Hardly any other subject can hit a freshman as hard as those strange “algebraic structures” during your first math course. You started out to solve real problems and now this professor explains to you, with a straight face, that there really exists such a thing as addition and multiplication and that surprisingly 0 is not equal to 1. What’s that got to do with the real world?
Applications like Google Maps or navigation systems calculate shortest paths or fastest paths all the time. There have been very efficient algorithms to solve this task since the 50s. The most reliable path can be found via the same methods – using a little trick.
The computer computes. Plus, minus, times and sometimes even divide. The results are always correct. Really? Computers usually use floating point numbers, which are very flexible, but still can cause quite substantial rounding errors – at least when you are doing complex calculations with many steps. These small inaccuracies can accumulate and become a creeping poison. Arbitrary precision arithmetic offers a way out.