I'm not a math person so please bear with me. This number called The
Golden Ratio entered my field of view and I took a brief interest. A
decimal approximation of the number for those who don't know it is:
1.618033988749895D0
It is calculated by taking two adjacent Fibonacci numbers and making a
ratio.
A neat thing about the number is seen when you take the inverse:
0.6180339887498948D0
The rounding error screws it up a bit. Call the Golden Ratio G, then G
satisfies this equation:
1/G = G - 1
I think this makes G and algebraic number. Since an infinite number of
terms are required to calculate it, I would think that this makes it
irrational. Am I right about this?
Another approximation from running my Fibonacci generator a number of
terms out:
4114000911454431885883343305337966369078499341559272017 /
2542592393026885507715496646813780220945054040571721231
(- (/ 1 G) (- G 1))
-1 /
10460227422369712082430101139343210978604418044476524004 <line-wrap>
619598073127602788214312390478193638313755137023092927
That number isn't too far off of zero.
This post was brought to you by a lack of sleep.
David Steuber wrote:
> I'm not a math person so please bear with me. This number called The
> Golden Ratio entered my field of view and I took a brief interest. A
> decimal approximation of the number for those who don't know it is:
>
> 1.618033988749895D0
>
> It is calculated by taking two adjacent Fibonacci numbers and making a
> ratio.
>
> A neat thing about the number is seen when you take the inverse:
>
> 0.6180339887498948D0
>
> The rounding error screws it up a bit. Call the Golden Ratio G, then G
> satisfies this equation:
>
> 1/G = G - 1
>
> I think this makes G and algebraic number. Since an infinite number of
> terms are required to calculate it, I would think that this makes it
> irrational. Am I right about this?
>
The golden ratio is (/ (+ 1 (sqrt 5)) 2), so yes it is irrational. The
reason that it is called a ratio has to do with the idea of the golden
section. _The Concise Oxford Dictionary of Mathematics_ says:
A line segment is divided in golden section if the ratio of the whole
length to the larger part is equal to the ratio of the larger part to
the smaller part.
Suppose the smaller part has unit length and the larger part has length
t. Then (t + 1)/t = t / 1, or t^2 = t + 1. Here t is your G, and you
stumbled across a related equation above.
According to SICP, pg. 38, Fib(n) is the closest integer to G^n / 5^(1/2).
Winston's _Lisp_ (2e) pg. 72 also introduces Binet's formula for
determining Fibonacci numbers:
f(n) = 1/(5^(1/2)) (G^(n+1) - G'^(n+1)]
where G = (/ (+ 1 (sqrt 5)) 2) and G' = (/ (- 1 (sqrt 5)) 2)
David Sletten
"David Steuber" <·····@david-steuber.com> writes:
> I think this makes G and algebraic number. Since an infinite number of
> terms are required to calculate it, I would think that this makes it
> irrational. Am I right about this?
Yes. Try to multiply the equation with G, and you end up with a regular
quadratic equation, whose only positive solution is
G= (/ (+ (sqrt 5) 1) 2)
PS: The mathematica pages are always a good resource for math
self-education:
http://mathworld.wolfram.com/GoldenRatio.html
--
(espen)
"David Steuber" <·····@david-steuber.com> writes:
> I'm not a math person so please bear with me. This number called The
> Golden Ratio entered my field of view and I took a brief interest. A
> decimal approximation of the number for those who don't know it is:
> 1.618033988749895D0
>This post was brought to you by a lack of sleep.
I think even the ancient Greek knew that the Golden Ratio is
(/ (1+ (sqrt 5)) 2) -- however expressed in ancient greek
so they knew it's irrational, and they had geometric means to
construct it (witness their temples, but I bet you know that already).
>It is calculated by taking two adjacent Fibonacci numbers and making a
>ratio.
I'm sure that's not its definition. Rather, one can proove that the
limit of this series of numbers happens to be the number known as Golden Ratio.
(Maybe I even did that once in my math studies :-)
Regards,
Jorg Hohle
Telekom/T-Systems Technology Center
[Joerg Hoehle <······@users.sourceforge.net>, 03 Mar 2005 10:51:21 +0100]:
> I think even the ancient Greek knew that the Golden Ratio is
> (/ (1+ (sqrt 5)) 2) -- however expressed in ancient greek
> so they knew it's irrational, and they had geometric means to
> construct it (witness their temples, but I bet you know that already).
As for the temples or for the idea that it is the most pleasing ratio,
this does not seem to be that clear:
<http://www.maa.org/devlin/devlin_06_04.html>
Albert.
"David Steuber" <·····@david-steuber.com> writes:
> 1/G = G - 1
>
> I think this makes G and algebraic number. Since an infinite number of
> terms are required to calculate it, I would think that this makes it
> irrational. Am I right about this?
Yes. PI too is an irrational ratio (circumference/diameter).
--
__Pascal Bourguignon__ http://www.informatimago.com/
The world will now reboot. don't bother saving your artefacts.
On Thu, 03 Mar 2005 13:02:01 +0100, Pascal Bourguignon wrote:
> "David Steuber" <·····@david-steuber.com> writes:
>> 1/G = G - 1
>>
>> I think this makes G and algebraic number. Since an infinite number of
>> terms are required to calculate it, I would think that this makes it
>> irrational. Am I right about this?
>
> Yes. PI too is an irrational ratio (circumference/diameter).
Phi's irrationality:
http://en.wikipedia.org/wiki/Golden_ratio#A_startlingly_quick_proof_of_irrationality
I love wikipedia :)
David Steuber wrote:
> I'm not a math person so please bear with me. This number called The
> Golden Ratio entered my field of view and I took a brief interest. A
> decimal approximation of the number for those who don't know it is:
...
> I think this makes G and algebraic number. Since an infinite number of
> terms are required to calculate it, I would think that this makes it
> irrational. Am I right about this?
You may already have seen this in your tour through the
web pages others have pointed you at. But, in a certain
sense, the "golden ratio" is either the least rational
number there is, or the least irrational irrational number
there is.
"Least rational": Think about trying to approximate it
by rational numbers. The best you can do, it turns out,
is to use the ratio of adjacent Fibonacci numbers. When
you do that, the error you get is close to 1 / sqrt(5)q^2
where q is the denominator of your rational approximation.
Well, it so happens that you can approximate *any*
irrational number by rational numbers with an error
that's no worse than about 1 / sqrt(5)q^2. And that
for any number other than the golden ratio and some
of its close friends[1], you can do strictly better;
i.e., you can use a larger constant than sqrt(5)
in the denominator.
[1] Those whose continued fraction ends with
an infinite string of 1s. That means only numbers
of the form (a+bg)/(c+dg) with a,b,c,d integers
and ad-bc = 1, or something like that.
So the golden ratio is harder to approximate with
rational numbers than any other irrational number
(other than a few that are basically exactly as
hard as the golden ratio). Hence, "least rational".
"Least irrational": That business about the continued
fraction of the golden ratio can be viewed as saying
that the golden ratio is as simple as an irrational
number can be. Similarly, look at the quadratic equation
it satisfies: x^2-x-1=0. You can't get any simpler than
that and still have a real solution. (I guess that's
a matter of definition; some people would say that
x^2-2=0 is simpler. I wouldn't.)
Continued fractions are way cool. If you enjoy any
of this stuff, you should find out about them.
--
Gareth McCaughan
.sig under construc
David Steuber schrieb:
> I'm not a math person so please bear with me. This number called The
> Golden Ratio entered my field of view and I took a brief interest....
> This post was brought to you by a lack of sleep.
Hi there.
There was also a sleepless night and here is the event that happens.
(I called it "plus-minus 42")
42 / 0 = X
43 / 7 = 6,7428577 41 / 7 = 5,8577429
44 / 14 = 3,2742857 40 / 14 = 2,8577429
45 / 21 = 2,7428577 39 / 21 = 1,8577429
46 / 28 = 1,6428577 38 / 28 = 1,3577429
47 / 35 = 1,3428577 37 / 35 = 1,0577429
48 / 42 = 1,1428577 36 / 42 = 0,8577429
49 / 49 = 1 35 / 49 = 0,7142857
50 / 56 = 0,8925857 34 / 56 = 0,6077429
51 / 63 = 0,8095238 33 / 63 = 0,5238095
52 / 70 = 0,7428577 32 / 70 = 0,4571429
53 / 77 = 0,6883117 31 / 77 = 0,4025974
54 / 84 = 0,6428571 30 / 84 = 0,3571429
... ...
... once the word reckon and scribe means the same.
stefan