Computer arithmetic can behave in apparently strange ways, even though each elementary floating point operation (+,-,/,*) is performed to high accuracy. We describe various examples of how rounding errors can have surprising or even calamitous consequences, including puzzling differences between results from similar Cray computers, and the Patriot missile software problem in the Gulf war. On the positive side, we describe IEEE standard arithmetic and its advantages for the programmer. As an example of how the effects of rounding errors can be understood we explain how to obtain a sharp bound for the error in summing floating point numbers.