อ้างอิง ของ ภาวะคู่หรือคี่ของ_0

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  1. ↑ Penner 1999, p. 34: Lemma B.2.2, The integer 0 is even and is not odd. แพนเนอร์ใช้สัญลักษณ์คณืตศาสตร์ ∃ ตัวบ่งปริมาณสำหรับตัวมีจริง ในการระบุข้อพิสูจน์: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0."
  2. ↑ Ball, Lewis & Thames (2008, p. 15) อภิปรายความท้าทายนี้สำหรับครูประถมศึกษา ผู้ต้องการให้เหตุผลทางคณิตศาสตร์แก่ข้อเท็จจริงทางคณิตศาสตร์ แต่นักเรียนไม่ใช้บทนิยามเดียวกัน หรือหากสอนแล้วจะไม่เข้าใจ
  3. ↑ เทียบกับ Lichtenberg (1972, p. 535) Fig. 1
  4. ↑ Lichtenberg 1972, pp. 535–536 "...numbers answer the question How many? for the set of objects ... zero is the number property of the empty set ... If the elements of each set are marked off in groups of two ... then the number of that set is an even number."
  5. ↑ Lichtenberg 1972, pp. 535–536 "Zero groups of two stars are circled. No stars are left. Therefore, zero is an even number."
  6. ↑ Dickerson & Pitman 2012, p. 191.
  7. ↑ Lichtenberg 1972, p. 537; compare her Fig. 3. "If the even numbers are identified in some special way ... there is no reason at all to omit zero from the pattern."
  8. ↑ Lichtenberg 1972, pp. 537–538 "At a more advanced level ... numbers expressed as (2 × ▢) + 0 are even numbers ... zero fits nicely into this pattern."
  9. ↑ Caldwell & Xiong 2012, pp. 5–6.
  10. ↑ Gowers 2002, p. 118 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012).
  11. 1 2 3 Partee 1978, p. xxi
  12. 1 2 Stewart 2001, p. 54 กำหนดกฎไว้ดังนี้ แต่ไม่ได้ยกมาคำต่อคำ
  13. ↑ Devlin 1985, pp. 30–33
  14. ↑ Penner 1999, p. 34.
  15. 1 2 Berlinghoff, Grant & Skrien 2001 For isolated vertices see p. 149; for groups see p. 311.
  16. ↑ Lovász, Pelikán & Vesztergombi 2003, pp. 127–128
  17. ↑ Starr 1997, pp. 58–62
  18. ↑ Border 1985, pp. 23–25
  19. ↑ Lorentz 1994, pp. 5–6; Lovas & Pfenning 2008, p. 115; Nipkow, Paulson & Wenzel 2002, p. 127
  20. ↑ Bunch 1982, p. 165
  21. ↑ Salzmann et al. 2007, p. 168
  22. ↑ Wise 2002, pp. 66–67
  23. ↑ Anderson 2001, p. 53; Hartsfield & Ringel 2003, p. 28
  24. ↑ Dummit & Foote 1999, p. 48
  25. ↑ Andrews 1990, p. 100
  26. ↑ Tabachnikova & Smith 2000, p. 99; Anderson & Feil 2005, pp. 437–438
  27. ↑ Barbeau 2003, p. 98
  28. ↑ Arnold 1919, p. 21 "By the same test zero surpasses all numbers in 'evenness.'"; Wong 1997, p. 479 "Thus, the integer b000⋯000 = 0 is the most 'even.'
  29. ↑ Wong 1997, p. 479
  30. ↑ Gouvêa 1997, p. 25 Of a general prime p: "The reasoning here is that we can certainly divide 0 by p, and the answer is 0, which we can divide by p, and the answer is 0, which we can divide by p…" (ellipsis in original)
  31. ↑ Krantz 2001, p. 4
  32. ↑ Salzmann et al. 2007, p. 224
  33. 1 2 Frobisher 1999, p. 41
  34. ↑ This is the timeframe in United States, Canada, Great Britain, Australia, and Israel; see Levenson, Tsamir & Tirosh (2007, p. 85)
  35. ↑ Frobisher 1999, pp. 37, 40, 42; results are from the survey conducted in the mid-summer term of 1992.
  36. ↑ Frobisher 1999, p. 41 "The percentage of Year 2 children deciding that zero is an even number is much lower than in the previous study, 32 per cent as opposed to 45 per cent"
  37. ↑ Frobisher 1999, p. 41 "The success in deciding that zero is an even number did not continue to rise with age, with approximately one in two children in each of Years 2 to 6 putting a tick in the 'evens' box ..."
  38. ↑ Frobisher 1999, pp. 40–42, 47; these results are from the February 1999 study, including 481 children, from three schools at a variety of attainment levels.
  39. ↑ Frobisher 1999, p. 41, attributed to "Jonathan"
  40. ↑ Frobisher 1999, p. 41, attributed to "Joseph"
  41. ↑ Frobisher 1999, p. 41, attributed to "Richard"
  42. ↑ Keith 2006, pp. 35–68 "There was little disagreement on the idea of zero being an even number. The students convinced the few who were not sure with two arguments. The first argument was that numbers go in a pattern ...odd, even, odd, even, odd, even... and since two is even and one is odd then the number before one, that is not a fraction, would be zero. So zero would need to be even. The second argument was that if a person has zero things and they put them into two equal groups then there would be zero in each group. The two groups would have the same amount, zero"
  43. ↑ Levenson, Tsamir & Tirosh 2007, pp. 83–95
  44. 1 2 Ball, Lewis & Thames 2008, p. 27, Figure 1.5 "Mathematical claims about zero."
  45. ↑ Ball, Lewis & Thames 2008, p. 16.
  46. ↑ Levenson, Tsamir & Tirosh 2007; Dickerson & Pitman 2012
  47. ↑ Dickerson & Pitman 2012.
  48. ↑ Ball, Hill & Bass 2005, pp. 14–16
  49. ↑ Hill et al. 2008, pp. 446–447.
  50. ↑ Lichtenberg 1972, p. 535
  51. ↑ Ball, Lewis & Thames 2008, p. 15. See also Ball's keynote for further discussion of appropriate definitions.
  52. ↑ As concluded by Levenson, Tsamir & Tirosh (2007, p. 93), referencing Freudenthal (1983, p. 460)
  53. ↑ Nuerk, Iversen & Willmes (2004, p. 851): "It can also be seen that zero strongly differs from all other numbers regardless of whether it is responded to with the left or the right hand. (See the line that separates zero from the other numbers.)"
  54. ↑ ดูข้อมูลทั้ง Dehaene, Bossini & Giraux (1993), และสรุปโดย Nuerk, Iversen & Willmes (2004, p. 837).
  55. ↑ Dehaene, Bossini & Giraux 1993, pp. 374–376
  56. ↑ Dehaene, Bossini & Giraux 1993, pp. 376–377
  57. ↑ Dehaene, Bossini & Giraux 1993, p. 376 "In some intuitive sense, the notion of parity is familiar only for numbers larger than 2. Indeed, before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical definition. The evidence, in brief, suggests that instead of being calculated on the fly by using a criterion of divisibility by 2, parity information is retrieved from memory together with a number of other semantic properties ... If a semantic memory is accessed in parity judgments, then interindividual differences should be found depending on the familiarity of the subjects with number concepts."
  58. ↑ Nuerk, Iversen & Willmes 2004, pp. 838, 860–861
  59. ↑ The Math Forum participants 2000; Straight Dope Science Advisory Board 1999; Doctor Rick 2001
  60. ↑ Grimes 1975, p. 156 "...one can pose the following questions to married couples of his acquaintance: (1) Is zero an even number? ... Many couples disagree..."
  61. ↑ Wilden & Hammer 1987, p. 104
  62. ↑ Snow 2001; Morgan 2001
  63. ↑ Steinberg 1999; Siegel 1999; Stingl 2006
  64. ↑ Sones & Sones 2002 "It follows that zero is even, and that 2/20/2000 nicely cracks the puzzle. Yet it's always surprising how much people are bothered by calling zero even..."; Column 8 readers 2006a "'...according to mathematicians, the number zero, along with negative numbers and fractions, is neither even nor odd,' writes Etan..."; Column 8 readers 2006b "'I agree that zero is even, but is Professor Bunder wise to 'prove' it by stating that 0 = 2 x 0? By that logic (from a PhD in mathematical logic, no less), as 0 = 1 x 0, it's also odd!' The prof will dispute this and, logically, he has a sound basis for doing so, but we may be wearing this topic a little thin ..."
  65. ↑ Kaplan Staff 2004, p. 227
  66. ↑ Graduate Management Admission Council 2005, pp. 108, 295–297; Educational Testing Service 2009, p. 1
  67. ↑ Arsham 2002; The quote is attributed to the heute broadcast of October 1, 1977. Arsham's account is repeated by Crumpacker (2007, p. 165).
  68. ↑ Sones & Sones 2002 "Penn State mathematician George Andrews, who recalls a time of gas rationing in Australia ... Then someone in the New South Wales parliament asserted this meant plates ending in zero could never get gas, because 'zero is neither odd nor even. So the New South Wales parliament ruled that for purposes of gas rationing, zero is an even number!'"
  69. ↑ A 1980 Maryland law specifies, "(a) On even numbered calendar dates gasoline shall only be purchased by operators of vehicles bearing personalized registration plates containing no numbers and registration plates with the last digit ending in an even number. This shall not include ham radio operator plates. Zero is an even number; (b) On odd numbered calendar dates ..." Partial quotation taken from Department of Legislative Reference (1974), Laws of the State of Maryland, Volume 2, p. 3236, สืบค้นเมื่อ 2 June 2013
  70. ↑ Cutler 2008, pp. 237–238
  71. ↑ Brisman 2004, p. 153
  72. ↑ Smock 2006; Hohmann 2007; Turner 1996
  73. ↑ Diagram Group 1983, p. 213
  74. ↑ Baroody & Coslick 1998, p. 1.33

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