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Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating pi to 1 in "base pi," ... Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique
Probably more maths than science, but I was thinking specifically of pi and the fact that the decimal places just seem to keep going. If we changed our number system from base-10 to something else like base-7 or base-19 for example, would it change the spectrum of irrational numbers and would we be able to find a better way to resolve something like pi?