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When is .999... less than 1?
, 2010
"... We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is “an infinite number of 9s ” merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone’s “semicolon ” notation? Is it p ..."
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We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is “an infinite number of 9s ” merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone’s “semicolon ” notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol.999... be inculcated in a prelimit teaching environment? The problem of the unital evaluation is hereby examined from the preR, prelim viewpoint of the student.
APPROXIMATE SPECTRAL SYNTHESIS IN THE BERGMAN SPACE
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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WEIGHTED SHIFTS OF FINITE MULTIPLICITY
, 2013
"... This DissertationOpen Access is brought to you for free and open access ..."
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Nonstandard Analysis and Applications UC Davis Mathematics StudentRun Seminar Presentation notes
, 2006
"... Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alterna ..."
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Leibniz and Newton, both independently credited as inventors of calculus, relied on the concept of an infinitesimal (nonzero “numbers ” that were “infinitely small”) in their development. Our standard rigorous treatment of calculus involves an “arbitrary epsilon ” limit definiton. There’s an alternative rigorous study of calculus beyond the limits of real analysis. In 1961, Robinson constructed the “hyperreal line ” as a direct consequence of the compactness theorem of first order logic. We will examine some typical proofs of known statements in advanced calculus and extend the nonstandard framework to other mathematical fields. 1 Introduction to FirstOrder Logic 1.1 FirstOrder Languages Define a firstorder language to be a set of symbols, as a base containing a symbol for the logical NAND, quantifiers ∃ and ∀, equality (=), grouping parenthesies/brackets, and variables (as many as are needed). Though NAND is all
A STRICT NONSTANDARD INEQUALITY.999... < 1
, 811
"... Abstract. Is.999... equal to 1? Some nonstandard thoughts on the ambiguity of the ellipsis. Contents 1. A geometric sum 1 2. Arguing by “I told you so ” 2 3. ’Fessing up 2 4. Squaring.999... < 1 with reality 3 ..."
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Abstract. Is.999... equal to 1? Some nonstandard thoughts on the ambiguity of the ellipsis. Contents 1. A geometric sum 1 2. Arguing by “I told you so ” 2 3. ’Fessing up 2 4. Squaring.999... < 1 with reality 3