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Re: infinities reformulated



======= At 2005-05-31, 09:59:56 Alex Shinn wrote: =======

>On 5/31/05, Chongkai Zhu <mathematica@xxxxxxxxx> wrote:
>> 
>> And your statement "the precision of an inexact (or exact) number
>> representation cannot be unlimited" is wrong. For example, Mathematica
>> implements "unlimited precision number", although it may have some
>> flaws.
>
>It cannot be "unlimited" in the sense that it is at least memory limited.
>No matter how much memory you have, you can never exactly represent
>the square root of 2 with a floating point representation - that is the
>definition of irrational, you don't even need to go to transcendental numbers
>for this.  In fact, irrational literally means "no ratio," and was only later
>applied by analogy to imply a lack of reasoning.

Yes, there is memory limited. But here the "unlimited" should mean the
implementation makes no limit on how big the number will be. Just as many
Scheme implementations provide "unlimited" big integer.

I mentioned Mathematica, only for the "inexact number" part of it, not the
"symbolic manipluation" part of it. For example, if you want to save the
square root of 2 as an inexact number, you can write:

v1=1.414

the precision or the inexact number v1 is 4 (decimal digits);

but you can also write

v2=1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573

and v2 will get precision 100 (all these digits are saved into memory).

> v1^2
1.9994

> v2^2
2.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

>
>But your original statement was "arbitrarily big," not "unlimited," and this
>should be allowed.  Infinity in this case could be defined as the range of
>all real numbers greater than the largest possible BigFloat using all of
>memory for the exponent.
>
>-- 
>Alex

= = = = = = = = = = = = = = = = = = = =
			
Chongkai Zhu
mathematica@xxxxxxxxx
2005-05-31