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Topic: Representing 1 without 1's in non-integer bases. (Read 556 times) |
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towr
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Representing 1 without 1's in non-integer bases.
« on: Mar 15th, 2010, 6:07am » |
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Can you find any real, non-integer number r, such that there exists a sequence of digits di for which
\sum i=0inf di*r-i = 1
0 <= di < r and di =/= 1
(With \pi I got to 0.3003333303 2002233032 0032030320 0203322202 0003203 before it broke down.)
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SMQ
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Re: Representing 1 without 1's in non-integer base
« Reply #1 on: Mar 15th, 2010, 6:48am » |
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Sure: just use the trick for converting repeating decimals to rationals in reverse.
e.g. 0.232323...b = 1
23.2323...b = b2
23.2323...b - 0.232323...b = 23b = b2 - 1
2b + 3 = b2 - 1
b2 - 2b - 4 = 0
b = 1 +/- sqrt(5)
so 0.232323... = 1 in base 2phi.
--SMQ
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--SMQ
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towr
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Re: Representing 1 without 1's in non-integer base
« Reply #2 on: Mar 15th, 2010, 6:58am » |
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Nice work. I should have thought of that.
How about in a non-algebraic base?
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| « Last Edit: Mar 15th, 2010, 6:58am by towr » |
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