Or that (Fraction) Trees Can Grow from (Continued) Fractions?
Suppose you wanted to make a list of all the rationals... How might you start?
One approach might be to begin with the denominators of the fractions: first, list those with denominator 1 - zero and 1: \(\frac{0}{1}, \frac{1}{1} \).
Next, add those with denominator 2: \(\frac{0}{1}, \frac{1}{2}, \frac{1}{1} \).
Add those with denominator 3 ( \(\frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1} \)) and you are on your way to the Farey Sequence! This last would be referred to as the Farey Sequence of Order 3.
Ford Circles offer a beautiful way to help us to better visualise these special numbers. Imagine each term of a Farey Sequence is plotted on a number line. Then draw a circle on each point with a particular radius(!?) which allows it to just touch (to kiss!) any adjacent circle.
There are, of course, other ways to approach the problem of listing (enumerating) the rationals.
A second approach may be found in the form of Fraction Trees - Farey Trees, Stern-Brocot and Calkin-Wilf Trees offer amazing insights into the nature and structure of rational numbers, and their relationship with continued fractions. (Did you know that these fraction trees provide simple ways, not only to list all the rationals, but to express them in continued fraction form?)
In particular, we will discover a special continued fraction which generates all the rational numbers in the Farey Tree, and from these to the Stern-Brocot and Calkin-Wilf Trees!
Can you see how the fractions at each level are calculated from theirparents?
How do the paths to each term lead to continued fractions?
How can you tell which level of the Stern-Brocot or Calkin-Wilf trees to look in to find a particular rational number?
Investigate the amazing Stern-Brocot continued fraction. Explore further using the buttons and sliders below - then try the quiz to predict the next level.
