Talk:Golden ratio base

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Needs reference material. Adam McCormick (talk) 00:50, 21 March 2008 (UTC)Reply[reply]

Added section on rationals, other revisions[edit]

Just added the section on rationals, which includes a comment I will repeat here:

Someone could expand on this by tidying up the next section. Basically the trading algorithm works fine, but the trade has to rely on φ2 = φ + 1

The example I have given on long division is probably good enough to be referred to in the following section, but I will leave that to someone else.

Another change I will probably make when I next return is to change "Numbers" to "Numerals" in several places, and change "convert" to "represent". The reason for this is that 10.01φ is still the same number as 2, it is just represented in a different numeration system. AndrewKepert

Aargh User:Dysprosia is trying to edit at the same time as me -- worthwhile changes, but I would prefer to finish so that the content I am trying to add sticks! --AndrewKepert 01:45, 19 Sep 2003 (UTC)

Sorry about that. I'm going for a little while, so it'll be all yours :) Dysprosia 01:46, 19 Sep 2003 (UTC)

Not a problem - the original page was horrible - way too much colloq stuff (unencyclopaedic as you put it), so I was doing a quick hack fixing up factual errors, omissions (e.g. recurring expansions, division) before letting it loos on someone else. Trouble was I was doing it section by section... --AndrewKepert 01:50, 19 Sep 2003 (UTC)

Okay - going home now. Thanks for fixing up the minor typos - I would have got to them after my first run. I do "preview" but don't edit in one session - especially if the server/connection is flaky. This edit session was a bit too much clashing and crashing (server down from what I could see). --AndrewKepert 08:37, 19 Sep 2003 (UTC)

p.s. for the record, here are today's changes, against the current TOC is

Intro (rewritten AndrewKepert)
1 Writing a φ-base number in standard form (original, maybe minor edits)
2 Representing integers as Golden mean base numbers (no change)
 2.1 Non-uniqueness (new)
3 Representing Rationals as Golden mean base numbers (new - what got me started)
4 Addition, subtraction, multiplication (rewritten)
 4.1 Calculate then convert to standard form (was old section 4)
 4.2 Avoiding digits other than 0 and 1 (a new point of view)
5 Division (rewritten)
6 A Close relation: Fibonacci Representation (new)
7 External Links (orig)
I like these changes made by Andrew with the sole exception of the base representation images added here and to a few of the other base articles. They are interesting computer-generated "art" of sorts, but I don't think they really enhance understanding (and would take more explanation than it's worth for any unfamiliar person to understand their meaning). Daniel Quinlan 04:31, Sep 20, 2003 (UTC)
Responded to this on talk:binary --AndrewKepert 05:12, 20 Sep 2003 (UTC)

Pi in Phi-base[edit]

hello, I see you gives representation of 1/2, √5.. in phi-base (so called "golden-base"), I really would like to see PI and e representation in phi-base. I've no skill for doing that myself, I had an old teacher in the past that was a fan of phi, and explained string relation between phi, pi and e. (this teacher is dead now, I know anyone who can help me). I'm sure seeing pi representation in golderbase will show the key :). regards.

OK, I've added π and e ... what's the relationship? I know there is a connection between π and e described at Euler's identity. But I need a little help seeing the connection with φ (phi). -- 05:11, 17 January 2007 (UTC)Reply[reply]

Not the same as Fibonacci coding!!![edit]

The edit summary of a recent change to this page states that golden ratio base and Fibonacci coding is the same thing, and that the articles should be merged. This is not correct - while they are connected, they are clearly different. So please don't merge!--Niels Ø 13:33, Apr 1, 2005 (UTC)

At first I was misled into thinking they were the same ... but now I see the difference.

   bit pattern
           interpreted as fibonacci number / Zeckendorf number
               interpreted as base φ
       1   1   1
      10   2   1.618 0...
     100   3   2.618 0...
     101   4   3.618 0...
    1000   5   4.236 1...
    1001   6   5.236 1...
    1010   7   5.854 1...
   10000   8   6.854 1...
   10001   9   7.854 1...
   10010  10   8.472 1...
   10100  11   9.472 1...
   10101  12  10.472 1...
  100000  13  11.090 2...
  100001  14  12.090 2...
  100010  15  12.708 2...
  100100  16  13.708 2...
  100101  17  14.708 2...
  101000  18  15.326 2...
  101001  19  16.326 2...
  101010  20  16.944 3...
 1000000  21  17.944 3...
 1000001  22  18.944 3...
 1000010  23  19.562 3...
...85321 (fibonacci value)

At first I thought that interpreting a bit pattern as a base phi value, then rounding to the nearest integer, would give the same integer as interpreting the bit pattern as a Fibonacci number / Zeckendorf number. (Similar to the same way that some people use a simplified version of Binet's formula, then round to the nearest integer, to get the n'th Fibonacci number).

It's very close -- since the ratio of the (n)th place to the (n+1)th place in base φ is *exactly* φ, while it is only *approximately* φ in fibonacci numbers (but increasingly accurate at high n).

I suspected that fiddling around with adding or subtracting 1, rounding or using floor and ceiling functions, I could make a function that converted the base phi value (in the last column) into the Fibonacci integer (in the middle column). I could get it to work from 1 to 6, or 6 to 14, or 14 to 20 ... but I can't seem to get it to work over the whole range. Does Binet's formula hold the key? -- 05:11, 17 January 2007 (UTC)Reply[reply]


I copyedited this article, mostly to make the terminology internally consistent and consistent with related WP articles.

  1. Changed golden mean to golden ratio throughout because (1) that is the term used in the title of this article; (2) that is the title of the underlying golden ratio article; and (3) golden ratio has only one meaning, the one used in this article, while golden mean is ambiguous: see Golden mean (a disambiguation page) and Golden mean (philosophy) (the other meaning). This reflects the consensus after extensive discussion at Wikipedia:Articles for deletion/Golden Mean.
  2. Wikify by using lowercase "g" in golden (none of these terms is a proper noun).
  3. Used "base-φ" throughout, to be internally consistent and consistent with the usages base-n and base-10.
  4. Changed the link Fibonacci representation to Fibonacci coding and used the latter term throughout. Fibonacci representation is merely a redirect page to Fibonacci coding. Also deleted the See also section because the only entry was Fibonacci coding, which became redundant when that link was used in the body of the article.

Please visit my Talk page if you have any comments or questions. Finell (Talk) 06:25, 18 October 2005 (UTC)Reply[reply]

An anonymous user has proposed merging Zeckendorf's theorem into the golden ratio base article. I oppose this merge, because the articles discuss two different concepts. For example, the golden base representation of 6, which is 1010.0001, is completely different from its Zeckendorf representation, which is 5+1. The "base" of Zeckendorf representations is the Fibonacci sequence, not the golden ratio. Gandalf61 09:48, 1 February 2006 (UTC)Reply[reply]

No-one has supported the merge proposal (or shown any interest at all !), so I will remove it in a few days, unless anyone objects. Gandalf61 11:48, 10 February 2006 (UTC)Reply[reply]
No objections (or any other response) so I have removed the merge proposal. Gandalf61 09:28, 14 February 2006 (UTC)Reply[reply]

Non-standard positional numeral systems[edit]

I have addressed certain issues by creating the article Non-standard positional numeral systems, and making related changes to Unary numeral system, Golden ratio base, Quater-imaginary base, Positional notation, Base (mathematics), and Category:Positional numeral systems. I suggest further discussion of these issues takes place at talk:Non-standard positional numeral systems.--Niels Ø 14:31, 26 February 2006 (UTC)Reply[reply]


I have removed the squared brackets in the following paragraph, added by user:, from the article:

Just as with any base-n system, numbers with a terminating representation have an alternative recurring representation [is this really true?]). In base-10, this relies on the observation that if x = 0.9999... then 10x = 9.99999... so that 9x = 9 and x=1. [This is not correct use of the operation + and x with a recurring 0.9999..., which are not defined for this.]

To answer the questions raised, it is true that e.g. 0.31827 is equal to both 0.31827000... and 0.31826999... . The example given in the text (1=0.999...) can also be argued by considering this "calculation": 1 = 9/9 = 9 × 1/9 = 9 × 0.111... = 0.999... . A formal mathematical argument (i.e. a proof) can also be given, though it is more complicated. A decimal number 0 . d1 d2 d3 ... has the value d1×10-1 + d2×10-2 + d3×10-3 + ..., i.e. or . Applying this formula to 0.999... yields the limit 1.--Niels Ø 01:16, 5 April 2006 (UTC) --- See Proof that 0.999... equals 1--Niels Ø 16:23, 21 April 2006 (UTC)Reply[reply]


is it usefull for anything or just a way of killing time for mathematitians? --Wan30ate 18:13, 3 June 2006 (UTC)Reply[reply]

The article definatly needs at least a couple or paragraphs explaining the meaning / uses / usefulnesss / motivation for the Golden Base.

The mechanics of HOW the base works are explained clearly and in detail. But not a single word on WHY such a weird base exists. —Preceding unsigned comment added by (talk) 04:23, 1 April 2011 (UTC)Reply[reply]

An established truth/fact in mathematics need NOT be accompanied by a motivation or an example of use. I thought everybody understood that. I.e. complex numbers had no "use" until the introduction of AC-electricity.Laelele (talk) 10:16, 5 November 2012 (UTC)Reply[reply]
Mm, they're useful in trigonometry. —Tamfang (talk) 10:23, 5 November 2012 (UTC)Reply[reply]

Which is still "within" matematics and hence just a way of killing time for mathematitians. If with "useful" we mean something outside of mathematics I think my example with AC-electricity stands. But why on earth would we at all consider uses?Laelele (talk) 11:03, 5 November 2012 (UTC)Reply[reply]

I assure you that trigonometry has applications in the real world. —Tamfang (talk) 19:25, 5 November 2012 (UTC)Reply[reply]

Probably the founder of this paragraph "use?" aimed to tease for some discussion. Suggesting it is killing time is a subtle way of judging the freedom of thought. Often history will judge different about initial useless issues. Usefulness and efficiency as a demand, sometimes kill creativity. This Golden ratio base number-system is just a spin off of the human mind and probably there are philosophers who can make more of that.

At the moment, there is a small chance this subject can help factorizing big integer numbers. More modest but with a reasonable change it will help demystify some other “useless” issues within math. Anyway it has a small place in the Wiki-pages and there is a big change it will inspire a few people also in the years to come.

Read these articles,(proof bpm and compensation) maybe the reader can help promote this Golden ratio base number-system getting out of the current “useless” status. --Swiersma (talk) 17:57, 28 January 2016 (UTC)Reply[reply]

Oh, I see the problem[edit]

The introduction to this article needs to make clearer that it isn't going to consider non-standard representations as valid. Theoretical studies of non-integer bases often do consider all representations -- see [1] for a very relevant example, particularly the bottom of p.5 -- in which case almost every number has infinitely many base-phi representations, not just one or two. Melchoir 16:54, 26 September 2006 (UTC)Reply[reply]

irrational number bases[edit]

The article claims that "Despite using an irrational number base, all integers have a unique representation as a terminating (finite) base-φ expansion." as if it were a fact worth mentioning.

But that is also true of base-sqrt(2), base-sqrt(3), base-sqrt(5), base-sqrt(6), base-sqrt(7), and base-sqrt(n) in general, right? And quater-imaginary base ?

What irrational number base does *not* represent integers as a unique finite-length series of digits?

What irrational number bases are worth mentioning in this encyclopedia?

I think the notable thing with the golden ratio base that I haven't encountered in any other base (is this really what the above sentence is trying to say? Is there a better way of saying this?) is that representing an integer requires a finite, but non-zero, amount of places after the "decimal" point. -- 05:11, 17 January 2007 (UTC)Reply[reply]

I’m pretty sure the base-e number system cannot represent any integer (other than zero, one, or negative one) as a finite-length series of digits. — Ti89TProgrammer 20:35, 30 July 2007 (UTC)Reply[reply]


Can someone please create a section about the significance of this numbering system. Particulary, can someone demonstrate how a phi-based numbering system is advantageous when solving a particular problem. It would be helpful to have some sort of walkthrough. It's an interesting, but purely academic article if you're into mathematics, which I'm sure anyone reading this is, but I don't see how this can be applied from the article or what value it might have to a non-mathematician. Thanks. 23:43, 1 April 2007 (UTC)Reply[reply]

George Bergman[edit]

Why does this article not mention George Bergman at all? He invented this system 50 years ago. Read "A Number System with an Irrational Base." Math. Mag. 31, 98-110, 1957. He didn't call it "phinary", but rather "tau". -- 00:01, 11 June 2007 (UTC)Reply[reply]

I just added this to Requests for expansion. — Ti89TProgrammer 20:22, 30 July 2007 (UTC)Reply[reply]

Writing golden ratio base in rational form[edit]

Excuse me, but the equality 010φ = 101φ is uncorrect, because
010φ = 0x(2+1Φ) + 1(1+1Φ) + 0x(1+0Φ) = 1+1Φ;
101φ = 1x(2+1Φ) + 0(1+1Φ) + 1x(1+0Φ) = 3+1Φ,
and 1+1Φ is not equal to 3+1Φ.

Forgive me for every grammatical error, i'm not english ; ) . ITA32 16:54, 26 September 2007 (UTC)Reply[reply]

The article actually says 010φ = 101φ. The underlines 1 are important because they mean -1, not 1. Starting from Φ2 = Φ+1 we have -Φ = -Φ2+1, so 010φ = 101φ. Gandalf61 20:33, 26 September 2007 (UTC)Reply[reply]

OK, now i understand the equality and the following example
ITA32 15:29, 28 September 2007 (UTC)Reply[reply]


Has been over 6 months. Has been moved to talk page. Please expand soon. Vinh1000 (talk) 12:12, 5 July 2008 (UTC)Reply[reply]


It seems that the page is the same as this page. —Preceding unsigned comment added by (talk) 19:28, 31 March 2009 (UTC)Reply[reply]

No, its the other way around. The page on says at the bottom that the page was copied from Wikipedia. Finell (Talk) 00:16, 1 April 2009 (UTC)Reply[reply]

Rephrased something because:[edit]

"all non-negative integers have a unique representation as a terminating (finite) base-φ expansion, but only if in the standard form." -- this seemed to me to be implying that non-negative integers DO NOT have a terminating representation other than their standard form one, which is false. (6 can be written as 1001.1001, which is not standard form). I'm not sure what the only meant, so I also took that out. Gabiteodoru (talk) 16:41, 2 January 2011 (UTC)Reply[reply]

Standard form to early in the article[edit]

If you wait a while with the introduction af standard form an elaborate with just 1 and 0 in ANY configuration first I think the article will flow better. For instance under non-uniqueness if you let go of "standard form" (decimal)2 = 10.01 = 1.11 = 1.1011 = ... You don't need recurring representation. I think phinary numbers are quite (not entirely) unique in being non-unique, and from that example anyone can see the "uniqueness in the non-uniqueness". It says in the article: "211.01φ is not a STANDARD base-φ numeral, since it contains a "11" and a "2", which isn't a "0" or "1", and contains a 1=-1, which isn't a "0" or "1" either." I would say that it is not even a base-φ numeral. It's a bit like saying that w28tr# is not a base-8 numeral. And again wait a while with the no-11-sequence rule for standard form and see what you can do from there.Laelele (talk) 10:55, 5 November 2012 (UTC)Reply[reply]

At they use the term "minimal form" for least number of 1:s or standard form (no consecutive 1:s) and "maximal form" for most number of 1:s(no consecutive 0:s). But as you can see from the above example with (decimal)2 there are more possibilites.Laelele (talk) 11:21, 5 November 2012 (UTC)Reply[reply]

If we for a while forget about the special "dialect" of Fibonacci number-base called "Fibonacci coding" and think of a number system with the LSB to the right. And if we drop the "11 is verboten" in both the Fibonacci- and the phi-based systems we see some properties in common:

1. The representation is not unique. 2ten = 10.01φ = 1.11φ = 1.1011φ = ... and 16ten = 100100Fib = 11100Fib = 11011Fib.
2. ...100... = ...011... in both the Fibonacci- and the phi-base.
3. ...0200... = ...1001... in both the Fibonacci- and the phi-base.Laelele (talk) 13:02, 14 November 2012 (UTC)Reply[reply]


"It is possible to mix base-φ arithmetic with Fibonacci integer sequences. The sum of numbers in a General Fibonacci integer sequence that correspond with the nonzero digits in the base-φ number, is the multiplication of the base-φ number and the element at the zero-position in the sequence. For example: product 10 (10100.0101 base-φ) and 25 (zero position) = 5 + 10 + 65 + 170 = 250"


Not sure if any of the numerous topics at Generalizations of Fibonacci numbers is supposed to help. DAVilla (talk) 21:02, 25 April 2021 (UTC)Reply[reply]