0.999...

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Numerus 0.999...

Numerus decimalis periodicus 0.999... (numeris 9 ad infinitum iterantibus, aliter 0.9̅, 0.(9), 0.9̇ scriptus) idem est ac numerus realis unus (1). Haec aequalitas, quamquam a populo aliquando ignorata, multis modis et variis gradibus rigoris demonstrata est.

Zero excepto, quisque numerus decimalis qui terminum (seu numerum infinitum 0 sequentium) habet, repraesentationem geminam habet in qua numero infinito 9 terminatur (exempli gratia, 8.32 = 8.31999...). Repraesentatio quae terminatur plerumque alteri praeponitur, sed ambae sunt validae. Eadem res in omnibus basibus vel similibus numerorum realium repraesentationibus occurrit.

Aequalitas 0.999... et 1 pertinet ad absentiam infinitesimorum praeter zerum in systemate numerorum realium, in analysi mathematica usitatissimo. Nonnulla alia systemata numeralia, ut numeri hyperreales, vere infinitesimos praeter zerum continent. In plerisque talibus systematibus, 0.999... par 1 intellegitur, sed in ceteris signum "0.999..." significare potest numerum qui numerum infinitum 9 habet sed quantitate infinitesima ab 1 discrepat.

Aequalitas 0.999... et 1 diu a mathematicis agnoscitur et pars est disciplinae mathematicae communis. Nihilominus aliqui discipuli eam interrogant, reiiciunt, aut paradoxum putant. Difficultas huius scepticismi superandi est materies plurium tractatuum institutionis mathematicae.

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