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        riddles >> medium >> Representing 1 without 1's in non-integer bases.
        
(Message started by: towr on Mar 15^th, 2010, 6:07am)

Title: Representing 1 without 1's in non-integer bases.
Post by towr on Mar 15^th, 2010, 6:07am

Can you find any real, non-integer number r, such that there exists a sequence of digits d_i for which
\sum _i=0^inf d_i*r^-i = 1
0 <= d_i < r and d_i =/= 1

(With \pi I got to 0.3003333303 2002233032 0032030320 0203322202 0003203 before it broke down.)

Title: Re: Representing 1 without 1's in non-integer base
Post by SMQ on Mar 15^th, 2010, 6:48am

Sure: just use [hide]the trick for converting repeating decimals to rationals[/hide] in reverse.

e.g. [hide]0.232323..._b = 1
23.2323..._b = b²
23.2323..._b - 0.232323..._b = 23_b = b² - 1
2b + 3 = b² - 1
b² - 2b - 4 = 0
b = 1 +/- sqrt(5)

so 0.232323... = 1 in base 2phi.
[/hide]

--SMQ

Title: Re: Representing 1 without 1's in non-integer base
Post by towr on Mar 15^th, 2010, 6:58am

Nice work. I should have thought of that.

How about in a non-algebraic base?