Can you give a definition of the Conway base-$13$ function better than the one actually presented on wikipedia, which isn't clear? Maybe with some examples?
A definition of Conway base-$13$ function
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calculus
number-theory
definition
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5Try this one: http://williewong.wordpress.com/2010/07/20/conways-base-13-function – 2010-07-29
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0Can you say what's unclear about the definition on Wikipedia, so that it can be fixed? – 2010-07-29
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0@ShreevatsaR: For a start, it could explain the use of the weird .-+ notation – 2010-07-29
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0@Casebash: It does: "expand x […] in base 13 using the symbols 0,1,2,3,4,5,6,7,8,9,.,-,+". It also says "Note: Here the symbols "+", "-" and "." are used as symbols of base 13 decimal expansion, and do not have the usual meaning". What else could you write? (I didn't write it and am not trying to defend it; I just think it would be nice if the Wikipedia article got improved as a result of this question, and am trying to understand how it can be improved.) – 2010-07-29
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0@ShreevatasaR: I had an attempt to motivate it more in my answer – 2010-07-29
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0@ShreevatsaR The symbol "." is used with three different meanings, this is unclear. The dot between a_n and b_1 hasn't the same meaning as the three dots after b_3, for example. – 2010-07-29
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0@zar: Ah, good point; I hadn't realised that. Although the ellipsis (…) is not the same as three dots, it does look very similar, and this can be confusing. You're right. – 2010-07-29
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0I would have used A, B, and C, like programmers do when writing numbers in hexadecimal. But then, a and b already have meaning in this context, so maybe T, E, and W (suggesting their values, Ten, Eleven, and tWelve). The p, d, and m, used by Niel de Beaudrap, is not bad either. – 2010-07-31
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1needless to say, we could also work from the other side. Just write the infinite decimal number, then declare a number well-formed if its decimal expansion eventually contains digits from 0 to 6, preceded by a 7, then a finite number (but at least one) of digits from 0 to 6, then either a 8 or a 9. Now the value of the anti-Conway function is 0 for non well-formed numbers; for well formed numbers, throw away the leftmost part, substitute + for 8, - for 9 and . for 7, and read the resulting number as if it were in base 7. – 2010-08-01