Re: Decimal versus binary arithmetic was Re: J4 - presentation/discussion

Richard wrote:
> 
> Robert wrote:
> 
>> In phinary, EVERY number can be represented exactly in a
>> finite string of 0s and 1s, including e, pi and square root of 2.
> 
> In a finite string of 0s and 1s of length n 2^n different values can
> be be represented. Regardless of what those representable values may
> be there will always be another number that is not in that set of
> values, in fact there will still be an infinite number of
> unrepresented values.

And, indeed, the very references Robert cited note a number of values 
that have repeating, but infinite, decimal expansions in phinary. 
Indeed, the Wikipedia page claims that only elements of the field 
Q[\sqrt 5] have repeating or terminating expressions in phinary.

There are some numbers which are not the sum of a rational number and 
the product of a rational number times the square root of five. I'm 
pretty sure I have one around here somewhere... oh, there you are, 
\sqrt 2!

So there isn't even finite *notation* for all numbers in phinary. As 
indeed there could not be, by the pigeonhole principle (as Richard 
notes above), the Berry Paradox, or any other version of that old saw. 
Countable versus uncountable infinity and diagonalization and all that.

And again according to Wackipedia, ALL non-integer rationals have 
non-finite phinary representations, though they all repeat.

All that said, phinary *is* kind of cool.

-- 
Michael Wojcik