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Abstract: Particle/wave, digital/analog, discrete/continuous, int/float. A duality between exactness and approximation arises in many areas of human activity. Somewhat paradoxically, the uses of digital computers have lead to great strides in the mathematical understanding of approximation. At the same time, but somewhat less visibly, great strides have been made in understanding how to do exact mathematical computations as well. I will give an overview of the status of exact mathematical computation and my views on its future. Examples will come primarily from linear algebra, such as computations with integer matrices.