and e

Here are some rational approximations to pi that I computed, each entry looks like: {fraction, decimal expansion of that fraction (to 15 digits), percent error}. Just a reminder- pi to 15 digits is 3.14159265358979.

{4, 4.00000000000000, 27.3}

{3, 3.00000000000000, 4.51}

{19/6, 3.16666666666667, 0.798}

{160/51, 3.13725490196078, 0.138}

{1744/555, 3.14234234234234, 0.0239}

{644/205, 3.14146341463415, 0.00411}

{2529/805, 3.14161490683230, 0.000708}

{183296/58345, 3.14158882509212, 0.000122}

{3763456/1197945, 3.14159331187993, 0.0000210}

{4317632/1374345, 3.14159254044654, 3.60*10^-6}

{54743776/17425485, 3.14159267303033, 6.19*10^-7}

{1013549056/322622685, 3.14159265025025, 1.06*10^-7}

{30594128896/9738413685, 3.14159265416337, 1.83*10^-8}

And I can go on. These fractions were generated using Mobius transformations, and as such, they have geometrical interpretations. I might talk about those interpretations in a later post.