Triangle of Positive Rational Numbers

This triangular array consists entirely of positive rational numbers, with one (1/1) on the first row, two on the second row, four on the third, and so on, with each rational number (the “parent”) on one row giving rise to two rational numbers (its “children”) on the next row.

The activities are set up to guide you to guess the rules that determine the left-hand child and the right-hand child. From the rules it should be easy to see why the product of all the numbers on each row is 1, and there are some important properties that follow:

(1) Starting with any positive fraction in reduced terms, there is a unique path back to 1/1, which means that every positive reduced rational number occurs exactly once in the array.

(2) Starting with a non-reduced positive fraction, it is impossible to get back to 1/1, which implies that every fraction in the Triangle of Rational Numbers appears in reduced form.

(3) It follows from (1) and (2) that the set of all positive rational numbers is countable (the array itself provides the correspondence, the list starting with Row 1, then through Row 2, and so on).