off the front - surely
this could claim to be an even more elemental continued fraction?
Fwiw, I have many times looked at real sunflowers - which grow well
here (at the edge of the Kanto Plain) - but never been able to find
the numbers claimed. Is this just bad experimental wossname?
d...@cmtzone.net
Chandler) wrote:
>> OK, but he doesn't explain further: what is this "most irrational"?
>
>Thanks for the responses. (It's a relief to know that people have time
>for other things in the midst of all this Harris dreck.)
I ignore that thread in order to preserve what little sanity I have
left after perusing other threads.
>
>I could have written the "Being *more irrational* is like being *more
>pregnant*" response myself; nevertheless, I could see intuitively that
I've seen pregnant women that I think I would describe as "more
pregnant" than other pregnant women, and I won't even get into the
irrationality bit with regard to women. ;-) (even though I'd say I'm
of the belief that flame mail from women is better than no mail from
women!)
>phi is "keeping away from simple fractions as much as possible" (as
>Stewart explains w.r.t. the sunflower seeds), but not how to give this
>a rigorous backing.
>
>I'm still not clear on the claim that phi (= 1+ 1/(1+1/(1+... ) is
>*the most* irrational. Surely, with a bit of hand waving and appeal to
>symmetry, phi-1, or 2-phi is *equally* irrational, in that except for
>the beginning the continued fraction is essentially the same? In
>particular, phi-1 is just phi with "1+" missed off the front - surely
>this could claim to be an even more elemental continued fraction?
I see your point, but it seems to me that as you are writing in terms
of phi anyway, that is your fundamental constant. So while 1+phi
might be "equally irrational" or whatever (and the way it sounds from
your quote, Stewart is talking sort of loosely and not giving a
rigorous definition/result anyway), phi would be the important thing,
or perhaps even phi - 1, the fractional part of phi. Take a look at:
http://mathworld.wolfram.com/ContinuedFraction.html
It talks about continued fractions and more specifically simple
continued fractions, maybe it would help?
-don
Keith Ramsay
There's something to be said for reversing the metaphor here of
"more" irrational for "harder to approximate by rationals". In some
ways it's the numbers worst approximable by rationals that are most
like rationals.
If we consider approximation by rationals *distinct* from the original
number, then rationals are themselves worst approximable, because if
p/q <> m/n for integers m,n,p,q, then |p/q-m/n| > 1/nq. That's worse
approximability behavior than one gets for any irrational.
Then the next worst approximable ones are certain quadratic
irrationals such as (1+sqrt(5))/2 or sqrt(2). The next worst include
a mix of quadratic irrationals and other irrationals that just happen
to have bounded terms in their continued fraction expansions.
Anything that is much better approximable has to be transcendental.
(If there is an epsilon>0 and C>0 and infinitely many p/q such that
|r-p/q|<1/q^{2+epsilon} then it's transcendental by the Thue-Siegel-
Roth theorem.) The very best approximable ones are easiest to prove
transcendental (Liouville's theorem, much easier to prove, is enough).
Keith Ramsay