> > If we assume that the room is infinitely large (and compared to the size of the beaker it might as well be) and at a constant temperature of 20șC, then the beaker will gradually rise in temperature back to 20șC (the mathematician in me now points out that it will only reach that temperature after an infinite time, but the engineer in me says "Who cares? It'll be 99.9% of the way there in a finite amount of time, and that's good enough")
>
> Okay, NOW I remember what I wanted to ask about this particular bit of math. :-)
>
> I've think I've heard it's a standard assumption to claim a rate-limiting process will eventually "get 99.9% of the way where it's supposed to go in a finite amount of time, and so 'that's good enough.'" Engineers are the ones who make this claim about real-world parameters; I've never really heard any other discipline talk in this way. Is this a joke which is commonly taught in engineering school? For what situations? I'm pretty sure the assumption doesn't hold for the contents of a beaker -- or a cup of coffee -- eventually achieving thermal homeostasis with its ambient environment (i.e. thermal equilibrium).
>
> Wolf "sounds like a modern variant of Zeno's Paradox involving the neverending race between the tortoise and Achilles" spirit

It is something usually mentioned in engineering school, though not usually as a joke so much as a matter of practicality. Basically, we are told that a pure scientist will talk about limits approaching infinity, but an engineer should not use that sort of precision because it is impractical to do so. Obviously, if I have a cup of water and I want it to be at room temperature so I can make bread, I would have to wait an infinite amount of time for this to be the case. However, I cannot wait an infinite amount of time to make my bread, and with fluctuations in room temperature and such, the water achieves room temperature for all measurable and practical purposes in a much shorter time. Basically, it is all a matter of making use of simplifying assumptions when it is reasonable to do so.

So when, exactly, do these simplifications apply? Whenever a practical need arises, like my water for making bread example above. In general, the application is made to real world systems. As I implied with the water example, there are numerous variables in the real world which will affect a system. Most scientific studies make use of simplifying assumptions, such as the infinitely large room with a constant temperature of 20șC. However, a real room with people moving around in it and weather conditions changing outside (as well as numerous other imfluences) will always be in a state of temperature flux, even if only by a thousandth of a degree per second. I guess what I'm getting at is that we are taught that perfect models of systems are fine for educational purposes, but the real world is not a perfect model and hence we should not be overly precise in our calculations of real world systems.

Don't even get me started on the principle of numerical precision and what sorts of error levels are considered "acceptable". (The field I work in right now considers a number calculated within ±50% to be extraordinarily accurate.) :-)

Don "OK, that was maybe a bit much. Sorry." Monkey