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So we all know that the continued fraction containing all $1$s...

$$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$

yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting quadratic equation and assuming that a continued fraction that only contains additions will give a positive number.

Now, a friend asked me what would happen if we replaced all additions with subtractions:

$$ x = 1 - \frac{1}{1 - \frac{1}{1 - \ldots}} $$

I thought "oh cool, I know how to solve this...":

\begin{align} x &= 1 - \frac{1}{x} \\ x^2 - x + 1 &= 0 \end{align}

And voila, I get...

$$ x \in \{e^{i\pi/3}, e^{-i\pi/3} \} $$

Ummm... why does a continued fraction containing only $1$s, subtraction and division result in one of two complex (as opposed to real) numbers?

(I have a feeling this is something like the $\sum_i (-1)^i$ thing, that the infinite continued fraction isn't well-defined unless we can express it as the limit of a converging series, because the truncated fractions $1 - \frac{1}{1-1}$ etc. aren't well-defined, but I thought I'd ask for a well-founded answer. Even if this is the case, do the two complex numbers have any "meaning"?)

Simon Fraser
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Martin Ender
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    As an aside, do continued fractions with subtraction instead of addition have a name? I feel like the term "continued fraction" is reserved the form that uses addition. – Martin Ender Mar 03 '16 at 20:48
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    Definitely an infinite continued fraction makes sense only under limit. It seems to suggest that this limiting procedure does not converge for real number input. It leaves a possibility that it converges for some complex number input. – Sangchul Lee Mar 03 '16 at 20:53
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    @MartinBüttner You got to be careful who you pick as your friends :) – imranfat Mar 03 '16 at 22:07
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    @imranfat Looking at this page, I couldn't ask for better ones. ;) – Martin Ender Mar 03 '16 at 22:08
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    “Why is $x$ complex“ is IMO not even a well-phrased question. A number is complex if it was introduced accordingly. If I start with “let $a\in\mathbb{C}$ such that etc etc, then do some maths and conclude that $a=3\pi$, it is still a complex number. What's interesting though is that it's _in the real subset_. Similarly, if you started out with some quaternion, then the question “is it complex” would make sense. But just juggling around algebraic expressions, using a solution technique _for quadratic equations in $ℂ$_ and then being surprised that the result is complex, is a bit ridiculous. – leftaroundabout Mar 03 '16 at 23:29
  • Yes, this is still considered a continued fraction; in general they can have complex partial numerators and partial denominators. *Simple* continued fractions are the ones whose partial numerators are all $1$. – J. M. ain't a mathematician Mar 03 '16 at 23:57
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    @leftaroundabout Of course the surprising part isn't that a quadratic equation can have complex roots, but that a (by analogy with the golden ratio case seemingly valid) way to transform a (seemingly) real expression results in a quadratic equation with complex solutions, just like you'd be surprised if you got [a negative rational result from summing natural numbers](https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Ramanujan_summation). My question wasn't "why am I getting complex roots for a quadratic" but "why do I seem to be able to deduct that this fraction is complex". – Martin Ender Mar 04 '16 at 08:56
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    ... with "complex" implying "complex and not real". – Martin Ender Mar 04 '16 at 08:57
  • somewhat related: https://en.m.wikipedia.org/wiki/Contraction_mapping – royhowie Mar 04 '16 at 17:20
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    So you know the method to find the fix-point of a sequence, but you didn't verify that there is a real fix-point in a first place. It's like throwing your fishing pole in to a random lake and expecting to fish a salmon without doing the homework and verifying that there is a salmon in that lake. As it happens, you got a trout instead.... – Uri Goren Mar 04 '16 at 17:46
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    @UriGoren Well, clearly I wasn't thinking of the solution in terms of a fixed point of a sequence. I just plugged the equation into itself and solved it. To me, having a sequence that would converge to the continued fraction was a completely separate thought which could be used a sanity check. I never considered $x = 1 - 1/x$ to mean $x_{n+1} = 1 - 1/x_n$ (so Patrick's answer has been quite enlightening). – Martin Ender Mar 04 '16 at 17:48
  • In 3-dimensional and 4-dimensional topology the only continued fractions that show up are ones with subtraction. They are usually called ... continued fractions. – PVAL-inactive Mar 04 '16 at 17:55
  • This is similar to what is called a [divergent series](https://en.wikipedia.org/wiki/Divergent_series). – ulatekh Mar 04 '16 at 22:50
  • It is not real, because it doesn't end. All real things in this universe end eventually. – hkBattousai Mar 10 '16 at 15:15
  • My first ever publication dealt with this topic: https://maa.tandfonline.com/doi/full/10.1080/07468342.2019.1534491 – Dark Malthorp Jan 25 '20 at 22:11

8 Answers8

530

You're attempting to take a limit.

$$x_{n+1} = 1-\frac{1}{x_n}$$

This recurrence actually never converges, from any real starting point. Indeed, $$x_2 = 1-\frac{1}{x_1}; \\ x_3 = 1-\frac{1}{1-1/x_1} = 1-\frac{x_1}{x_1-1} = \frac{1}{1-x_1}; \\ x_4 = x_1$$

So the sequence is periodic with period 3. Therefore it converges if and only if it is constant; but the only way it could be constant is, as you say, if $x_1$ is one of the two complex numbers you found.

Therefore, what you have is actually basically a proof by contradiction that the sequence doesn't converge when you consider it over the reals.

However, you have found exactly the two values for which the iteration does converge; that is their significance.

Alternatively viewed, the map $$z \mapsto 1-\frac{1}{z}$$ is a certain transformation of the complex plane, which has precisely two fixed points. You might find it an interesting exercise to work out what that map does to the complex plane, and examine in particular what it does to points on the real line.

Patrick Stevens
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    It's coincidental that this 'continued fraction' has such nice periodic nature. Incidentally, the 'original continued fraction' has non-trivial convergence behaviour, as I show at http://math.stackexchange.com/a/624037/21820. (No idea who downvoted my answer though it is the other one that is incorrect.) – user21820 Mar 04 '16 at 04:29
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    Fantastic answer! One point: it’s not quite a transformation of the complex plane; but rather of the *extended* complex plane $\mathbb{C} \cup \{\infty\}$, aka the Riemann sphere, or alternatively (more elementarily but less elegantly) of $\mathbb{C} \setminus \{ 0, 1 \}$. – Peter LeFanu Lumsdaine Mar 05 '16 at 21:15
  • @MartinBüttner: the Golden Ratio goes to $\phi - 2$, not to 1. From $\mathbb{C}_\infty$, one can exclude any orbit or union of orbits, and get another domain on which the transformation acts (totally and invertibly); and $\{0,1,\infty\}$ is a complete (cyclic) orbit, so $\mathbb{C}\setminus\{0,1\}$ gives a suitable domain. – Peter LeFanu Lumsdaine Mar 06 '16 at 08:39
  • @PeterLeFanuLumsdaine never mind, I mixed this up with another similar map I've been looking into yesterday. – Martin Ender Mar 06 '16 at 08:47
  • Thinking about this some more... if $f(x) = 1 - 1/x$, then couldn't we also view the continued fraction in the question as the limit of $x_{n+1} = f(f(f(x)))$, of course this is just a constant sequence for any $x_0$, which would mean mean we could show the continued fraction to have any value we want. Is that just another way of saying its value isn't well-defined? – Martin Ender Mar 07 '16 at 07:49
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    @MartinBüttner Yes: assuming the value exists, you can show it is anything using your method. Therefore (by contradiction) it can't exist. – Patrick Stevens Mar 07 '16 at 07:57
  • Give the series to a computer, with $x_0=1$ Then $x_1=1-1/x_0=0$ and $x_2=1-1/x_1=$?. You get the same difficulty when you try to evaluate partial sums in the continued-fraction expression. – DanielWainfleet Mar 12 '16 at 06:48
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    @user254665 That doesn't mean the series *could never* converge; only that it doesn't do so from this particular starting point. – Patrick Stevens Mar 12 '16 at 10:48
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I guess that what you're asking for is how the the imaginary unit, i.e. the square root of $-1$ is involved. Indeed it comes from the known identity between continuous fraction and continuous square roots, i.e. $$ \sqrt{a-b\sqrt{a-b\sqrt{a-b\sqrt{a-b\sqrt{\cdots}}}}}= -\cfrac{a}{b-\cfrac{a}{b-\cfrac{a}{b-\cfrac{a}{\ddots}}}} $$ Then you have in your case $a=1$ and $b=1$ you have $$ \sqrt{1-1\sqrt{1-1\sqrt{1-1\sqrt{1-1\sqrt{\cdots}}}}}= -\cfrac{1}{1-\cfrac{1}{1-\cfrac{1}{1-\cfrac{1}{1-\ddots}}}} $$ The solution is the well known solution of the equation $$ x=a/(b+x)$$ which brings to the result you found.

Dac0
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    I _hate_ hearing it called the "imaginary root". Damn you Descartes! Nice answer anyway :) – MichaelChirico Mar 04 '16 at 16:03
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    Don't tell me! I wrote "imaginary unit" and someone edited it! – Dac0 Mar 04 '16 at 17:47
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    You don't seem to have written "unit": the edit history doesn't have a single occurrence of this word. – Ruslan Mar 04 '16 at 19:35
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    strange... I never use the term "imaginary root", anyway now I change it :) – Dac0 Mar 04 '16 at 20:19
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    “how the the imaginary unit, i.e. the square root of −1−1 is involved” - FWIW, remember that, via the closed form formula for the solution of the general cubic equation, non-real numbers may be encountered, even though the roots are real. – Mike Jones Mar 09 '16 at 17:56
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When you substitute $a_n=a_{n+1}=x$ in $$a_{n+1}=1-\frac{1}{a_n}$$ you assume that the sequence converges to a fixed-point.

If this assumption is true (as in the + case), this method will help you find the fixed-point.

However, since the sequence does not converge, the solution of $$x=1-\frac{1}{x}$$ cannot be the fixed-point (since there is none).

Uri Goren
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Have you even looked at the approximations, i.e., what happens if you stop filling up the "$\ldots$"? You get things that look like this: $$ 1-\frac1{1-\frac1{1-\frac1{\underbrace{\color{red}{1-\frac1{1}}}_0}}}$$ and that means that there is always some division by zero lurking deep inside. Hence we certainly cannot define this sort-of continued fraction as the limit of its finite approximants.

S.C.B.
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Hagen von Eitzen
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    Yes I have (and said so in the last paragraph of the question). But the infinite continued fraction could still have been the limit of some other sequence of fractions. (In fact it is as Patrick's answer shows. Replace the 0 with one of the two complex roots and you get a constant and well-defined sequence.) – Martin Ender Mar 04 '16 at 17:30
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    This is not a valid explanation. The fraction continues indefinitely, so it cannot be evaluated in this sense. It may, however, still be divergent. It is more accurate to say we have an expression of the form $1 - \frac{1}{x}$, with $x$ still possibly be undefined. – ThisIsNotAnId Mar 06 '16 at 22:02
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    Instead of just leaving something away to terminate the "…", you could as well set it to some arbitrary value, e. g. `2`. Then there would never "lurk" a DBZ deep inside. – Alfe Mar 10 '16 at 16:03
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    @ThisIsNotAnId For the *usual* continued fractions, the value is *defined* as limit of the finite continued fractions obtained precisely by chopping off at some point as I did. But I agree that in order to make sense of the OP's expression, one *must* somehow adjust the definition. Adjusting well-known definitions may lead to interesting results (such as $1+2+3+\ldots=-\frac1{12}$), but one must be aware that one has stretched things and be prepared for questions of why that makes sense and how that relates to the "traditional" but unapplicable definition (again, cf. $1+2+\ldots=-\frac1{12}$) – Hagen von Eitzen Mar 10 '16 at 20:43
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    is this indefiniteness the cause of complex root? – pooja somani May 19 '17 at 12:45
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    This is better understood if you consider $\infty$ to be there: everything else work, and you will get alternatively $0, 1,\infty$ ! – Andrea Marino Feb 01 '21 at 08:35
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This general method really is used. In the 1981 book Zetafunktionen und Quadratischer Körper by D. B. Zagier, he uses $$ x = n_0 - \frac{1}{n_1 - \frac{1}{n_2 - \frac{1}{n_3 - \ldots}}} $$ with $n_1,n_2,n_3,\ldots \geq 2$ as his basic way to represent quadratic irrationals. In the question above, the OP has all the $n_j = 1,$ which Zagier forbids. Zagier begins with this on page 126. It is necessary for him to do this because he wants to define "reduced" indefinite binary quadratic forms (page 122) as $A x^2 + B xy + C y^2$ with $B^2 - 4AC> 0$ but not a square, and $A>0, C>0, B > A+C.$ Here we go, page 126: the real number $w$ is the larger root of $Ax^2 - Bx + C=0,$ where $\langle A,B,C\rangle$ is reduced, if and only if the continued fraction (with the minus signs) for $w$ is purely periodic. Good, that is exactly how this should work. Meanwhile, just as with ordinary continued fractions, for finite continued fractions we do not want $n_j = 1,$ as that just replaces the integer $n_{j-1}$ by $n_{j-1} - 1.$ Of course, for infinite fractions we need $n_j \geq 2$ for convergence.

Oh, Gauss-Lagrange reduced indefinite forms, integer coefficients, have $AC<0, B > |A+C|.$ These go together with traditional continued fractions. Very similar theorem about purely periodic continued fractions.

Will Jagy
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For any continued fraction we have some inequalities, that $[a]< [a,b,c]<[a,b]$ or with big fractions:

$$ a < a + \frac{1}{b + \frac{1}{c}} < a + \frac{1}{b} \tag{$\ast$}$$

What should the values of $a,b,c$ be in your case. Let's try adjusting the signs little bit:

$$ x = 1 - \cfrac{1}{1 - \cfrac{1}{x}} = 1 + \cfrac{1}{-1 + \cfrac{1}{x}} $$

Looks like $a = 1, b = -1, c = x$. Do we have that $[1] < [1,-1,x] < [1,x]$ ? We need $a,b,c > 0$. In your case we have the sequence:

$$ 1 \to 1 - \frac{1}{1} = 0 \to 1 - \frac{1}{0}= \infty \to 1 + \frac{1}{\infty}= 1$$

Then $1 \to 0 \to \infty \to 1$ is a cycle of size 3.

This still seems like a cop-out to me. We should look for an explanation why normal operations like $x \to x+1$ and $x \mapsto \frac{1}{x}$ takes us outside the realm of the fractions $\mathbb{Q}$.

cactus314
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    A comment to the last paragraph (below the rule): Also $\lim_{n\to \infty} \sum_{i=0}^n \frac{1}{n!} = \mathrm e \not\in \mathbb Q$, even though it is just the addition of rational numbers. The connection is the limit to infinity, I think. – Martin Ueding Mar 05 '16 at 17:39
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    In your second code block, shouldn't the sign-adjusted version $x = 1 + 1/(-1 + 1/x)$ be negated? Meaning shouldn't it be $x = -(1 + 1/(-1 + 1/x))$ or $-x = 1 + 1/(-1 + 1/x)$? – Justin Morgan Mar 14 '16 at 15:46
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Note that the answer, $x =1-\frac{1}{1-x}$.

$(x-1)(1-x)=1$

$-x^2+2x^2-1=1$

$-x^2+2x^2-2=0$

Now, take the quadratic formula. You get $x=1\pm i $

Lets look into the quadratic, and take the rest for real within the quadratic formula.

If you have a formula in the form of $ax^2+bx+c=0$, you can take $b^2-4ac$. If that value is positive, it is real. If not, it is imaginary. Since $a=-1, b=2,c=-2$, $b^2-4ac=4-(4\cdot(-1)\cdot(-2))=-4$, which means that the square root of that number will be an imaginary number, as shown above.

Hanul Jeon
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Xetrov
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  • Thanks but this doesn't really answer my question. From one of the comments on the question: "Of course the surprising part isn't that a quadratic equation can have complex roots, but that a (by analogy with the golden ratio case seemingly valid) way to transform a (seemingly) real expression results in a quadratic equation with complex solutions, just like you'd be surprised if you got a negative rational result from summing natural numbers. My question wasn't "why am I getting complex roots for a quadratic" but "why do I seem to be able to deduct that this fraction is complex"." – Martin Ender May 07 '17 at 10:20
  • Is this appreciation or is that critism? I don't understand with my bad English :P – Xetrov May 07 '17 at 13:20
  • I appreciate that you're trying to help, but I feel that your answer misses the point of my question. I already know that quadratic equations can give complex results (and when that happens) but the question was why the continued fraction in the question evaluates to a complex value although it looks like it really shouldn't. Patrick Stevens has already answered that part to my satisfaction though. :) – Martin Ender May 07 '17 at 14:18
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what have you done can be seen as a proof of divergence, we assume that the given continued fraction converges to a finite real value $x$, thus once you prove that $x$ is a complex number with $ Im(x) \neq 0$ you get the contradiction.

Ahmed Bachir
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