0,999…

A matematikában a 0,999… visszatérő tizedes tört – amely felírható a következő alakokban is: $0.{\bar {9}},0.{\dot {9}}$ vagy $\ 0.(9)$ – egy olyan valós számot jelöl, amely megegyezik az 1-gyel. Más szavakkal a "0,999…" szimbólum ugyanazt a számot jelképezi, mint az "1" szimbólum. Ezt az egyenlőséget a matematikusok már régóta elfogadták és a tankönyvekben is ezt tanítják. Számos matematikai bizonyítás született már erről az azonosságról, különböző szigorral, a valós számok preferált fejlődésével, háttér-feltevésekkel, történelmi köntextussal és célközönséggel.

Az utóbbi náhány évtizedben a matemaoktatás kutatói vizsgálni kezdték, mennyire fogadják el ezt az egyenlőséget a matematikushallgatók. Jelentős azoknak a száma, akik megkérdőjelezik vagy elutasítják ezt az egyenlőséget – elsőre legalábbis. Sokukat meggyőz a tankönyvek, a tanárok és az alábbi aritmetikai érvelés arról, hogy a kettő azonos.Sokan azonban eléggé kitartóak ahhoz, hogy további bizonyítást kérnek.A diákok érvelése általában egy vagy több gyakori téves feltételezésen alapul a valós számok tekintetében. például arra, hogy minden egyes tizedes bővítésnek meg kell felelnie egyetlen számnak, hogy léteznie kell nemnulla infinitezimális mennyiségeknek,vagy hogy a 0,999… bővítése végül befejezödik.

Az ilyen bővítések nem-egyedisége nem korlátozódik a valós számok rendszerére. Ugyanez a jelenség megfigyelhető a 10-től eltérő egész szám alapú bázisoknál és a matematikusok számszerűsítették az egy írásmódját nem-egész számú bázisoknál is. A jelenség nem csak az 1 esetén igaz, minden nemnulla véges tizes számrendszerű számnak van 9-esekkel végződő ikerpárja. Az egyszerűség kedéért majdnem mindig a véges tizedes számot használjuk a megjelenítésnél, ami csak tovább erősíti azt a téves feltételezést, hogy ez az egyetlen leírási mód. Valójában abban az esetben, ha megengedük a végtelen bővítést, minden pozicionális számrendszerben végtelen számú kétértelmű szám jön létre. Például a 28,3287 ugyanaz a szám, mint a 28,3286999…, a 28,3287000 vagy számos egyéb írott alak. Ezeket a különböző azonosságokat alkalmazták a a törtek tizedes bővítésémek és az egyszerű fraktálok és a Cantor teszt mintázatainak jobb megértéséhez is. Előfordulnak a valós számok teljes halmazának végtelenségéét érintő klasszikus vizsgálatban is.

Létrehozható olyan számrendszer is, melyben a 0,999… szigorúan kisebb mint 1, de ezek kívül állnak a standard valós számok rendszerén, amelyet az elemi matematika használ.

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Kőslő hivatkozások

A Wikimédia Commons tartalmaz 0,999… témájú médiaállományokat.

• .999999… = 1? from cut-the-knot
• Why does 0.9999… = 1 ?
• Ask A Scientist: Repeating Decimals
• Proof of the equality based on arithmetic
• Repeating Nines
• Point nine recurring equals one
• David Tall's research on mathematics cognition