DWilson>I would tend to group approximations into two general categories: Definition-based approximations, which follow mathematical properties, for example π ~= 3 π ~= ln(640320^3+744)/sqrt(163) The first follows from the approximate equality of the perimeter of a circle and its circumscribed hexagon, the second follows from a truncated series expansion in class theory. There are reasons for these approximations. Value-based approximations have no apparent direct relationship to mathematical definitions. For example, π ~= 22/7 π ~= 355/113 π ~= sqrt(9.87) π ~= cbrt(10) π^4 + π^4 ~= e^6 e^π - π ~= 20 all seem to be value-based. These relationships are commonly found by observation or numerical analysis of the numerical values, not derived from mathematical properties. This is clearly a subjective distinction. Tomorrow we may find reasons for approximations that baffle us today. ---------------- E.g., where would you put Out[426]= 22/7 + Sin[22/7] In[427]:= N[% - π, 2] Out[427]= 3.4*10^-10 ?-) rwg