Other than 355/113 ~ pi, I always liked that sqrt(3/4) ~ 13/15.
It turns out
sqrt(4/5) is near 17/19 sqrt(5/6) is near 21/23 sqrt(6/7) is near 25/27
and so on. This is very useful when picking out gear ratios or divider ratios etc.
Tim.
A good slide rule will give you lots of rational approximations...
[1] c:\\>ratapprx 0.8660254037844386 Usage: ratvalue [number [maxnumerator]] number defaults to PI, maxnumerator to 500 Rational approximation to 0.866025403784439 1 / 1 = 1.000000000000000 with error 0.133974596215561 4 / 5 = 0.800000000000000 with error 0.066025403784439 5 / 6 = 0.833333333333333 with error 0.032692070451105 6 / 7 = 0.857142857142857 with error 0.008882546641581 13 / 15 = 0.866666666666667 with error 0.000641262882228 45 / 52 = 0.865384615384615 with error 0.000640788399823 58 / 67 = 0.865671641791045 with error 0.000353761993394 71 / 82 = 0.865853658536585 with error 0.000171745247853 84 / 97 = 0.865979381443299 with error 0.000046022341140 181 / 209 = 0.866028708133971 with error 0.000003304349533
See my published code earlier in the thread. Note the sharp improvements at 13/15, 71/82, 84/97, and 181/209.
--
[mail]: Chuck F (cbfalconer at maineline dot net)
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