I have been looking at concise ways to represent irrational numbers using only integers.
I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the rationals with $\sqrt 5$ (i.e. $\mathbb Q[\sqrt 5]$) in a finite (potentially infinite but periodic) representation. For example, $5 + \sqrt 5 = 10000.01$ exactly.
I also noticed that $\sqrt a$ always results in a periodic continued fraction (CF) when $a$ is a square-free integer. The golden ratio $\phi$ is the least accurate irrational number to approximate by truncating its continued fraction, as it is [1, 1, 1, …] (see Hurwitz's theorem and the Wikipedia Continued Fractions page).
Since truncating $\phi$'s CF is not as accurate a rational approximation as for other irrationals, and given that base $\phi$ allows us to represent $\phi$ concisely ($\phi = 10$) and allows a representation of integers (e.g. $2 = 10.01$), which is what CF's already use,
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Are there any potential problems with using base $\phi$ as the base for a CF representation considering it gets around the accuracy issue for $\phi$ and allows continued fractions involving $\sqrt 5$ to be finite (I think)?
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Is the continued fraction of the square root of a base $\phi$ number periodic when the CF is expressed in base $\phi$, in much the same manner as a square free integer?
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Are there any other advantages/disadvantages to representing continued fractions in this base instead of another more typical base?
This is my first post, so please let me know if I need to correct anything.
Best Answer
I'm not entirely sure what you mean by 'as the base for a CRF representation'; continued fraction representations, at heart, don't use bases at all - just integers.
Assuming that what you mean is allowing the coefficients of a standard continued fraction representation to be elements of $\mathbb{Z}[\phi]$ rather than just $\mathbb{Z}$, then you run into trouble of a different source: in that case, there's no canonical CF representatation! The usual algorithm for generating a canonical continued fraction uses the floor operation $\lfloor x\rfloor$ (that is, 'the largest integer less than $x$') : $a_0 = \lfloor x\rfloor, a_1=\lfloor\frac{1}{x-a_0}\rfloor$, etc. But this operation can't be defined in $\mathbb{Z}[\phi]$ because the values in that ring are dense in the reals; there's no 'largest member less than $x$' for any $x$.
On the other hand, if you're interested in generalizing some of the properties of continued fractions to the ring of 'golden integers', there's another approach you could investigate:
As I'm sure you're already aware, the sequence of partial convergents to a continued fraction for some irrational number $x$ offers the best approximations to $x$, in a very canonical sense: each term $\frac{p}{q}$ in the sequence is the best approximation to $x$ of height less than $q$, where we define the height of a fraction $\frac{c}{d}$ in simplest terms to be just the denominator of the fraction. In essence, the height represents the simplicity of the number, and the best rational approximation property says that no 'simpler' number than any of the partial convergents of the CF can be a better approximation.
You could try and generalize this property by defining a canonical height for golden integers (for instance, $\mathrm{ht}(a+b\phi) = \max(|a|, |b|)$ or even just $\mathrm{ht}(a+b\phi) = |b|$) and then studying the sequence of 'convergents' to a number $x$ using this height and the best-approximation property; this is in some sense similar to the continued fraction approach in that both are looking at the best approximation to $x$ of the form $f(a,b)$ for particular functions $f$ and integers $a,b$ less than some specified bound (and studying how that approximation changes as the bound moves) — in the case of continued fractions then $f(a,b) = \frac{a}{b}$ while here it's $f(a,b) = a+b\phi$. It's very possible that the similarity of the problem means that these convergents have some canonical structure (similar to the way that the continued-fraction structure unites the best rational approximations) which you could use to define a continued-fraction equivalent for the representation by golden integers. Speaking personally, I'd certainly be curious to see whether anything interesting came out of it!