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*To*: will@xxxxxxxxxxx*Subject*: Re: inexactness vs. exactness*From*: Aubrey Jaffer <agj@xxxxxxxxxxxx>*Date*: Sun, 24 Jul 2005 17:13:12 -0400 (EDT)*Cc*: srfi-70@xxxxxxxxxxxxxxxxx*Delivered-to*: srfi-70@xxxxxxxxxxxxxxxxx*In-reply-to*: <y9loe8sbnhh.fsf@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> (message from William D Clinger on Sun, 24 Jul 2005 10:46:18 +0200)*References*: <y9loe8sbnhh.fsf@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>

| From: William D Clinger <will@xxxxxxxxxxx> | Date: Sun, 24 Jul 2005 10:46:18 +0200 | | Aubrey Jaffer claims to have proved that the language of | the R5RS not only regards inexact numbers as neighborhoods, No, it claims that inexact numbers are in one-to-one correspondence with neighborhoods around their nominal values. Although the proof dealt only with reals, the same is true for complex numbers; and in that case, the neighborhoods are not all rectangular. | but that no other interpretations of the R5RS are tenable. I don't think so; how could one disprove a proposition before it is made? | Jaffer's alleged proof contains many errors of logic, which | I will happily enumerate if anyone claims to remain convinced | by the alleged proof. Yes I would. I have appended a detailed proof of the first part (finite number of inexacts). | Below I merely offer a simple argument that the "inexact numbers | denote neighborhoods" interpretation is itself untenable. ... Clinger's alleged counterexample assumes interval arithmetic on inexacts, which was never part of SRFI-70. In SRFI-70 inexact calculations are performed on the nominal mathematical values around which the neighborhoods are based. The result of a calculation is also single valued; which neighborhood it lies in determining which inexact number is returned. For the finite number of inexacts case, the assertion that mathematical numbers close to the nominal value project onto that inexact number is neither profound nor revolutionary. As SRFI-70 demonstrates: In an implementation which represents inexact real numbers with IEEE-754 64-bit flonums: (= 3.14159265358979323846 3.1415926535897932384626433 3.141592653589793238462643383279 3.14159265358979323846264338327950288) ==> #t -=-=-=-=- In the following proof, "inexact number" refers to a member of an equivalence class (under `=') of inexact numbers represented by the R5RS-compliant implementation. Other numbers are mathematical numbers. Given: {a} A R5RS-compliant implementation has a finite number greater than 1 of inexact number equivalence classes under the transitive R5RS predicate `='. {b} That implementation contains an inexact number class #i1.0 such that (eqv? #i1.0 (string->number "1.0")) {c} procedure: string->number string Returns a [inexact] number of the maximally precise representation expressed by the given STRING. {d} for any inexact number x: (= x #i1.0) if-and-only-if (zero? (- x #i1.0)) Consider the mathematical sequence indexed by positive integer k: E[k] = 1 + (-1/10)^k {e} 0.9, 1.01, 0.999, 1.0001, 0.99999, 1.000001, ... E[k+2] is strictly between E[k] and E[k+1] {e}. {f} The limit of E[k] as k tends to infinity is 1 {e}. {g} Let S[k] be the string containing a decimal representation of E[k]. That sequence begins: {h} "0.9", "1.01", "0.999", "1.0001", "0.99999", "1.000001", ... Let I[k] be (STRING->NUMBER S[k]) {c,h}. {i} Because the implementation has a finite number (> 1) of distinct (under `=') inexact numbers classes {a}, the lower bound, L[1], of the distance between #i1.0 and members of any other inexact number class must be a nonzero {a,d}. {j} Thus there must exist an integer j such that (abs (- E[k] 1)) is less than L[1]/4 for all k > j {f,g,i,j}. {k} I[k] must be #i1.0 for k > j, because no other inexact number class can be closer {c,j}; and because STRING->NUMBER of the string representation of the limiting value, "1.0", is #i1.0 {b,g,k}. {l} Thus the projection of mathematical numbers between E[j] and E[j+1] into inexact number classes is #i1.0. -=-=-=-=- Note that the neighborhood between E[j] and E[j+1] is a subset of the full neighborhood which STRING->NUMBER maps to inexact number class #i1.0. There was nothing special about #i1.0 in this proof. It will apply to any other finite inexact number y satisfying: (eqv? y (string->number (number->string y))) where the string representation of y, (number->string y), is "decimal" according to R5RS-6.2.6. {m}

**References**:**Re: inexactness vs. exactness***From:*William D Clinger

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