In this video, the instructor asks which arbitrary value of [imath]ĉ[/imath] produces as simple a system as possible. The choice made was [imath]K_n[/imath], but my initial reaction was to make [imath]ĉ[/imath] very large, i.e., a googolplex. For any fixed values of the other parameters, one could find a value of [imath]ĉ[/imath] large enough to make a good portion of the complication in the system practically disappear. This would make solutions to the system slightly imprecise, but in applied problems such as the one being considered, this should be much less error-inducing than any empirical data used as parameters.
This feels like cheating, because it makes hard problems easier than it feels like they should be. Nonetheless, if the scaling constants are truly arbitrary, then they should be able to made arbitrarily close to infinity. This might change the nature of analytic solutions, but numeric solutions at least should remain accurate.
Is this a valid trick? If not, why doesn't it work?
This feels like cheating, because it makes hard problems easier than it feels like they should be. Nonetheless, if the scaling constants are truly arbitrary, then they should be able to made arbitrarily close to infinity. This might change the nature of analytic solutions, but numeric solutions at least should remain accurate.
Is this a valid trick? If not, why doesn't it work?