Improper fractional base
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The usual methods of converting from base 10 to another base do not work for improper fractional bases: most integers, when we convert them using this method, have an infinite representation in an improper fractional base. (Note that this means there is not a unique representation for each number in an improper fractional base.)
Improper fractional bases were first discovered by A. J. Kempner in 1936, but were not investigated deeply.