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This short note is meant to give an idea of what Scientific Computing is and what its goal is. In particular, I will focus on the role of Mathematics inside Scientific Computing. On an extremely simple example I will indicate the various steps that lead from the model of the physical phenomenon to the practical solution. I will provide a list of few problems of interest for the applications, among those closer to my field of expertise, and finally I will show some famous cases of failure due to small, almost negligible, errors in one of the steps.