Math spinoffs

NASA has fueled the development of lots of spinoff technologies: smoke detectors, memory foam, infrared ear thermometers, etc. NASA didn't pursue these things directly, but they were a useful side effect.

Something analogous happens in mathematics. While pursuing one goal, mathematicians spin off tools that are useful for other purposes. Algebraic topology might be the NASA of pure mathematics. It's worth taking a course in algebraic topology even if you don't care about topology, just to see a lot of widely used ideas in their original habitat.

Number theory has developed an enormous amount of technical machinery. If a graduate student in number theory described to you what she's working on, you probably wouldn't recognize it as number theory. But as far as I know, not much of the machinery of number theory has been applied to much besides number theory. Maybe there's a arbitrage opportunity, not to apply number theory per se, but to apply the machinery of number theory.