0,999...
Catatan kaki

1 2  cf. with the binary version of the same argument in Silvanus P. Thompson, Calculus made easy, St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.
↑  Rudin p.61, Theorem 3.26; J. Stewart p.706
↑  Euler p.170
↑  Grattan-Guinness p.69; Bonnycastle p.177
↑  For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31
↑  The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) Thomas' Calculus: Early Transcendentals 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).
↑  Davies p.175; Smith and Harrington p.115
↑  Beals p.22; I. Stewart p.34
↑  Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46
↑  Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27
↑  Apostol p.12
↑  Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number x can be named by giving an infinite set of rationals, namely all the rationals less than x. We will in effect define x to be the set of rationals smaller than x. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"
↑  Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".
↑  Richman p.399
1 2  J J O'Connor and E F Robertson (2005). "History topic: The real numbers: Stevin to Hilbert". MacTutor History of Mathematics. Diarsipkan dari versi asli tanggal 2007-09-29. Diakses tanggal 2006-08-30.  Parameter |url-status= yang tidak diketahui akan diabaikan (bantuan); Parameter |month= yang tidak diketahui akan diabaikan (bantuan)
↑  "Mathematics Magazine:Guidelines for Authors". Mathematical Association of America. Diarsipkan dari versi asli tanggal 2012-02-10. Diakses tanggal 2006-08-23.
↑  Richman pp.398–399
↑  Griffiths & Hilton §24.2 "Sequences" p.386
↑  Griffiths & Hilton pp.388, 393
↑  Griffiths & Hilton pp.395
↑  Petkovšek p.408
↑  Protter and Morrey p.503; Bartle and Sherbert p.61
↑  Komornik and Loreti p.636
↑  Kempner p.611; Petkovšek p.409
↑  Petkovšek pp.410–411
↑  Leavitt 1984 p.301
↑  Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98
↑  Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.
↑  Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.
↑  Rudin p.50, Pugh p.98
↑  Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."
↑  Tall and Schwarzenberger pp.6–7; Tall 2000 p.221
↑  Tall and Schwarzenberger p.6; Tall 2000 p.221
↑  Tall 2000 p.221
↑  Tall 1976 pp.10–14
↑  Pinto and Tall p.5, Edwards and Ward pp.416–417
↑  Mazur pp.137–141
↑  As observed by Richman (p.396). Hans de Vreught (1994). "sci.math FAQ: Why is 0.9999… = 1?". Diarsipkan dari versi asli tanggal 2007-09-29. Diakses tanggal 2006-06-29.
↑  Cecil Adams (2003-07-11). "An infinite question: Why doesn't  .999~ = 1?". The Straight Dope. Chicago Reader. Diarsipkan dari versi asli tanggal 2012-01-10. Diakses tanggal 2006-09-06.
1 2  "Blizzard Entertainment Announces .999~ (Repeating) = 1". Press Release. Blizzard Entertainment. 2004-04-01. Diarsipkan dari versi asli tanggal 2007-01-20. Diakses tanggal 2006-09-03.
1 2  Gowers p.60
↑  Berz 439–442
↑  For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis.
↑  Lightstone pp.245–247
↑  Gardiner p.98; Gowers p.60
↑  Fjelstad p.11
↑  Fjelstad pp.14–15
↑  DeSua p.901
↑  DeSua pp.902–903