For a positive integer n, the Farey sequence of order n is the set
We have developed a mathlet that displays the elements of a Fn at their locations in the unit interval, where n is chosen by the user. At p/q, a line segment of height 1/q is drawn, for each p/q in Fn. This method of display illustrates the self-similar structure of the rational numbers in a compelling manner.
The mathlet can be used as a teaching and exploration
tool at many levels, from children in middle school learning
relative sizes of fractions to real analysis students studying the
classical example of a function whose set of points of
continuity consists of exactly the irrational numbers.
Web browser capable of Java 1.1 (Internet Explorer 5, Mozilla Firefox, or Netscape 4, or higher) with Unicode character set selected.
Open the Farey Sequence Mathlet.