Chapter 2.7, Problem 26MS

Farey fractions. Let F be the collection of all rational numbers between 0 and 1 (we write 0 as 0ⁿ/1 and 1 as 1/1) whose numerators and denominators do not exceed n. So, for example,

$F_1=\left\{\frac{0}{1},\frac{1}{1}\right\},F_2=\left\{\frac{0}{1},\frac{1}{2},\frac{1}{1}\right\},F_3=\left\{\frac{0}{1},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{1}{1}\right\},$

F_n is called the nth Farey fraction. List F₄, F₅, F₆, F₇, and F₈. Make a large number line segment between 0 and 1 and write in the Farey fractions. How can you generate F₈ using F₇? Generalize your observations and describe how to generate F_n. (Hint: Try adding fractions a wrong way.)