1108:. A couple of fourth-years realized that zero can be split into equal parts. Another fourth-year reasoned "1 is odd and if I go down it's even." The interviews also revealed the misconceptions behind incorrect responses. A second-year was "quite convinced" that zero was odd, on the basis that "it is the first number you count". A fourth-year referred to 0 as "none" and thought that it was neither odd nor even, since "it's not a number". In another study, Annie Keith observed a class of 15 second-graders who convinced each other that zero was an even number based on even-odd alternation and on the possibility of splitting a group of zero things in two equal groups.
2273:, 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."
2077:, 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"
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years of primary education may not yet have learned what "integer" or "multiple" means, much less how to multiply with 0. Additionally, stating a definition of parity for all integers can seem like an arbitrary conceptual shortcut if the only even numbers investigated so far have been positive. It can help to acknowledge that as the number concept is extended from positive integers to include zero and negative integers, number properties such as parity are also extended in a nontrivial way.
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1416:. Half of the numbers in a given range end in 0, 2, 4, 6, 8 and the other half in 1, 3, 5, 7, 9, so it makes sense to include 0 with the other even numbers. However, in 1977, a Paris rationing system led to confusion: on an odd-only day, the police avoided fining drivers whose plates ended in 0, because they did not know whether 0 was even. To avoid such confusion, the relevant legislation sometimes stipulates that zero is even; such laws have been passed in
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287:. The concept of parity is used for making groups of two objects. If the objects in a set can be marked off into groups of two, with none left over, then the number of objects is even. If an object is left over, then the number of objects is odd. The empty set contains zero groups of two, and no object is left over from this grouping, so zero is even.
2425:"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!'"
2363:"'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 ..."
1101:. This time the number of children in the same age range identifying zero as even dropped to 32%. Success in deciding that zero is even initially shoots up and then levels off at around 50% in Years 3 to 6. For comparison, the easiest task, identifying the parity of a single digit, levels off at about 85% success.
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scores after taking the teachers' classes. In a more in-depth 2008 study, the researchers found a school where all of the teachers thought that zero was neither odd nor even, including one teacher who was exemplary by all other measures. The misconception had been spread by a math coach in their
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of parity and their concept definitions. Levenson et al.'s sixth-graders both defined even numbers as multiples of 2 or numbers divisible by 2, but they were initially unable to apply this definition to zero, because they were unsure how to multiply or divide zero by 2. The interviewer eventually led
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Making an exception for zero in the definition of evenness forces one to make such exceptions in the rules for even numbers. From another perspective, taking the rules obeyed by positive even numbers and requiring that they continue to hold for integers forces the usual definition and the evenness of
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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
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Some of the contexts where the parity of zero makes an appearance are purely rhetorical. Linguist Joseph Grimes muses that asking "Is zero an even number?" to married couples is a good way to get them to disagree. People who think that zero is neither even nor odd may use the parity of zero as proof
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Dehaene's experiments were not designed specifically to investigate 0 but to compare competing models of how parity information is processed and extracted. The most specific model, the mental calculation hypothesis, suggests that reactions to 0 should be fast; 0 is a small number, and it is easy to
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were divided into two groups: those in literary studies and those studying mathematics, physics, or biology. The slowing at 0 was "essentially found in the group", and in fact, "before the experiment, some L subjects were unsure whether 0 was odd or even and had to be reminded of the mathematical
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have included the true-or-false prompt "0 is an even number" in a database of over 250 questions designed to measure teachers' content knowledge. For them, the question exemplifies "common knowledge ... that any well-educated adult should have", and it is "ideologically neutral" in that the answer
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analyzed US third grade students' ideas about even and odd numbers and zero, which they had just been discussing with a group of fourth-graders. The students discussed the parity of zero, the rules for even numbers, and how mathematics is done. The claims about zero took many forms, as seen in the
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The above rules would therefore be incorrect if zero were not even. At best they would have to be modified. For example, one test study guide asserts that even numbers are characterized as integer multiples of two, but zero is "neither even nor odd". Accordingly, the guide's rules for even and odd
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It is also possible to explain why zero is even without referring to formal definitions. The following explanations make sense of the idea that zero is even in terms of fundamental number concepts. From this foundation, one can provide a rationale for the definition itself—and its applicability to
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Mathematically, proving that zero is even is a simple matter of applying a definition, but more explanation is needed in the context of education. One issue concerns the foundations of the proof; the definition of "even" as "integer multiple of 2" is not always appropriate. A student in the first
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importantly, there is a leftover object, so 5 is odd. In the group of four objects, there is no leftover object, so 4 is even. In the group of just one object, there are no pairs, and there is a leftover object, so 1 is odd. In the group of zero objects, there is no leftover object, so 0 is even.
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These ideas can be illustrated by drawing objects in pairs. It is difficult to depict zero groups of two, or to emphasize the nonexistence of a leftover object, so it helps to draw other groupings and to compare them with zero. For example, in the group of five objects, there are two pairs. More
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This strong dependence on familiarity again undermines the mental calculation hypothesis. The effect also suggests that it is inappropriate to include zero in experiments where even and odd numbers are compared as a group. As one study puts it, "Most researchers seem to agree that zero is not a
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There is another concrete definition of evenness: if the objects in a set can be placed into two groups of equal size, then the number of objects is even. This definition is equivalent to the first one. Again, zero is even because the empty set can be divided into two groups of zero items each.
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More in-depth investigations were conducted by Esther
Levenson, Pessia Tsamir, and Dina Tirosh, who interviewed a pair of sixth-grade students in the USA who were performing highly in their mathematics class. One student preferred deductive explanations of mathematical claims, while the other
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them to conclude that zero was even; the students took different routes to this conclusion, drawing on a combination of images, definitions, practical explanations, and abstract explanations. In another study, David
Dickerson and Damien Pitman examined the use of definitions by five advanced
1638:, 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
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works because if the ray never crosses the polygon, then its crossing number is zero, which is even, and the point is outside. Every time the ray does cross the polygon, the crossing number alternates between even and odd, and the point at its tip alternates between outside and inside.
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Hill, Heather C.; Blunk, Merrie L.; Charalambous, Charalambos Y.; Lewis, Jennifer M.; Phelps, Geoffrey C.; Sleep, Laurie; Ball, Deborah
Loewenberg (2008), "Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study",
1078:. The data is from Len Frobisher, who conducted a pair of surveys of English schoolchildren. Frobisher was interested in how knowledge of single-digit parity translates to knowledge of multiple-digit parity, and zero figures prominently in the results.
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It would be possible to similarly redefine the term "even" in a way that no longer includes zero. However, in this case, the new definition would make it more difficult to state theorems concerning the even numbers. Already the effect can be seen in
235:. Class discussions can lead students to appreciate the basic principles of mathematical reasoning, such as the importance of definitions. Evaluating the parity of this exceptional number is an early example of a pervasive theme in mathematics: the
1562:, 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."
1267:, in a 1972 study reported that when a group of prospective elementary school teachers were given a true-or-false test including the item "Zero is an even number", they found it to be a "tricky question", with about two thirds answering "False".
131:. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if
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records the time it takes the subject to push one of two buttons to identify the number as odd or even. The results showed that 0 was slower to process than other even numbers. Some variations of the experiment found delays as long as 60
551:, which is not an integer. Since zero is not odd, if an unknown number is proven to be odd, then it cannot be zero. This apparently trivial observation can provide a convenient and revealing proof explaining why an odd number is nonzero.
1453:" is also affected: if both players cast zero fingers, the total number of fingers is zero, so the even player wins. One teachers' manual suggests playing this game as a way to introduce children to the concept that 0 is divisible by 2.
1537:, p. 15) 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.
1345:. Both the sequence of powers of two and the sequence of positive even numbers 2, 4, 6, 8, ... are well-distinguished mental categories whose members are prototypically even. Zero belongs to neither list, hence the slower responses.
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composed of all odd digits that would occur for a very long time, and that "2000/02/02" was the first all-even date to occur in a very long time. Since these results make use of 0 being even, some readers disagreed with the idea.
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preferred practical examples. Both students initially thought that 0 was neither even nor odd, for different reasons. Levenson et al. demonstrated how the students' reasoning reflected their concepts of zero and division.
2220:, 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.)"
996:, or the more times it is divisible by 2, the sooner it appears. Zero's bit reversal is still zero; it can be divided by 2 any number of times, and its binary expansion does not contain any 1s, so it always comes first.
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list on the right. Ball and her coauthors argued that the episode demonstrated how students can "do mathematics in school", as opposed to the usual reduction of the discipline to the mechanical solution of exercises.
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It is uncertain how many teachers harbor misconceptions about zero. The
Michigan studies did not publish data for individual questions. Betty Lichtenberg, an associate professor of mathematics education at the
1337:.) The results of the experiments suggested that something quite different was happening: parity information was apparently being recalled from memory along with a cluster of related properties, such as being
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experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some teachers—and some children in mathematics classes—think that zero is odd, or both even and odd, or neither. Researchers in
2017:, 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 ..."
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is divisible by 2. This description does not work for 0; no matter how many times it is divided by 2, it can always be divided by 2 again. Rather, the usual convention is to set the 2-order of 0 to be
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invoke the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above choices have far-reaching consequences. For example, the fact that positive numbers have unique
1225:. They found that the undergraduates were largely able to apply the definition of "even" to zero, but they were still not convinced by this reasoning, since it conflicted with their concept images.
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of the group of all integers; this is an elementary example of the subgroup concept. The earlier observation that the rule "even − even = even" forces 0 to be even is part of a general pattern: any
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has a subsimplex that contains every color. Rather than directly construct such a subsimplex, it is more convenient to prove that there exists an odd number of such subsimplices through an
1333:. (Subjects are known to compute and name the result of multiplication by zero faster than multiplication of nonzero numbers, although they are slower to verify proposed results like
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Ball, Deborah
Loewenberg; Hill, Heather C.; Bass, Hyman (2005), "Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide?",
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from infinity to the point and counts the number of times the ray crosses the edge of polygon. The crossing number is even if and only if the point is outside the polygon. This
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form, spelled out, and spelled in a mirror image. Dehaene's group did find one differentiating factor: mathematical expertise. In one of their experiments, students in the
3532:"Analysis: Today's date is signified in abbreviations using only odd numbers. 1-1, 1-9, 1-9-9-9. The next time that happens will be more than a thousand years from now."
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Nuerk, Hans-Christoph; Iversen, Wiebke; Willmes, Klaus (July 2004), "Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect",
2005:, 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"
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391:", so 1 is not prime. This definition can be rationalized by observing that it more naturally suits mathematical theorems that concern the primes. For example, the
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means that one can determine whether a number has an even or odd number of distinct prime factors. Since 1 is not prime, nor does it have prime factors, it is a
2355:"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...";
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Repeated experiments have shown a delay at zero for subjects with a variety of ages and national and linguistic backgrounds, confronted with number names in
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Although 0 is divisible by 2 more times than any other number, it is not straightforward to quantify exactly how many times that is. For any nonzero integer
2297:, p. 156 "...one can pose the following questions to married couples of his acquaintance: (1) Is zero an even number? ... Many couples disagree..."
1598:, 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."
1393:, if a question asks about the behavior of even numbers, it might be necessary to keep in mind that zero is even. Official publications relating to the
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Keith, Annie (2006), "Mathematical
Argument in a Second Grade Class: Generating and Justifying Generalized Statements about Odd and Even Numbers",
2029:, pp. 40–42, 47; these results are from the February 1999 study, including 481 children, from three schools at a variety of attainment levels.
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Adults who do believe that zero is even can nevertheless be unfamiliar with thinking of it as even, enough so to measurably slow them down in a
682:. With this definition, the evenness of zero is not a theorem but an axiom. Indeed, "zero is an even number" may be interpreted as one of the
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Levenson, Esther; Tsamir, Pessia; Tirosh, Dina (2007), "Neither even nor odd: Sixth grade students' dilemmas regarding the parity of zero",
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side, but zero is reserved for compartments that intersect the centerline. That is, the numbers read 6-4-2-0-1-3-5 from port to starboard.
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that zero is even. A number is called "even" if it is an integer multiple of 2. As an example, the reason that 10 is even is that it equals
302:. When even and odd numbers are distinguished from each other, their pattern becomes obvious, especially if negative numbers are included:
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This definition has the conceptual advantage of relying only on the minimal foundations of the natural numbers: the existence of 0 and of
2359:"'...according to mathematicians, the number zero, along with negative numbers and fractions, is neither even nor odd,' writes Etan...";
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The fact that zero is even, together with the fact that even and odd numbers alternate, is enough to determine the parity of every other
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has an even degree.) In order to prove the statement, it is actually easier to prove a stronger result: any odd-order graph has an
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are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as
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586:: any graph has an even number of vertices of odd degree. Finally, the even number of odd vertices is naturally explained by the
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In interviews, Frobisher elicited the students' reasoning. One fifth-year decided that 0 was even because it was found on the 2
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and itself is 0, and 0 is even, the base vertex is colored differently from its neighbors, which lie at a distance of 1.
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of even degree vertices. The appearance of this odd number is explained by a still more general result, known as the
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distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This implies that the
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1574:, pp. 535–536 "Zero groups of two stars are circled. No stars are left. Therefore, zero is an even number."
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Frobisher, Len (1999), "Primary School
Children's Knowledge of Odd and Even Numbers", in Anthony Orton (ed.),
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argument. A stronger statement of the lemma then explains why this number is odd: it naturally breaks down as
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Given a set of objects, one uses a number to describe how many objects are in the set. Zero is the count of
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being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer
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as a special case. This convention is not peculiar to the 2-order; it is one of the axioms of an additive
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of no transpositions, is an even permutation since zero is even; it is the identity element of the group.
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Proceedings of the 36th
Conference of the International Group for the Psychology of Mathematics Education
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when the outcome depends on whether some randomized number is odd or even, and it turns out to be zero.
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Inserting appropriate values into the left sides of these rules, one can produce 0 on the right sides:
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Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as
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plates. Zero is an even number; (b) On odd numbered calendar dates ..." Partial quotation taken from
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There is a sense in which some multiples of 2 are "more even" than others. Multiples of 4 are called
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wrote that 1 was prime. The modern definition of "prime number" is "positive integer with exactly 2
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Around the year 2000, media outlets noted a pair of unusual milestones: "1999/11/19" was the last
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The chart on the right depicts children's beliefs about the parity of zero, as they progress from
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Lovas, William; Pfenning, Frank (2008-01-22), "A Bidirectional
Refinement Type System for LF",
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Mathematical
Conventions for the Quantitative Reasoning Measure of the GRE revised General Test
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propose that these misconceptions can become learning opportunities. Studying equalities like
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is a more advanced application of the same strategy. The lemma states that a certain kind of
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up or down by twos reaches the other even numbers, and there is no reason to skip over zero.
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This is the timeframe in United States, Canada, Great Britain, Australia, and Israel; see
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The subject of the parity of zero is often treated within the first two or three years of
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of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a
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2610:"Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives"
2191:, p. 15. See also Ball's keynote for further discussion of appropriate definitions.
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typical even number and should not be investigated as part of the mental number line."
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Turner, Julian (1996-07-13), "Sports Betting – For Lytham Look to the South Pacific",
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or about 10% of the average reaction time—a small difference but a significant one.
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the former numbers are even and the latter are odd. For example, 1 is odd because
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rely on zero being even. Not only is 0 divisible by 2, it is divisible by every
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Teachers Engaged in Research: Inquiry in Mathematics Classrooms, Grades Pre-K-2
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3118:(11th ed.), McLean, Virginia, USA: Graduate Management Admission Council,
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an advantage on such bets. Similarly, the parity of zero can affect payoffs in
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when asked the parity of zero. A follow-up investigation offered more choices:
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One of the themes in the research literature is the tension between students'
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Among the general public, the parity of zero can be a source of confusion. In
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Making a table of these facts then reinforces the number line picture above.
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Statistical analysis of experimental data, showing separation of 0. In this
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The Classical Fields: Structural Features of the Real and Rational Numbers
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was created from a revision of this article dated 27 August 2013
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Salzmann, Helmut; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer (2007),
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Hohmann, George (2007-10-25), "Companies let market determine new name",
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Fostering Children's Mathematical Power: An Investigative Approach to K-8
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1989:, pp. 37, 40, 42; results are from the survey conducted in the mid-
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Math Forum » Discussions » History » Historia-Matematica
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Lichtenberg, Betty Plunkett (November 1972), "Zero is an even number",
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Dickerson, David S.; Pitman, Damien J. (July 2012), Tai-Yih Tso (ed.),
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Ball, Deborah Loewenberg; Lewis, Jennifer; Thames, Mark Hoover (2008),
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Smock, Doug (2006-02-06), "The odd bets: Hines Ward vs. Tiger Woods",
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1412:
on alternate days, according to the parity of the last digit in their
1058:, as the concept of even and odd numbers is introduced and developed.
211:
used by computers. In this sense, 0 is the "most even" number of all.
2384:
1674:, p. 54 These rules are given, but they are not quoted verbatim.
1439:
927:
The rule "even × integer = even" means that the even numbers form an
228:
1292:, only the clustering of data is meaningful; the axes are arbitrary.
935:
of integers, and the above equivalence relation can be described as
836:
subtraction must be a subgroup, and in particular, must contain the
187:
of even integers, and it is the starting case from which other even
2729:
Fixed Point Theorems with Applications to Economics and Game Theory
765:
and coloring every vertex black or white, depending on whether its
3490:
Discrete Mathematics: Proof Techniques and Mathematical Structures
2799:
2569:
Markedness Theory: the union of asymmetry and semiosis in language
2405:
1427:
On U.S. Navy vessels, even-numbered compartments are found on the
1283:
1045:
881:
is even. Here, the evenness of zero is directly manifested as the
848:
784:
761:, then a bipartition can be constructed by choosing a base vertex
736:
704:
634:
270:
31:
1509:, to state the proof: "To see that 0 is even, we must prove that
1081:
In a preliminary survey of nearly 400 seven-year-olds, 45% chose
312:
The even and odd numbers alternate. Starting at any even number,
247:
The standard definition of "even number" can be used to directly
2872:
Perfect Figures: The Lore of Numbers and How We Learned to Count
2709:
Berlinghoff, William P.; Grant, Kerry E.; Skrien, Dale (2001),
721:
applies the above ideas. To determine if a point lies within a
2409:
broadcast of October 1, 1977. Arsham's account is repeated by
843:
Since the even integers form a subgroup of the integers, they
2951:
Devlin, Keith (April 1985), "The golden age of mathematics",
2912:"The mental representation of parity and numerical magnitude"
2551:
A First Course in Abstract Algebra: Rings, Groups, And Fields
753:, such that neighboring vertices have different colors. If a
304:
255:. In the same way, zero is an integer multiple of 2, namely
58:
2388:
1934:
1901:: "The reasoning here is that we can certainly divide 0 by
1790:
824:. Moreover, the group of even integers under addition is a
674:. As such, it is useful for computer logic systems such as
566:(having an odd number of vertices) always has at least one
283:; in more formal terms, it is the number of objects in the
2164:
2104:, p. 27, Figure 1.5 "Mathematical claims about zero."
939:. In particular, even integers are exactly those integers
3392:, The Mathematical Association of America, archived from
1722:
1614:
are even numbers ... zero fits nicely into this pattern."
1438:, the number 0 does not count as even or odd, giving the
1308:, led a series of such experiments in the early 1990s. A
3658:"01:02:03 04/05/06; We can count on some things in life"
3112:
Graduate Management Admission Council (September 2005),
1713:
For isolated vertices see p. 149; for groups see p. 311.
908:
letters. The elements of the alternating group, called
812:(zero) is even, together with the evenness of sums and
3625:
Steinberg, Neil (1999-11-30), "Even year, odd facts",
3416:
Isabelle/Hol: A Proof Assistant for Higher-Order Logic
1710:
1250:. In a 2000–2004 study of 700 primary teachers in the
808:
that require the inclusion of zero. The fact that the
570:. (The statement itself requires zero to be even: the
950:
This formulation is useful for investigating integer
3741:
The rules are no game: the strategy of communication
3164:
Pearls in Graph Theory: A Comprehensive Introduction
2270:
2258:
2246:
2230:
2201:
2125:
2086:
1962:
40:
contain zero objects, divided into two equal groups.
3262:
Dictionary of algebra, arithmetic, and trigonometry
3056:
Pattern in the Teaching and Learning of Mathematics
3047:
Didactical phenomenology of mathematical structures
2967:
The Official World Encyclopedia of Sports and Games
2282:
2234:
2217:
1766:
974:, so it surpasses all other numbers in "evenness".
395:is easier to state when 1 is not considered prime.
275:
The box with 0 objects has no red object left over.
3738:
3509:
3487:
3434:The Quarterly Journal of Experimental Psychology A
3238:
3133:
2711:A Mathematics Sampler: Topics for the Liberal Arts
1667:
1665:
401:the algebraic rules governing even and odd numbers
2360:
2356:
1854:
820:of addition, means that the even integers form a
3779:Computational Methods in Physics and Engineering
3339:Electronic Notes in Theoretical Computer Science
1818:
1026:The powers of two—1, 2, 4, 8, ...—form a simple
239:of a familiar concept to an unfamiliar setting.
2713:(5th rev. ed.), Rowman & Littlefield,
2188:
2140:
2129:
2113:
2101:
1583:
1534:
227:can address students' doubts about calling 0 a
2893:(Centennial ed.), Naval Institute Press,
2781:Caldwell, Chris K.; Xiong, Yeng (2012-12-27),
2514:
1945:
1943:
1909:, and the answer is 0, which we can divide by
1905:, and the answer is 0, which we can divide by
1706:
1704:
1179:Zero is not always going to be an even number.
985:used by some computer algorithms, such as the
298:Numbers can also be visualized as points on a
71:
2656:Journal for Research in Mathematics Education
2502:
1762:
1639:
1623:
1505:. Penner uses the mathematical symbol ∃, the
1467:
1465:
749:is a graph whose vertices are split into two
639:Recursive definition of natural number parity
179:, require 0 to be even. Zero is the additive
8:
3666:(Final ed.), p. B1, archived from
3004:Dummit, David S.; Foote, Richard M. (1999),
2747:Mensa Guide to Casino Gambling: Winning Ways
2437:Department of Legislative Reference (1974),
2372:
2306:
2152:
1858:
977:One consequence of this fact appears in the
3609:General Equilibrium Theory: An Introduction
3470:Fundamentals of Mathematics for Linguistics
3368:Discrete Mathematics: Elementary and Beyond
2919:Journal of Experimental Psychology: General
2891:The Bluejacket's Manual: United States Navy
2860:: CS1 maint: numeric names: authors list (
2842:Column 8 readers (2006-03-16), "Column 8",
2833:: CS1 maint: numeric names: authors list (
2815:Column 8 readers (2006-03-10), "Column 8",
2205:
2176:
1830:
1607:
1595:
1571:
1559:
1547:
773:is even or odd. Since the distance between
532:. One way to prove that zero is not odd is
2422:
2410:
2385:Graduate Management Admission Council 2005
2352:
2097:
2095:
1170:Zero is always going to be an even number.
3350:
3195:
3167:, Mineola, New York, USA: Courier Dover,
2910:; Bossini, Serge; Giraux, Pascal (1993),
2798:
2691:Baroody, Arthur; Coslick, Ronald (1998),
2333:
2062:
2050:
2038:
2026:
2014:
2002:
1986:
1974:
1949:
1652:
1650:
1648:
1030:of numbers of increasing 2-order. In the
3737:Wilden, Anthony; Hammer, Rhonda (1987),
1814:
1114:
88:, and does not reflect subsequent edits.
3583:Sones, Bill; Sones, Rich (2002-05-08),
2487:
2471:
2440:Laws of the State of Maryland, Volume 2
1870:
1842:
1758:
1723:Lovász, Pelikán & Vesztergombi 2003
1671:
1461:
851:. These cosets may be described as the
3092:Mathematics: A Very Short Introduction
2853:
2826:
2533:A First Course in Discrete Mathematics
2491:
2459:
2400:
2341:
2337:
2322:
2294:
2041:, p. 41, attributed to "Jonathan"
1922:
1894:
1746:
1695:
1683:
1656:
1635:
1494:
1471:
1408:, in which cars may drive or purchase
912:, are the products of even numbers of
195:. Applications of this recursion from
3472:, Dordrecht, Netherlands: D. Reidel,
3058:, London, UK: Cassell, pp. 31–48
3008:(2e ed.), New York, USA: Wiley,
2549:Anderson, Marlow; Feil, Todd (2005),
2483:
2074:
2065:, p. 41, attributed to "Richard"
1778:
1734:
647:. This idea can be formalized into a
7:
3703:The Math Forum participants (2000),
3585:"To hide your age, button your lips"
3281:The Journal of Mathematical Behavior
3022:Educational Testing Service (2009),
2765:Mathematical Fallacies and Paradoxes
2318:
2231:Dehaene, Bossini & Giraux (1993)
2053:, p. 41, attributed to "Joseph"
1882:
1802:
1711:Berlinghoff, Grant & Skrien 2001
1503:The integer 0 is even and is not odd
1475:
832:subset of an additive group that is
651:of the set of even natural numbers:
623:when one considers the two possible
497:, which is necessary for it to be a
27:Quality of zero being an even number
2649:"Making mathematics work in school"
2202:Levenson, Tsamir & Tirosh (2007
1963:Levenson, Tsamir & Tirosh (2007
1521:and this follows from the equality
3115:The Official Guide for GMAT Review
2271:Dehaene, Bossini & Giraux 1993
2259:Dehaene, Bossini & Giraux 1993
2247:Dehaene, Bossini & Giraux 1993
2235:Nuerk, Iversen & Willmes (2004
2218:Nuerk, Iversen & Willmes (2004
2126:Levenson, Tsamir & Tirosh 2007
2087:Levenson, Tsamir & Tirosh 2007
1404:The parity of zero is relevant to
403:. The most relevant rules concern
25:
3073:(2nd ed.), Springer-Verlag,
2571:, Durham: Duke University Press,
2403:; The quote is attributed to the
2283:Nuerk, Iversen & Willmes 2004
1767:Nipkow, Paulson & Wenzel 2002
1478:, p. 479 "Thus, the integer
1401:tests both state that 0 is even.
804:, the even integers form various
393:fundamental theorem of arithmetic
3871:
3856:
3811:, The Math Forum, archived from
3494:, River Edge: World Scientific,
3049:, Dordrecht, Netherlands: Reidel
2608:Arsham, Hossein (January 2002),
2389:Educational Testing Service 2009
1897:, p. 25 Of a general prime
900:is a subgroup of index 2 in the
688:extends the definition of parity
70:
3838:from the original on 2022-07-14
3777:Wong, Samuel Shaw Ming (1997),
3732: grdn000020011017ds7d00bzg
3633: chi0000020010826dvbu0119h
3558: CGAZ000020060207e226000bh
3212: CGAZ000020071027e3ap0001l
2850: SMHH000020060315e23g0004z
2823: SMHH000020060309e23a00049
2695:, Lawrence Erlbaum Associates,
2673:Barbeau, Edward Joseph (2003),
1312:is flashed to the subject on a
3805:"Zero odd/even: Is Zero Even?"
3745:, Routledge Kegan & Paul,
3611:, Cambridge University Press,
3516:, Cambridge University Press,
3259:Krantz, Steven George (2001),
3071:-adic numbers: an introduction
2846:(First ed.), p. 20,
2819:(First ed.), p. 18,
2731:, Cambridge University Press,
2585:Arnold, C. L. (January 1919),
1535:Ball, Lewis & Thames (2008
1050:Percentage responses over time
517:is odd if there is an integer
1:
3803:Matousek, John (2001-03-28),
3530:Siegel, Robert (1999-11-19),
3468:Partee, Barbara Hall (1978),
3220:Kaplan SAT 2400, 2005 Edition
3031:, Educational Testing Service
2783:"What is the Smallest Prime?"
2189:Ball, Lewis & Thames 2008
2114:Ball, Lewis & Thames 2008
2102:Ball, Lewis & Thames 2008
1855:Tabachnikova & Smith 2000
937:equivalence modulo this ideal
698:is even, including zero, and
162:always have the same parity.
3629:(5XS ed.), p. 50,
3384:Morgan, Frank (2001-04-05),
3319:Lorentz, Richard J. (1994),
3293:10.1016/j.jmathb.2007.05.004
2787:Journal of Integer Sequences
2591:The Ohio Educational Monthly
1819:Hartsfield & Ringel 2003
1304:, a pioneer in the field of
1271:Implications for instruction
662:+ 1) is even if and only if
444:numbers contain exceptions:
3884:Is Zero Even? - Numberphile
3638:Stewart, Mark Alan (2001),
3352:10.1016/j.entcs.2007.09.021
2931:10.1037/0096-3445.122.3.371
2204:, p. 93), referencing
2141:Dickerson & Pitman 2012
2130:Dickerson & Pitman 2012
1640:Caldwell & Xiong (2012)
1584:Dickerson & Pitman 2012
1265:University of South Florida
1161:Zero is not an even number.
207:, which is relevant to the
3931:
3886:, video with James Grime,
3663:Milwaukee Journal Sentinel
3656:Stingl, Jim (2006-04-05),
3486:Penner, Robert C. (1999),
3132:Grimes, Joseph E. (1975),
2889:Cutler, Thomas J. (2008),
2869:Crumpacker, Bunny (2007),
2515:Baroody & Coslick 1998
2153:Ball, Hill & Bass 2005
1034:, such sequences actually
1011:to be the number of times
741:Constructing a bipartition
702:of even ordinals are odd.
574:has an even order, and an
36:The weighing pans of this
3563:Snow, Tony (2001-02-23),
3446:10.1080/02724980343000512
3414:; Wenzel, Markus (2002),
3197:10.1080/07370000802177235
3184:Cognition and Instruction
2844:The Sydney Morning Herald
2817:The Sydney Morning Herald
2767:, Van Nostrand Reinhold,
2553:, London, UK: CRC Press,
2387:, pp. 108, 295–297;
1913:…" (ellipsis in original)
1763:Lovas & Pfenning 2008
1624:Caldwell & Xiong 2012
319:With the introduction of
3888:University of Nottingham
3705:"A question around zero"
3689:, London, UK: Springer,
3390:Frank Morgan's Math Chat
3364:Vesztergombi, Katalin L.
3064:Gouvêa, Fernando Quadros
2763:Bunch, Bryan H. (1982),
2745:Brisman, Andrew (2004),
2535:, London, UK: Springer,
2307:Wilden & Hammer 1987
1859:Anderson & Feil 2005
1374:, or as an example of a
1354:École Normale Supérieure
1228:
1125:Zero is not even or odd.
1118:Claims made by students
1076:English education system
1061:
816:of even numbers and the
503:Möbius inversion formula
3681:Tabachnikova, Olga M.;
3640:30 Days to the GMAT CAT
3136:The Thread of Discourse
3096:Oxford University Press
2727:Border, Kim C. (1985),
2285:, pp. 838, 860–861
1831:Dummit & Foote 1999
1290:smallest space analysis
1206:Deborah Loewenberg Ball
1152:Zero has to be an even.
680:Isabelle theorem prover
499:multiplicative function
145:has the same parity as
3828:"Is zero odd or even?"
3759:Wise, Stephen (2002),
3687:Topics in Group Theory
3301:The Arithmetic Teacher
3223:, Simon and Schuster,
2964:Diagram Group (1983),
2567:Andrews, Edna (1990),
2531:Anderson, Ian (2001),
2423:Sones & Sones 2002
2361:Column 8 readers 2006b
2357:Column 8 readers 2006a
2353:Sones & Sones 2002
1507:existential quantifier
1370:that every rule has a
1293:
1242:does not vary between
1239:University of Michigan
1051:
990:fast Fourier transform
797:
793:(blue) as subgroup of
742:
719:computational geometry
710:
640:
335:and 0 is even because
309:
276:
201:computational geometry
111:. In other words, its
66:
46:Listen to this article
41:
3905:Elementary arithmetic
3826:Adams, Cecil (1999),
3642:, Stamford: Thomson,
3542:National Public Radio
3537:All Things Considered
3217:Kaplan Staff (2004),
3206:Charleston Daily Mail
3140:, Walter de Gruyter,
1550:, p. 535) Fig. 1
1287:
1235:mathematics education
1049:
1003:, one may define the
979:bit-reversed ordering
788:
740:
709:Point in polygon test
708:
638:
568:vertex of even degree
477:Countless results in
473:Mathematical contexts
425:even × integer = even
308:
274:
221:mathematics education
209:binary numeral system
65:
35:
3910:Parity (mathematics)
3880:at Wikimedia Commons
3781:, World Scientific,
3412:Paulson, Lawrence C.
3321:Recursive Algorithms
3313:10.5951/AT.19.7.0535
2970:, Paddington Press,
2229:See data throughout
1935:Salzmann et al. 2007
1791:Salzmann et al. 2007
857:equivalence relation
806:algebraic structures
649:recursive definition
631:Even-odd alternation
554:A classic result of
97:More spoken articles
3711:, Drexel University
3570:Jewish World Review
3362:; Pelikán, József;
3323:, Intellect Books,
2809:2012arXiv1209.2007C
2662:: 13–44 and 195–200
2614:The Pantaneto Forum
2167:, pp. 446–447.
1485:is the most 'even.'
1306:numerical cognition
1280:Numerical cognition
1229:Teachers' knowledge
1134:Zero could be even.
1062:Students' knowledge
1023:in higher algebra.
853:equivalence classes
745:In graph theory, a
448:even ± even = even
193:recursively defined
115:—the quality of an
3551:Charleston Gazette
2908:Dehaene, Stanislas
2503:Diagram Group 1983
2462:, pp. 237–238
2261:, pp. 376–377
2249:, pp. 374–376
1861:, pp. 437–438
1725:, pp. 127–128
1406:odd–even rationing
1391:standardized tests
1294:
1248:reform mathematics
1052:
983:integer data types
847:the integers into
798:
781:Algebraic patterns
743:
711:
641:
588:degree sum formula
419:even ± even = even
310:
277:
267:Basic explanations
67:
42:
3876:Media related to
3864:Arithmetic portal
3832:The Straight Dope
3788:978-981-02-3043-2
3770:978-0-415-24651-4
3752:978-0-7100-9868-9
3696:978-1-85233-235-8
3649:978-0-7689-0635-6
3627:Chicago Sun-Times
3618:978-0-521-56473-1
3523:978-0-521-86516-6
3501:978-981-02-4088-2
3479:978-90-277-0809-0
3425:978-3-540-43376-7
3377:978-0-387-95585-8
3330:978-1-56750-037-0
3272:978-1-58488-052-3
3252:978-1-59311-495-4
3230:978-0-7432-6035-0
3174:978-0-486-43232-8
3147:978-90-279-3164-1
3125:978-0-9765709-0-5
3105:978-0-19-285361-5
3080:978-3-540-62911-5
3043:Freudenthal, Hans
3015:978-0-471-36857-1
2977:978-0-448-22202-8
2900:978-1-55750-221-6
2882:978-0-312-36005-4
2774:978-0-442-24905-2
2756:978-1-4027-1300-2
2738:978-0-521-38808-5
2720:978-0-7425-0202-4
2702:978-0-8058-3105-4
2684:978-0-387-40627-5
2633:American Educator
2587:"The Number Zero"
2578:978-0-8223-0959-8
2560:978-1-58488-515-3
2542:978-1-85233-236-5
2373:Kaplan Staff 2004
2233:, and summary by
2206:Freudenthal (1983
1548:Lichtenberg (1972
1365:Everyday contexts
1302:Stanislas Dehaene
1256:standardized test
1203:
1202:
1056:primary education
910:even permutations
898:alternating group
896:Analogously, the
855:of the following
814:additive inverses
810:additive identity
757:graph has no odd
584:handshaking lemma
454:odd ± odd = even
259:so zero is even.
63:
16:(Redirected from
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3006:Abstract Algebra
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2933:, archived from
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2200:As concluded by
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2180:
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2174:
2168:
2165:Hill et al. 2008
2162:
2156:
2155:, pp. 14–16
2150:
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2089:, pp. 83–95
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1197:Zero is special.
1143:Zero is not odd.
1115:
994:binary expansion
949:
880:
868:
802:abstract algebra
715:point in polygon
622:
550:
543:
534:by contradiction
531:
520:
516:
496:
493:takes the value
422:odd ± odd = even
338:
337:0 = (2 × 0) + 0.
334:
333:1 = (2 × 0) + 1,
330:
326:
258:
254:
243:Why zero is even
231:and using it in
226:
181:identity element
178:
161:
155:
148:
144:
134:
130:
103:In mathematics,
87:
85:
74:
73:
64:
54:
52:
47:
21:
18:Evenness of zero
3930:
3929:
3925:
3924:
3923:
3921:
3920:
3919:
3895:
3894:
3862:
3857:
3855:
3852:
3841:
3839:
3825:
3818:
3816:
3802:
3799:
3797:Further reading
3794:
3789:
3776:
3771:
3758:
3753:
3736:
3721:
3714:
3712:
3702:
3697:
3683:Smith, Geoff C.
3680:
3673:
3671:
3655:
3650:
3637:
3624:
3619:
3603:
3596:
3594:
3582:
3575:
3573:
3565:"Bubba's fools"
3562:
3554:, p. P1B,
3547:
3529:
3524:
3507:
3502:
3485:
3480:
3467:
3431:
3426:
3406:
3399:
3397:
3383:
3378:
3358:
3336:
3331:
3318:
3298:
3278:
3273:
3258:
3253:
3236:
3231:
3216:
3208:, p. P1C,
3203:
3180:
3175:
3159:Ringel, Gerhard
3153:
3148:
3131:
3126:
3111:
3106:
3088:Gowers, Timothy
3086:
3081:
3062:
3053:
3041:
3034:
3032:
3028:
3021:
3016:
3003:
2988:
2983:
2978:
2963:
2950:
2943:
2941:
2937:
2914:
2906:
2901:
2888:
2883:
2868:
2852:
2841:
2825:
2814:
2780:
2775:
2762:
2757:
2744:
2739:
2726:
2721:
2708:
2703:
2690:
2685:
2672:
2665:
2663:
2651:
2646:
2630:
2623:
2621:
2607:
2600:
2598:
2584:
2579:
2566:
2561:
2548:
2543:
2530:
2526:
2521:
2513:
2509:
2501:
2497:
2482:
2478:
2470:
2466:
2458:
2454:
2446:
2444:
2436:
2433:
2429:
2421:
2417:
2413:, p. 165).
2399:
2395:
2383:
2379:
2371:
2367:
2351:
2347:
2332:
2328:
2317:
2313:
2305:
2301:
2293:
2289:
2281:
2277:
2269:
2265:
2257:
2253:
2245:
2241:
2237:, p. 837).
2228:
2224:
2216:
2212:
2199:
2195:
2187:
2183:
2175:
2171:
2163:
2159:
2151:
2147:
2139:
2135:
2124:
2120:
2112:
2108:
2100:
2093:
2085:
2081:
2073:
2069:
2061:
2057:
2049:
2045:
2037:
2033:
2025:
2021:
2013:
2009:
2001:
1997:
1985:
1981:
1973:
1969:
1960:
1956:
1948:
1941:
1933:
1929:
1921:
1917:
1893:
1889:
1881:
1877:
1869:
1865:
1853:
1849:
1841:
1837:
1829:
1825:
1813:
1809:
1801:
1797:
1789:
1785:
1777:
1773:
1765:, p. 115;
1757:
1753:
1745:
1741:
1733:
1729:
1721:
1717:
1709:
1702:
1694:
1690:
1682:
1678:
1670:
1663:
1655:
1646:
1634:
1630:
1626:, pp. 5–6.
1622:
1618:
1611:
1606:
1602:
1594:
1590:
1582:
1578:
1570:
1566:
1558:
1554:
1545:
1541:
1533:
1529:
1522:
1510:
1493:
1489:
1479:
1470:
1463:
1459:
1434:In the game of
1418:New South Wales
1367:
1334:
1330:
1282:
1273:
1233:Researchers of
1231:
1064:
1044:
964:
944:
902:symmetric group
887:binary relation
870:
860:
783:
747:bipartite graph
692:ordinal numbers
690:to transfinite
633:
613:
594:Sperner's lemma
576:isolated vertex
545:
537:
522:
518:
514:
511:
494:
491:Möbius function
475:
345:
343:Defining parity
336:
332:
328:
324:
269:
256:
252:
245:
224:
189:natural numbers
166:
157:
150:
146:
136:
132:
128:
127:, specifically
101:
100:
89:
83:
81:
78:This audio file
75:
68:
59:
56:
50:
49:
45:
28:
23:
22:
15:
12:
11:
5:
3928:
3926:
3918:
3917:
3912:
3907:
3897:
3896:
3891:
3890:
3881:
3878:Parity of zero
3868:
3867:
3851:
3850:External links
3848:
3847:
3846:
3823:
3798:
3795:
3793:
3792:
3787:
3774:
3769:
3756:
3751:
3734:
3728:, p. 23,
3719:
3700:
3695:
3678:
3653:
3648:
3635:
3622:
3617:
3605:Starr, Ross M.
3601:
3580:
3560:
3545:
3527:
3522:
3505:
3500:
3483:
3478:
3465:
3440:(5): 835–863,
3429:
3424:
3408:Nipkow, Tobias
3404:
3381:
3376:
3360:Lovász, László
3356:
3334:
3329:
3316:
3307:(7): 535–538,
3296:
3276:
3271:
3256:
3251:
3234:
3229:
3214:
3201:
3190:(4): 430–511,
3178:
3173:
3151:
3146:
3129:
3124:
3109:
3104:
3084:
3079:
3060:
3051:
3039:
3019:
3014:
3001:
2981:
2976:
2961:
2948:
2925:(3): 371–396,
2904:
2899:
2886:
2881:
2866:
2839:
2812:
2778:
2773:
2760:
2755:
2742:
2737:
2724:
2719:
2706:
2701:
2688:
2683:
2670:
2644:
2628:
2605:
2582:
2577:
2564:
2559:
2546:
2541:
2527:
2525:
2522:
2520:
2519:
2517:, p. 1.33
2507:
2495:
2476:
2464:
2452:
2443:, p. 3236
2427:
2415:
2393:
2377:
2365:
2345:
2334:Steinberg 1999
2326:
2311:
2299:
2287:
2275:
2263:
2251:
2239:
2222:
2210:
2208:, p. 460)
2193:
2181:
2169:
2157:
2145:
2133:
2118:
2106:
2091:
2079:
2067:
2063:Frobisher 1999
2055:
2051:Frobisher 1999
2043:
2039:Frobisher 1999
2031:
2027:Frobisher 1999
2019:
2015:Frobisher 1999
2007:
2003:Frobisher 1999
1995:
1987:Frobisher 1999
1979:
1975:Frobisher 1999
1967:
1965:, p. 85).
1954:
1950:Frobisher 1999
1939:
1927:
1915:
1887:
1875:
1863:
1857:, p. 99;
1847:
1835:
1823:
1817:, p. 53;
1807:
1795:
1783:
1771:
1751:
1739:
1727:
1715:
1700:
1688:
1676:
1661:
1644:
1628:
1616:
1600:
1588:
1586:, p. 191.
1576:
1564:
1552:
1539:
1527:
1497:, p. 34:
1487:
1460:
1458:
1455:
1451:odds and evens
1414:license plates
1376:trick question
1372:counterexample
1366:
1363:
1281:
1278:
1272:
1269:
1230:
1227:
1214:concept images
1201:
1200:
1192:
1191:
1183:
1182:
1174:
1173:
1165:
1164:
1156:
1155:
1147:
1146:
1138:
1137:
1129:
1128:
1120:
1119:
1063:
1060:
1043:
1040:
1032:2-adic numbers
963:
960:
914:transpositions
782:
779:
725:, one casts a
668:
667:
656:
645:natural number
632:
629:
627:of a simplex.
558:states that a
510:
507:
483:factorizations
474:
471:
466:
465:
464:integer = even
458:
452:
441:
440:
437:
434:
427:
426:
423:
420:
413:multiplication
344:
341:
321:multiplication
268:
265:
244:
241:
90:
76:
69:
57:
44:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3927:
3916:
3913:
3911:
3908:
3906:
3903:
3902:
3900:
3893:
3889:
3885:
3882:
3879:
3874:
3870:
3869:
3865:
3854:
3849:
3837:
3833:
3829:
3824:
3815:on 2020-11-29
3814:
3810:
3806:
3801:
3800:
3796:
3790:
3784:
3780:
3775:
3772:
3766:
3763:, CRC Press,
3762:
3757:
3754:
3748:
3743:
3742:
3735:
3731:
3727:
3726:
3720:
3710:
3706:
3701:
3698:
3692:
3688:
3684:
3679:
3670:on 2006-04-27
3669:
3665:
3664:
3659:
3654:
3651:
3645:
3641:
3636:
3632:
3628:
3623:
3620:
3614:
3610:
3606:
3602:
3593:, p. C07
3592:
3591:
3586:
3581:
3572:
3571:
3566:
3561:
3557:
3553:
3552:
3546:
3543:
3539:
3538:
3533:
3528:
3525:
3519:
3514:
3513:
3506:
3503:
3497:
3492:
3491:
3484:
3481:
3475:
3471:
3466:
3463:
3459:
3455:
3451:
3447:
3443:
3439:
3435:
3430:
3427:
3421:
3417:
3413:
3409:
3405:
3396:on 2009-01-08
3395:
3391:
3387:
3382:
3379:
3373:
3369:
3365:
3361:
3357:
3353:
3348:
3344:
3340:
3335:
3332:
3326:
3322:
3317:
3314:
3310:
3306:
3302:
3297:
3294:
3290:
3286:
3282:
3277:
3274:
3268:
3265:, CRC Press,
3264:
3263:
3257:
3254:
3248:
3243:
3242:
3235:
3232:
3226:
3222:
3221:
3215:
3211:
3207:
3202:
3198:
3193:
3189:
3185:
3179:
3176:
3170:
3166:
3165:
3160:
3156:
3152:
3149:
3143:
3138:
3137:
3130:
3127:
3121:
3117:
3116:
3110:
3107:
3101:
3097:
3093:
3089:
3085:
3082:
3076:
3072:
3068:
3065:
3061:
3057:
3052:
3048:
3044:
3040:
3027:
3026:
3020:
3017:
3011:
3007:
3002:
2998:
2994:
2987:
2982:
2979:
2973:
2969:
2968:
2962:
2958:
2954:
2953:New Scientist
2949:
2940:on 2011-07-19
2936:
2932:
2928:
2924:
2920:
2913:
2909:
2905:
2902:
2896:
2892:
2887:
2884:
2878:
2875:, Macmillan,
2874:
2873:
2867:
2863:
2857:
2849:
2845:
2840:
2836:
2830:
2822:
2818:
2813:
2810:
2806:
2801:
2796:
2792:
2788:
2784:
2779:
2776:
2770:
2766:
2761:
2758:
2752:
2748:
2743:
2740:
2734:
2730:
2725:
2722:
2716:
2712:
2707:
2704:
2698:
2694:
2689:
2686:
2680:
2676:
2671:
2661:
2657:
2650:
2645:
2642:
2641:2027.42/65072
2638:
2634:
2629:
2620:on 2007-09-25
2619:
2615:
2611:
2606:
2596:
2592:
2588:
2583:
2580:
2574:
2570:
2565:
2562:
2556:
2552:
2547:
2544:
2538:
2534:
2529:
2528:
2523:
2516:
2511:
2508:
2505:, p. 213
2504:
2499:
2496:
2493:
2489:
2485:
2480:
2477:
2474:, p. 153
2473:
2468:
2465:
2461:
2456:
2453:
2442:
2441:
2431:
2428:
2424:
2419:
2416:
2412:
2408:
2407:
2402:
2397:
2394:
2390:
2386:
2381:
2378:
2375:, p. 227
2374:
2369:
2366:
2362:
2358:
2354:
2349:
2346:
2343:
2339:
2335:
2330:
2327:
2324:
2320:
2315:
2312:
2309:, p. 104
2308:
2303:
2300:
2296:
2291:
2288:
2284:
2279:
2276:
2272:
2267:
2264:
2260:
2255:
2252:
2248:
2243:
2240:
2236:
2232:
2226:
2223:
2219:
2214:
2211:
2207:
2203:
2197:
2194:
2190:
2185:
2182:
2179:, p. 535
2178:
2173:
2170:
2166:
2161:
2158:
2154:
2149:
2146:
2142:
2137:
2134:
2131:
2127:
2122:
2119:
2116:, p. 16.
2115:
2110:
2107:
2103:
2098:
2096:
2092:
2088:
2083:
2080:
2076:
2071:
2068:
2064:
2059:
2056:
2052:
2047:
2044:
2040:
2035:
2032:
2028:
2023:
2020:
2016:
2011:
2008:
2004:
1999:
1996:
1992:
1988:
1983:
1980:
1976:
1971:
1968:
1964:
1958:
1955:
1951:
1946:
1944:
1940:
1937:, p. 224
1936:
1931:
1928:
1924:
1919:
1916:
1912:
1908:
1904:
1900:
1896:
1891:
1888:
1885:, p. 479
1884:
1879:
1876:
1872:
1867:
1864:
1860:
1856:
1851:
1848:
1845:, p. 100
1844:
1839:
1836:
1832:
1827:
1824:
1820:
1816:
1815:Anderson 2001
1811:
1808:
1804:
1799:
1796:
1793:, p. 168
1792:
1787:
1784:
1781:, p. 165
1780:
1775:
1772:
1769:, p. 127
1768:
1764:
1760:
1755:
1752:
1748:
1743:
1740:
1736:
1731:
1728:
1724:
1719:
1716:
1712:
1707:
1705:
1701:
1698:, p. 34.
1697:
1692:
1689:
1685:
1680:
1677:
1673:
1668:
1666:
1662:
1659:, p. xxi
1658:
1653:
1651:
1649:
1645:
1641:
1637:
1632:
1629:
1625:
1620:
1617:
1609:
1604:
1601:
1597:
1592:
1589:
1585:
1580:
1577:
1573:
1568:
1565:
1561:
1556:
1553:
1549:
1543:
1540:
1536:
1531:
1528:
1518:
1514:
1508:
1504:
1500:
1496:
1491:
1488:
1482:
1477:
1473:
1468:
1466:
1462:
1456:
1454:
1452:
1449:The game of "
1447:
1445:
1441:
1437:
1432:
1430:
1425:
1423:
1419:
1415:
1411:
1407:
1402:
1400:
1396:
1392:
1387:
1384:
1383:calendar date
1379:
1377:
1373:
1364:
1362:
1358:
1357:definition".
1355:
1351:
1346:
1344:
1340:
1326:
1324:
1319:
1315:
1311:
1307:
1303:
1299:
1298:reaction time
1291:
1286:
1279:
1277:
1270:
1268:
1266:
1260:
1257:
1253:
1252:United States
1249:
1245:
1240:
1236:
1226:
1224:
1220:
1219:undergraduate
1215:
1210:
1207:
1198:
1194:
1193:
1189:
1188:Zero is even.
1185:
1184:
1180:
1176:
1175:
1171:
1167:
1166:
1162:
1158:
1157:
1153:
1149:
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1079:
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1033:
1029:
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1022:
1018:
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1010:
1006:
1002:
997:
995:
991:
988:
984:
980:
975:
973:
969:
961:
959:
957:
953:
947:
942:
938:
934:
930:
925:
923:
922:empty product
919:
915:
911:
907:
903:
899:
894:
892:
888:
884:
878:
874:
867:
863:
858:
854:
850:
846:
841:
839:
835:
831:
827:
823:
819:
818:associativity
815:
811:
807:
803:
796:
792:
787:
780:
778:
776:
772:
768:
764:
760:
756:
752:
748:
739:
735:
732:
728:
724:
720:
716:
707:
703:
701:
697:
696:limit ordinal
693:
689:
685:
681:
677:
673:
665:
661:
657:
654:
653:
652:
650:
646:
637:
630:
628:
626:
621:
617:
611:
607:
603:
602:triangulation
599:
595:
591:
589:
585:
581:
577:
573:
569:
565:
561:
557:
552:
548:
541:
535:
529:
525:
509:Not being odd
508:
506:
504:
500:
492:
488:
484:
480:
479:number theory
472:
470:
463:
459:
457:
453:
451:
447:
446:
445:
438:
435:
432:
431:
430:
424:
421:
418:
417:
416:
414:
410:
406:
402:
396:
394:
390:
386:
382:
378:
374:
370:
366:
365:Prime numbers
362:
358:
354:
350:
342:
340:
322:
317:
315:
307:
303:
301:
296:
292:
288:
286:
282:
273:
266:
264:
260:
250:
242:
240:
238:
234:
230:
222:
217:
216:reaction time
212:
210:
206:
202:
198:
194:
190:
186:
182:
177:
173:
169:
163:
160:
154:
143:
139:
135:is even then
126:
122:
118:
114:
110:
106:
98:
94:
79:
39:
38:balance scale
34:
30:
19:
3892:
3840:, retrieved
3831:
3817:, retrieved
3813:the original
3809:Ask Dr. Math
3808:
3778:
3760:
3740:
3725:The Guardian
3723:
3713:, retrieved
3708:
3686:
3672:, retrieved
3668:the original
3661:
3639:
3626:
3608:
3595:, retrieved
3590:Deseret News
3588:
3574:, retrieved
3568:
3549:
3535:
3511:
3489:
3469:
3437:
3433:
3418:, Springer,
3415:
3398:, retrieved
3394:the original
3389:
3370:, Springer,
3367:
3342:
3338:
3320:
3304:
3300:
3287:(2): 83–95,
3284:
3280:
3261:
3240:
3219:
3205:
3187:
3183:
3163:
3135:
3114:
3091:
3070:
3067:
3055:
3046:
3033:, retrieved
3024:
3005:
2996:
2992:
2966:
2956:
2952:
2942:, retrieved
2935:the original
2922:
2918:
2890:
2871:
2843:
2816:
2790:
2786:
2764:
2749:, Sterling,
2746:
2728:
2710:
2692:
2677:, Springer,
2674:
2664:, retrieved
2659:
2655:
2632:
2622:, retrieved
2618:the original
2613:
2599:, retrieved
2594:
2590:
2568:
2550:
2532:
2524:Bibliography
2510:
2498:
2488:Hohmann 2007
2479:
2472:Brisman 2004
2467:
2455:
2445:, retrieved
2439:
2430:
2418:
2404:
2396:
2380:
2368:
2348:
2329:
2314:
2302:
2290:
2278:
2266:
2254:
2242:
2225:
2213:
2196:
2184:
2172:
2160:
2148:
2136:
2121:
2109:
2082:
2070:
2058:
2046:
2034:
2022:
2010:
1998:
1982:
1970:
1957:
1952:, p. 41
1930:
1918:
1910:
1906:
1902:
1898:
1890:
1878:
1873:, p. 98
1871:Barbeau 2003
1866:
1850:
1843:Andrews 1990
1838:
1833:, p. 48
1826:
1821:, p. 28
1810:
1798:
1786:
1774:
1759:Lorentz 1994
1754:
1742:
1730:
1718:
1691:
1679:
1672:Stewart 2001
1631:
1619:
1603:
1591:
1579:
1567:
1555:
1542:
1530:
1516:
1512:
1502:
1490:
1480:
1448:
1433:
1426:
1403:
1388:
1380:
1368:
1359:
1347:
1343:power of two
1327:
1323:milliseconds
1300:experiment.
1295:
1274:
1261:
1232:
1221:mathematics
1211:
1204:
1196:
1187:
1178:
1169:
1160:
1151:
1142:
1133:
1124:
1110:
1103:
1098:
1094:
1090:
1086:
1082:
1080:
1065:
1053:
1025:
1012:
1008:
1005:2-adic order
1000:
998:
987:Cooley–Tukey
976:
965:
962:2-adic order
948:≡ 0 (mod 2).
945:
940:
926:
918:identity map
905:
895:
876:
872:
865:
861:
842:
834:closed under
799:
794:
790:
774:
770:
762:
744:
713:The classic
712:
684:Peano axioms
669:
666:is not even.
663:
659:
642:
625:orientations
619:
615:
592:
579:
556:graph theory
553:
549:= −1/2
546:
539:
527:
523:
512:
501:and for the
487:product of 0
476:
467:
461:
455:
449:
442:
428:
397:
347:The precise
346:
329:(2 × ▢) + 1;
318:
311:
297:
293:
289:
280:
278:
261:
246:
213:
197:graph theory
175:
171:
167:
164:
158:
152:
141:
137:
102:
29:
3386:"Old Coins"
3345:: 113–128,
2675:Polynomials
2492:Turner 1996
2460:Cutler 2008
2401:Arsham 2002
2391:, p. 1
2342:Stingl 2006
2338:Siegel 1999
2323:Morgan 2001
2295:Grimes 1975
1991:summer term
1925:, p. 4
1923:Krantz 2001
1895:Gouvêa 1997
1747:Border 1985
1696:Penner 1999
1684:Devlin 1985
1657:Partee 1978
1636:Gowers 2002
1612:(2 × ▢) + 0
1495:Penner 1999
1483:000⋯000 = 0
1472:Arnold 1919
1244:traditional
1106:times table
968:doubly even
956:polynomials
883:reflexivity
572:empty graph
409:subtraction
325:(2 × ▢) + 0
300:number line
237:abstraction
109:even number
3915:0 (number)
3899:Categories
3842:2013-06-06
3819:2013-06-06
3761:GIS Basics
3715:2007-09-25
3674:2014-06-21
3597:2014-06-21
3576:2009-08-22
3400:2009-08-22
3035:2011-09-06
2944:2007-09-13
2666:2010-03-04
2624:2007-09-24
2601:2010-04-11
2597:(1): 21–22
2484:Smock 2006
2447:2013-06-02
2075:Keith 2006
1779:Bunch 1982
1735:Starr 1997
1457:References
1329:calculate
1259:building.
1099:don't know
972:power of 2
717:test from
700:successors
672:successors
655:0 is even.
580:odd number
521:such that
436:−3 + 3 = 0
361:degenerate
353:convention
349:definition
281:no objects
233:arithmetic
205:power of 2
93:Audio help
84:2013-08-27
2999:: 187–195
2800:1209.2007
2319:Snow 2001
1883:Wong 1997
1803:Wise 2002
1523:0 = 2 ⋅ 0
1476:Wong 1997
1444:prop bets
1335:2 × 0 = 0
1331:0 × 2 = 0
1042:Education
1038:to zero.
1021:valuation
845:partition
755:connected
731:algorithm
610:induction
513:A number
505:to work.
456:(or zero)
450:(or zero)
439:4 × 0 = 0
433:2 − 2 = 0
385:Kronecker
285:empty set
225:0 × 2 = 0
149:—indeed,
3836:archived
3685:(2000),
3607:(1997),
3462:10672272
3454:15204120
3366:(2003),
3161:(2003),
3090:(2002),
3066:(1997),
3045:(1983),
2856:citation
2829:citation
1993:of 1992.
1546:Compare
1436:roulette
1422:Maryland
1410:gasoline
1318:computer
1316:, and a
1036:converge
1028:sequence
1017:infinity
838:identity
830:nonempty
826:subgroup
767:distance
694:: every
678:and the
598:coloring
495:μ(1) = 1
405:addition
377:Legendre
369:Goldbach
314:counting
121:multiple
95: ·
3730:Factiva
3631:Factiva
3556:Factiva
3245:, IAP,
3210:Factiva
2848:Factiva
2821:Factiva
2805:Bibcode
1501:B.2.2,
1350:numeral
1314:monitor
1310:numeral
1237:at the
1091:neither
1074:of the
931:in the
885:of the
723:polygon
618:+ 1) +
606:simplex
562:of odd
462:nonzero
460:even ×
389:factors
373:Lambert
363:cases.
357:trivial
183:of the
117:integer
82: (
53:minutes
3785:
3767:
3749:
3693:
3646:
3615:
3520:
3498:
3476:
3460:
3452:
3422:
3374:
3327:
3269:
3249:
3227:
3171:
3144:
3122:
3102:
3077:
3012:
2974:
2959:(1452)
2897:
2879:
2771:
2753:
2735:
2717:
2699:
2681:
2575:
2557:
2539:
1515:(0 = 2
1440:casino
1223:majors
1097:, and
1072:Year 6
1068:Year 1
952:zeroes
943:where
916:. The
849:cosets
759:cycles
751:colors
469:zero.
411:, and
383:, and
381:Cayley
263:zero.
257:0 × 2,
229:number
113:parity
107:is an
3458:S2CID
3029:(PDF)
2989:(PDF)
2938:(PDF)
2915:(PDF)
2795:arXiv
2793:(9),
2652:(PDF)
2406:heute
1499:Lemma
1341:or a
1339:prime
1085:over
929:ideal
920:, an
891:index
822:group
769:from
604:of a
600:on a
564:order
560:graph
544:then
538:0 = 2
536:: if
253:5 × 2
249:prove
185:group
129:0 × 2
3783:ISBN
3765:ISBN
3747:ISBN
3691:ISBN
3644:ISBN
3613:ISBN
3518:ISBN
3496:ISBN
3474:ISBN
3450:PMID
3420:ISBN
3372:ISBN
3325:ISBN
3267:ISBN
3247:ISBN
3225:ISBN
3169:ISBN
3142:ISBN
3120:ISBN
3100:ISBN
3075:ISBN
3010:ISBN
2972:ISBN
2895:ISBN
2877:ISBN
2862:link
2835:link
2769:ISBN
2751:ISBN
2733:ISBN
2715:ISBN
2697:ISBN
2679:ISBN
2573:ISBN
2555:ISBN
2537:ISBN
1429:port
1420:and
1397:and
1395:GMAT
1246:and
1095:both
1083:even
933:ring
191:are
176:even
172:even
168:even
156:and
151:0 +
105:zero
3442:doi
3347:doi
3343:196
3309:doi
3289:doi
3192:doi
2957:106
2927:doi
2923:122
2660:M14
2637:hdl
1399:GRE
1389:In
1087:odd
1070:to
1007:of
981:of
954:of
904:on
893:2.
869:if
800:In
727:ray
542:+ 1
530:+ 1
526:= 2
359:or
327:or
199:to
123:of
3901::
3834:,
3830:,
3807:,
3707:,
3660:,
3587:,
3567:,
3540:,
3534:,
3456:,
3448:,
3438:57
3436:,
3410:;
3388:,
3341:,
3305:19
3303:,
3285:26
3283:,
3188:26
3186:,
3157:;
3098:,
3094:,
2995:,
2991:,
2955:,
2921:,
2917:,
2858:}}
2854:{{
2831:}}
2827:{{
2803:,
2791:15
2789:,
2785:,
2658:,
2654:,
2635:,
2612:,
2595:68
2593:,
2589:,
2490:;
2486:;
2340:;
2336:;
2321:;
2128:;
2094:^
1942:^
1703:^
1664:^
1647:^
1525:."
1519:),
1464:^
1424:.
1378:.
1199:"
1190:"
1181:"
1172:"
1163:"
1154:"
1145:"
1136:"
1127:"
1093:,
958:.
875:−
864:~
859::
840:.
676:LF
590:.
415::
407:,
379:,
375:,
371:,
174:=
170:−
140:+
51:31
3444::
3349::
3311::
3291::
3194::
3069:p
2997:2
2929::
2864:)
2837:)
2807::
2797::
2639::
2143:.
1911:p
1907:p
1903:p
1899:p
1642:.
1517:k
1513:k
1511:∃
1481:b
1195:"
1186:"
1177:"
1168:"
1159:"
1150:"
1141:"
1132:"
1123:"
1013:n
1009:n
1001:n
946:k
941:k
906:n
879:)
877:y
873:x
871:(
866:y
862:x
795:Z
791:Z
789:2
775:v
771:v
763:v
664:n
660:n
658:(
620:n
616:n
614:(
547:k
540:k
528:k
524:n
519:k
515:n
159:x
153:x
147:x
142:x
138:y
133:y
125:2
99:)
91:(
86:)
55:)
48:(
20:)
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