Re: BigDecimal and trigonometrics

Roedy Green wrote: 
On Mon, 14 Nov 2005 21:10:08 GMT, Jeffrey Schwab
<jeff@xxxxxxxxxxxxxxxx> wrote, quoted or indirectly quoted someone who
said :


That's only true if one assumes a traditional floating-point representation. There are plenty of techniques (continued fractions, symbolic algebra, etc.) for representing irrational numbers with infinite precision in finite memory.


I think you are confusing rational repeaters (which can be accurately
represented by a rational fraction) with irrationals that at best
could be described as a limit of a series, e.g. pi, e. 

Remember the proof that there are non-rational numbers, discovered,
IIRC by the ancient Greeks.  There is no way you can map all
rationals, much less all irrationals with perfect accuracy into a
finite address space. There an infinite but countably many rationals
and uncountably many irrationals.  (One of my pet peeves is
newscasters who use the term "uncountable" to mean "a large number" or
"more than I can count on my fingers".)

"Countable" to mathematicians means there exists a 1-1 mapping between
the set and the set of integers.

Can we assume that each rational number we need to represent must have be arrived at by some sequence of mathematical operations? If so, each such number can be represented by the sequence of operations that invokes it.
.

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