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With positional notation, is there any practical usage for negative bases? I recently learned that numbers can be represented in positional notation using a negative base, but I don't see any real practical application of this. So, is there any?

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I don't know what you mean by real application, but in Quantum Computation vectors are used as basis. There are bunch of links on that Wikipedia page that point to non-integer bases and how they simplify representation of some odd things, after all positional notation is only one way to represent numbers and there are many more ways, some are more useful for machines and programs than humans. –  Arjang Jan 6 '11 at 9:14
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According to the history section of that wikipedia link yes: Negabinary was implemented in the early Polish computer BINEG, built 1957-59, based on ideas by Z. Pawlak and A. Lazarkiewicz from the Mathematical Institute in Warsaw.[2] Implementations since then have been rare.

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