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*To*: Paul Schlie <schlie@xxxxxxxxxxx>*Subject*: Re: inexactness vs. exactness*From*: bear <bear@xxxxxxxxx>*Date*: Tue, 9 Aug 2005 09:10:07 -0700 (PDT)*Cc*: Aubrey Jaffer <agj@xxxxxxxxxxxx>, will@xxxxxxxxxxx, srfi-70@xxxxxxxxxxxxxxxxx*Delivered-to*: srfi-70@xxxxxxxxxxxxxxxxx*In-reply-to*: <BF1D2A73.B1DB%schlie@xxxxxxxxxxx>*References*: <BF1D2A73.B1DB%schlie@xxxxxxxxxxx>

On Mon, 8 Aug 2005, Paul Schlie wrote: > Thanks, I guess my point/question was predominantly related to the > observation that there seems often little need for truly "exact" > values beyond theoretical geometry and/or combinatorial mathematics, > which often themselves only require a determinable finite precision; This is a point on which you're going to lose. Combinatorial mathematics has developed subfields called cryptography, compression, and correction codes which are fundamental to modern networking. If you're doing any of those and you round anything off, you lose. > while simultaneously observing there's often broader need for more > precise potentially "inexact" values than typically supported by > double precision floating point implementations; so it just seemed > that in practice that it may be more useful to define that "exact" > values need only be as precise as necessary to support the exact > representation of the integer values bounded by the dynamic range of > the implementation's "inexact" implementation, and their > corresponding reciprocal values in practice (as you've implied); NACK! If you have limited precision, and the limited precision affects the answer, then the answer is inexact. PERIOD. There is no such thing as "exact numbers limited in precision" to *ANY* limit of precision. Once you go beyond a limit of precision and round something, you aren't talking about exact numbers anymore. Exact numbers are, by definition, *infinitely* precise. You may be talking about limiting the representation size of exact numbers, thereby decreasing the size of the set of exact numbers you can represent; but that's not the same thing. Infinite precision in finite memory arises when the number happens to match our representational scheme very well; integers and ratios of integers happen to be infinitely precise things we can represent in finite memory - but the finiteness of our memory means that we can only represent an infinitesimal fraction of those in any fixed amount of space. Things work because our usual calculations tend to give us results that are in the set of things we can represent; and when they don't, we can throw an error, if it's last-bit critical, or return an inexact number, if it isn't. > thereby both providing a likely reasonably efficient "inexact" (aka > double) >and a likely reasonably precise corresponding "exact" > representation, I will say it again. Exact numbers aren't "reasonably" precise. they are *exact*, which is to say "infinitely" precise. You are arguing for extended-precision inexact numbers, and I agree with you that these are needed and useful - but to call them exact is to confuse the issue and does not help. Bear

**Follow-Ups**:**Re: inexactness vs. exactness***From:*Paul Schlie

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