Finding Real Number Approximants by Direct Methods

This Demonstration shows two simple methods for finding fractions that approximate a target real number (rational or irrational). The respective algorithms will increment numerator () or denominator () or else will increment denominator () only. Several target numbers are considered, with sliders selecting different parts of the sequences and graphs to clarify the algorithms and their results. Using the method increment or , the red line on the numerator versus denominator graph represents the target; the dots give rational approximations. In each column of dots the approximant (the closest possible approximation to the target for that denominator) is shown in bold. Using the method increment only, only the approximants are shown, with errors and relative errors plotted. Here, bold points indicate members of the gold list of increasingly better approximations