I think the truly surprising thing is just how well floating point numbers work in many practical applications despite how different they are from the real numbers. One could call it the "unreasonable effectiveness of floating point mathematics".
There are many situations where you have something you want to compute to within low number of units in the last place, that seem fairly involved, but there are very often clever methods that let you do it without having to go to extended or arbitrary precision. Maybe not that surprising, but it's something I find interesting.
