"A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." -- John von Neumann (b. 1903-12-28, d. 1957-02-08), "The Role of Mathematics in the Sciences and in Society" (1954)
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The best overall answer I've seen is along the lines of, "Because it makes your brain stronger and more flexible, and teaches you to think logically, better than anything else", which is a sound argument, but not especially compelling to most students.
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