Perhaps you have seen a proof that the cardinality of the set of fractions is the same as that of the natural numbers. It relies on creating an ordered list of all the fractions. If you can talk about the nth fraction, then there must be as many as there are natural numbers.

If you order the fractions and arrange them in the following way, you get a Pascal's triangle - like deal with just as many patterns and interesting features.

## 1 comment:

From the 360 blog:

Ξ Says:

April 16, 2009 at 5:01 am

Ross,

I really like the visual nature of your triangle. In a similar vein, Have you seen the series Recounting the Rationals (Part I starts here) over on The Math Less Traveled? It’s a description of the paper Recounting the Rationals by Neil Calkin and Herbert Wilf, and it gives a completely different tree of fractions, but one in which all the fractions are in reduced form.

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