1 2 Arnold 1919, tr. 21Lỗi harv: không có mục tiêu: CITEREFArnold1919 (trợ giúp) "By the same test zero surpasses all numbers in 'evenness.'"; Wong 1997, tr. 479Lỗi harv: không có mục tiêu: CITEREFWong1997 (trợ giúp) "Thus, the integer b000⋯000 = 0 is the most 'even.'
↑ Penner 1999, tr. 34Lỗi harv: không có mục tiêu: CITEREFPenner1999 (trợ giúp): Bổ đề B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0."
↑ Ball, Lewis & Thames (2008, tr. 15)Lỗi harv: không có mục tiêu: CITEREFBallLewisThames2008 (trợ giúp) discuss this challenge for the elementary-grades teacher, who wants to give mathematical reasons for mathematical facts, but whose students neither use the same definition, nor would understand it if it were introduced.
↑ Lichtenberg 1972, tr. 535–536Lỗi harv: không có mục tiêu: CITEREFLichtenberg1972 (trợ giúp) "...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."
↑ Lichtenberg 1972, tr. 535–536Lỗi harv: không có mục tiêu: CITEREFLichtenberg1972 (trợ giúp) "Zero groups of two stars are circled. No stars are left. Therefore, zero is an even number."
↑ Lichtenberg 1972, tr. 537Lỗi harv: không có mục tiêu: CITEREFLichtenberg1972 (trợ giúp); 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."
↑ Lichtenberg 1972, tr. 537–538Lỗi harv: không có mục tiêu: CITEREFLichtenberg1972 (trợ giúp) "At a more advanced level... numbers expressed as (2 × ▢) + 0 are even numbers... zero fits nicely into this pattern."
↑ Gowers 2002, tr. 118Lỗi harv: không có mục tiêu: CITEREFGowers2002 (trợ giúp) "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)Lỗi harv: không có mục tiêu: CITEREFCaldwellXiong2012 (trợ giúp).
↑ Gouvêa 1997, tr. 25Lỗi harv: không có mục tiêu: CITEREFGouvêa1997 (trợ giúp) 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)
↑ This is the timeframe in United States, Canada, Great Britain, Australia, and Israel; see Levenson, Tsamir & Tirosh (2007, tr. 85)Lỗi harv: không có mục tiêu: CITEREFLevensonTsamirTirosh2007 (trợ giúp).
↑ Frobisher 1999, tr. 31 (Introduction); 40–41 (The number zero); 48 (Implications for teaching)Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp)
↑ Frobisher 1999, tr. 37, 40, 42Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp); results are from the survey conducted in the mid-summer term of 1992.
↑ Frobisher 1999, tr. 41Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp) "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"
↑ Frobisher 1999, tr. 41Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp) "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..."
↑ Frobisher 1999, tr. 40–42, 47Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp); these results are from the February 1999 study, including 481 children, from three schools at a variety of attainment levels.
↑ Frobisher 1999, tr. 41Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp), attributed to "Jonathan"
↑ Frobisher 1999, tr. 41Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp), attributed to "Joseph"
↑ Frobisher 1999, tr. 41Lỗi harv: không có mục tiêu: CITEREFFrobisher1999 (trợ giúp), attributed to "Richard"
↑ Keith 2006, tr. 35–68Lỗi harv: không có mục tiêu: CITEREFKeith2006 (trợ giúp) "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"
↑ Ball, Lewis & Thames 2008, tr. 15Lỗi harv: không có mục tiêu: CITEREFBallLewisThames2008 (trợ giúp). See also Ball's keynote for further discussion of appropriate definitions.
↑ Nuerk, Iversen & Willmes (2004, tr. 851)Lỗi harv: không có mục tiêu: CITEREFNuerkIversenWillmes2004 (trợ giúp): "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.)"
↑ Dehaene, Bossini & Giraux 1993, tr. 376Lỗi harv: không có mục tiêu: CITEREFDehaeneBossiniGiraux1993 (trợ giúp) "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."
↑ Grimes 1975, tr. 156Lỗi harv: không có mục tiêu: CITEREFGrimes1975 (trợ giúp) "...one can pose the following questions to married couples of his acquaintance: (1) Is zero an even number?... Many couples disagree..."
↑ Sones & Sones 2002Lỗi harv: không có mục tiêu: CITEREFSonesSones2002 (trợ giúp) "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 2006aLỗi harv: không có mục tiêu: CITEREFColumn_8_readers2006a (trợ giúp) "'...according to mathematicians, the number zero, along with negative numbers and fractions, is neither even nor odd,' writes Etan..."; Column 8 readers 2006bLỗi harv: không có mục tiêu: CITEREFColumn_8_readers2006b (trợ giúp) "'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..."
↑ Arsham 2002Lỗi harv: không có mục tiêu: CITEREFArsham2002 (trợ giúp); The quote is attributed to the heute broadcast of ngày 1 tháng 10 năm 1977. Arsham's account is repeated by Crumpacker (2007, tr. 165)Lỗi harv: không có mục tiêu: CITEREFCrumpacker2007 (trợ giúp).
↑ Sones & Sones 2002Lỗi harv: không có mục tiêu: CITEREFSonesSones2002 (trợ giúp) "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!'"
↑ 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, tr. 3236, truy cập ngày 2 tháng 6 năm 2013
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