Is every pair of rational numbers a pair of consecutive convergents of some continued fraction?

Question

I've been trying to use Gosper's continued fraction algorithms to work backwards from some randomly chosen pairs of rational numbers but it seems impossible to get arbitrary pairs to be consecutive convergents of a single continued fraction, particularly if they don't have the same integer part. Is there some way of determining which pairs do and which do not have this property?

Answer 1

It might help to think about Rationals geometrically? If you have a fraction $\frac{a}{b}$ and you were to plot $a$ on the y-axis of a chart, and b on the x-axis, then because the $a$'s & $b$'s are integers, you create an integer lattice of points on which the Rational could "land" on.

So if you were to take a pair of Rationals, the shortest possible distance between them would be 1 horizonally or vertically, creating a square of unit side. Now, the Determinant of a 2x2 matrix (here, our two fractions) can be considered the area of a parallelogram p.133, Proofs without Words, or this "square". That is, when the $|Det|=1$ we have neighbouring Rationals.

You can probably see where this is leading us from the diagram in the link? - the Mediant, which is another way of computing CFs.

Take (any) two Rationals $\frac{a}{b}$, $\frac{c}{d}$ and compute their Mediant $m=\frac{(a+c)}{b+d}$ and you get the next Convergent of a CF, geometrically shown in the link as the "addition" of two squares.

You can continue in this way until the CF terminates (when $|Det|=1$), or not! (infinite CF). Cook has a nice example of using the Mediant in this way.

There are many references in the literature on Best Rational Approximations, and every book on CF's explains this (Determinant concept) mathematically, much better than I! A really nice "easy" explanation of this is in p.184, Strange Curves, Counting Rabbits. I apologise for the non-mathematical explanation, but I've certainly tried to give an intuitive explanation to your question, and I hope this helps.