For many problems it would be miraculous and totally unexpected if somebody were to find a precise formula for the solution. ‘Estimates and approximations’ shows that most of the time one must settle for a rough estimate instead. The ways of approximating are outlined using an example of creating a simple formula to approximate a sequence of numbers. The concept of approximate logarithms and square roots is introduced to ease mathematical calculations and enable comparisons to be made. The prime number theorem is then discussed followed by sorting algorithms of computer programs, such as Quicksort, which is another area of mathematics that is full of rough estimates.