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Mathematics, study of relationships among quantities, magnitudes, and properties and of logical operations by which unknown quantities, magnitudes, and properties may be deduced. In the past, mathematics was regarded as the science of quantity, whether of magnitudes, as in geometry, or of numbers, as in arithmetic, or of the generalization of these two fields, as in algebra. Toward the middle of the 19th century, however, mathematics came to be regarded increasingly as the science of …

…of a large-scale computer at the University of Illinois. Mathematical knowledge in the modern world is advancing at a faster rate than ever before. Theories that were once separate have been incorporated into theories that are both more comprehensive and more abstract. Although many important problems have been solved, other hardy perennials, such as the Riemann hypothesis, remain, and new and equally challenging problems arise. Even the most abstract mathematics seems to be finding applications.