Evenness of zero

Evenness of zero

The number 0 is even. There are several ways to determine whether an integer is even or odd, all of which indicate that 0 is an even number: it is a multiple of 2, it is evenly divisible by 2, it is surrounded on both sides by odd integers, and it is the sum of an integer with itself. These proofs follow immediately from the definition of the term "even number", which does not allow in zero arbitrarily; it can be further motivated by the familiar rules for sums and products of even numbers. Within the even numbers, zero plays a central role: it is the identity element of the group of even integers, and it is the starting case from which all other even natural numbers are recursively generated. Every integer divides 0, including each power of 2; in this sense, 0 is the most even number of all.

On the other hand, psychologically speaking 0 is often the least even number of all. In reaction time experiments, most subjects are slower to call 0 even than other even numbers. Both students and teachers in primary education are prone to a common misconception that the parity of zero is ambiguous, or simply that zero is odd. Several researchers in mathematics education write that such misconceptions represent an opportunity for exploration. Class discussions can highlight the necessity of reasoning from agreed-upon definitions. Reviewing sentences like nowrap|1=0 × 2 = 0 can expose students' apprehensions about calling 0 a number and using it in arithmetic. While understanding zero is a worthy end in itself, the particular consideration of parity is an early example of extending a familiar concept to an unfamiliar and perhaps unexpected setting — a pervasive theme throughout mathematics.

In education

A number is called even if it is an integer multiple of 2. Zero is an integer multiple of 2, namely nowrap|0 × 2, so zero is even. [cite book |first=Robert C. |last=Penner |year=1999 |chapter=Lemma B.2.2, "The integer 0 is even and is not odd" |title=Discrete Mathematics: Proof Techniques and Mathematical Structures |location=River Edje |publisher=World Scientific |isbn=ISBN 981-02-4088-0 |pages=p. 34]

Mathematically, no further proof is required, but in the usual educational context, a little more explanation helps. The subject of the parity of zero is often tackled within the first two or three years of primary education, as the concept of even and odd numbers is introduced and developed. [This is the timeframe in United States, Canada, Great Britain, Australia, and Israel; see Levenson p. 85.] A student at this level may not yet have learned what "integer" or "multiple" means, much less how to multiply with 0. [See Ball's keynote for further discussion of appropriate definitions.] Age-appropriate explanations that zero is even, then, return to the concrete interpretation of parity in terms of paired objects, or they emphasize the even-odd-alternation between numbers. In fact, children may use the same explanations to convince each other. Meanwhile a number of misconceptions about 0 must be combatted, such as the belief that 0 means nothing and has no properties.

Explanations

Early in elementary school, numbers are used to count how many objects belong to a set. Zero is understood as the number of objects in the empty set, if not in so many words. Parity may then be introduced by making groups of two objects. If the objects in a set can be marked off into groups of two with none left over, the number of objects is even. Otherwise, one object is left over and the number is odd. The empty set contains zero groups of two, and no object is left over from this grouping, so zero is even. Although it is difficult to depict no groups of two, or to draw attention to the nonexistence of a leftover object, this conception of the evenness of zero can be illustrated by comparing the empty set with other sets, as on the right. See below for a set-theoretic formalization of this approach.

The connection of pairing with multiplication can be made more explicit by depicting elements in two rows . This picture has the advantage of simultaneously showcasing the properties of parity under addition: placing blocks next to each other, one sees that the sum of two even numbers is even, and the sum of an even and an odd is odd. Taking zero to be even preserves these two patterns.

Misconceptions

Len Frobisher conducted a pair of surveys of UK schoolchildren to determine how knowledge of single-digit parity translates to knowledge of multiple-digit parity, and zero figures prominently in the results. In a preliminary survey of nearly 400 seven-year-olds, 45% chose "even" over "odd" when asked the parity of zero. [Results are from the survey conducted in the mid-summer term of 1992; see Frobisher pp.37, 40, 42] A follow-up investigation offered more choices: "neither", "both", and "don't know". This time the number of children in the same age range identifying zero as even dropped to 32%. The latter study was differentiated by class, covering Years 1 to 6; results are on the right. 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. [These results are from the February 1999 study, including 481 children, from three schools at a variety of attainment levels; see Frobisher pp.40-42, 47]

In interviews, Frobisher elicited some of the students' reasoning. One fifth-year decided that 0 was even because it was found on the 2 times table. A couple fourth-years realized that zero can be split into equal parts: "no one gets owt if it's shared out." Another fourth-year reasoned "1 is odd and if I go down it's even." The interviews also revealed the misconceptions behind some 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".

Esther Levenson, Pessia Tsamir, and Dina Tirosh contrast "mathematically-based explanations", like the above examples, with "practically-based explanations", such as the following example for the number 14:

The researchers interviewed a pair of sixth-grade students who were performing highly in their mathematics class. Johnny preferred the mathematical explanations for 14, while Miri preferred the practical. Upon being asked to substitute 0 for 14, both students initially thought that 0 was neither even nor odd, for different reasons. Levenson et al's report in the Journal of Mathematical Behavior details the students' reasoning; one of the themes is that their beliefs about the parity of zero are consistent with their concepts of zero and division.

Johnny displays a common underlying error: he states that zero is not divisible by two or any other number. It is not that he confuses 0/2 with division by zero; after a discussion with the interviewer, he can write correct division sentences with 0. But he persists that 0 isn't divisible by two, apparently equating the number zero with nothing: "You can divide [0] by two, but you don’t get any answers." After further discussion, instead of conceding divisibility of zero, Johnny independently switches to a separate explanation to convince himself that 0 is even: 0 + 0 = 0.

Group discussions

Often curious students will directly ask if zero is even; the Israel National Mathematics Curriculum reminds first grade teachers that zero is even, but advises that it is unnecessary to mention this unless the class brings it up. [Levenson p.86, referring to the 2005 INMC] One study observed a class of 15 second grade students:blockquote|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. [cite conference |booktitle=Teachers Engaged in Research: Inquiry in Mathematics Classrooms, Grades Pre-K-2 |title=Mathematical Argument in a Second Grade Class: Generating and Justifying Generalized Statements about Odd and Even Numbers |first=Annie |last=Keith |pages=35-68 |year=2006 |publisher=IAP |id=ISBN 1593114958 |url=http://www.madison.k12.wi.us/sod/car/abstracts/217.pdf |accessdate=2007-08-27|format=PDF]

In another class of 22 third graders, Deborah Ball asked her students to reflect on "a particularly long and confusing discussion on even and odd numbers". One student commented that hearing other ideas had helped her understanding, and she now believed for the first time that zero was even. At the same time, another student had originally thought zero to be even but "got sort of mixed up" and wasn't sure whom to agree with. Ball finds it significant that the latter student expressed a desire to listen further to the discussion: in this sense, both students have learned something valuable about their own learning process. [cite journal |title=With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics |first=Deborah Loewenberg |last=Ball |journal=The Elementary School Journal |volume=93 |issue=4 |year=1993 |month=March |pages=373–397 |url=http://links.jstor.org/sici?sici=0013-5984%28199303%2993%3A4%3C373%3AWAEOTM%3E2.0.CO%3B2-F |doi=10.1086/461730 Quotation on p.392; emphasis is the author's.]

Later on, during a discussion on fractions, Ball asked the class whether or not voting would be a good way to prove what is true in mathematics. One of the students returned to her experience of the discussion on zero:

Teachers' knowledge

The National Council of Teachers of Mathematics's "Principles and Standards for School Mathematics" records a first grader's argument that zero is an even number: "If zero were odd, then 0 and 1 would be two odd numbers in a row. Even and odd numbers alternate. So 0 must be even…" In a survey of 10 college students preparing to teach mathematics, none of them thought that the argument sufficed as a mathematical proof. When they were told that it had been written by a first grader, most agreed that it was acceptable reasoning for that age level. [cite conference |last=Dickerson |first=David |title=Aspects of preservice teachers' understandings of the purposes of mathematical proof |booktitle=Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education |editor=Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds) |year=2006 |pages=pp. 710-716 |url=http://www.pmena.org/2006/cd/preservice.htm |location=Mérida, Mexico |publisher=Universidad Pedagógica Nacional |isbn=970-702-202-7]

Betty Lichtenberg, an associate professor of mathematics education at the University of South Florida, draws on her experience with a course she and her colleagues taught on methods for teaching arithmetic. She reports in 1972 that several sections 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", and about two thirds answered "False". [Lichtenberg p.535]

Researchers of mathematics education at the University of Michigan used the true-or-false prompt "0 is an even number", among many similar questions, in a 2000-2004 study of 700 primary teachers in the United States. For them the question exemplifies "common knowledge … that any well-educated adult should have", and it is "ideologically neutral" in that the answer does not vary between traditional and reform mathematics. Although the authors do not describe results for individual questions, overall performance in the study significantly predicted improvements in students' standardized test scores after taking the teachers' classes. [cite journal |author=Ball, Deborah Loewenberg, Heather C. Hill, and Hyman Bass |year=2005 |month=Fall |title=Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? |journal=American Educator |pages=14–46 |url=http://www.aft.org/pubs-reports/american_educator/issues/fall2005/BallF05.pdf |accessdate=2007-09-16|format=PDF]

Numerical cognition

Even among adults who believe that zero is even, the fact can be unfamiliar enough to measurably delay its recall. This phenomenon is known from one of the simplest tools in the field of numerical cognition: the reaction time experiment. To investigate the task of parity determination, a numeral or a number word is flashed to the subject on a monitor, and a computer records the time it takes the subject to identify the number as odd or even by striking an appropriate button, such as a Morse key. A series of such experiments led by Stanislas Dehaene in the early 1990s found that 0 was significantly slower to process than other even numbers. Some variations of the experiment found delays as long as 60 milliseconds or about 10% of the average reaction time. [See data throughout Dehaene et al., and summary by Nuerk et al. p.837]

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 extraction model, the mental calculation hypothesis, suggests that reactions to 0 should be fast: 0 is a small number, and it is easy to calculate nowrap|1=0 × 2 = 0. (There is some subtlety here: subjects are known to compute and name the result of multiplication by zero faster than multiplication of nonzero numbers, but they are slower to verify proposed results like nowrap|1=2 × 0 = 0.) 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 prime or a power of two. Both the sequence of powers of two and the sequence of positive evens 2, 4, 6, 8, … are well-distinguished mental categories whose members are prototypically even. Zero belongs to neither list, hence the slower responses. [Dehaene et al. 374-376]

Repeated experiments have showed a delay at zero for subjects from a variety of national and linguistic backgrounds, representing both left to right and right to left writing systems; almost all right-handed; from 17-53 years of age; confronted with number names in numeral 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 École Normale Supérieure 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 [literary] 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 definition". [Dehaene et al. 376-377] This strong dependence on familiarity again undermines the mental calculation hypothesis. In the view of some researchers, the effect also implies that it is inappropriate to include zero in parity judgment experiments at all. [Nuerk et al. 860-861]

Nominal status

Everyday use

From time to time, the question "Is zero even?" makes an appearance outside of elementary school. It provides material for Internet message boards, [cite web |url=http://mathforum.org/kb/message.jspa?messageID=1178542 |title=A question around zero
work=Math Forum » Discussions » History » Historia-Matematica |publisher=Drexel University |accessdate=2007-09-25 |author=Forum participants
] ask-the-expert websites, [cite web |url=http://www.straightdope.com/mailbag/mzeroeven.html |title=Is zero odd or even? |work=The Straight Dope Mailbag |author=Straight Dope Science Advisory Board |accessdate=2007-09-24] [cite web |url=http://mathforum.org/library/drmath/view/57188.html |title=Is Zero Even? |work=Ask Dr. Math |publisher=The Math Forum |author=Doctor Rick |accessdate=2007-09-24] and bored linguists at cocktail parties. [cite book |first=Joseph E. |last=Grimes |title=The Thread of Discourse |year=1975 |publisher=Walter de Gruyter |isbn=902793164X |pages=156 "...one can pose the following questions to married couples of his acquaintance: (1) Is zero an even number? ... Many couples disagree..."] There are also some situations where calling zero even or not has more serious consequences.

The above example of non-standard usage is found in a third-party study guide for the GMAT. Although that guide states that 0 is not even, the test's authors publish an official study guide that explicitly includes 0 in the even numbers. The correct answers to some of the GMAT's data sufficiency questions assume that the usual rules for even numbers, such as "n" being even if nowrap|("n" + 2) is even, hold without exception for 0. [cite book |title=The Official Guide for GMAT Review |edition=11th Edition |pages=pp.108, 295-297 |author=Graduate Management Admission Council |month=September |year=2005 |isbn=0976570904] Even on other standardized tests, if a question asks about the behavior of even numbers, it might be necessary to keep in mind that zero is even. [cite book |title=Kaplan SAT 2400, 2005 Edition |author=Kaplan Staff |year=2004 |publisher=Simon and Schuster |isbn=074326035X |pages=227]

The nominal evenness of zero is relevant to odd-even rationing systems. Cars might be allowed to drive or to purchase gasoline on alternate days, according to the parity of the last digit in their license plates. 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. The relevant legislation sometimes stipulates that zero is even to avoid confusion. [For example, 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 [http://books.google.com/books?q=%22ham+radio+operator+plates.+Zero+is+an+even+number%22 Google book search] , accessed on 2008-02-22.] In fact, an odd-even restriction on driving in 1977 Paris did lead to confusion when the rules were unclear. On an odd-only day, the police avoided fining drivers whose plates ended in 0, because they did not know whether 0 was even. [cite web |url=http://www.pantaneto.co.uk/issue5/arsham.htm |title=Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives |first=Hossein |last=Arsham |work=The Pantaneto Forum |year=2002 |month=January |accessdate=2007-09-24 The quote is attributed to the "heute" broadcast of October 1, 1977. Arsham's account is repeated in cite book |title=Perfect Figures: The Lore of Numbers and How We Learned to Count |first=Bunny |last=Crumpacker |year=2007 |publisher=Macmillan |isbn=0312360053 |pages=165]

In other situations, it can make sense to separate 0 from the other even numbers. On U.S. Navy vessels, even-numbered compartments are found on the port side, but zero is reserved for compartments on the centerline. That is, the numbers read ...6420135... from port to starboard. [cite book |title=The Bluejacket's Manual: United States Navy |year=2008 |edition=Centennial Edition |first=Thomas J. |last=Cutler |publisher=Naval Institute Press |isbn=1557502218 |pages=237-238] In the game of roulette, the casino has an interest in making sure that less than half of the numbers are counted as even. Thus the number 0 does not count as even or odd; a bet placed on either even or odd does not win if the ball falls on "0" or "00". The exact result depends on local rules, but the overall effect is to give the house an edge on even-money bets. [cite book |first=Andrew |last=Brisman |title=Mensa Guide to Casino Gambling: Winning Ways |publisher=Sterling |year=2004 |isbn=1402713002 |pages=153] The game odds and evens is also affected: if both players cast zero fingers, who wins? The strategy of the game varies with the agreed-upon answer, although generally zero is counted as even. [cite book |title=The Official World Encyclopedia of Sports and Games |author=Diagram Group, David Heidenstam, Paulin Meier, Jack Wilkinson |publisher=Paddington Press |year=1983 |isbn=0448222027 |pages=213] In fact, playing this game has been suggested as a way of introducing children to the concept that 0 is divisible by 2. [cite book |author=Baroody, Arthur and Ronald Coslick |title=Fostering Children's Mathematical Power: An Investigative Approach to K-8 |year=1998 |publisher=Lawrence Erlbaum Associates |isbn=0805831053 |pages=1.33]

History

Children's development of numerical understanding parallels the historical development; even and odd numbers were known before the number zero was introduced. Ancient Greek mathematicians generally considered 2 to be the first even number and 3 the first odd number, and some did not even recognize 2 as even. [cite book |author=Plato and Reginald E. Allen |title=Plato's Parmenides |year=1997 |publisher=Yale University Press |isbn=0300077297 |pages=262-264] The number 1 was not a number at all, but a component of all other numbers; as such it had to be both even and odd, and therefore neither truly even nor truly odd. This dual role for 1 was a source of metaphysical discomfort; one historian asserts that the Greeks could have avoided the issue had they known about 0. [cite book |first=W. K. C. |last=Guthrie |title=A History of Greek Philosophy: The Earlier Presocratics and the Pythagoreans |isbn=0521294207 |pages=239-242]

The algebraic properties of 0 were first systematically explored by Brahmagupta. According to two modern writers, the recognition of zero as an even number followed soon after. An economics book attributes the first statement to Indian mathematicians in the time of Brahmagupta, [cite book |title=Macroeconomic Foundations Of Macroeconomics |first=Alvaro |last=Cencini |year=2003 |location=London |publisher=Routledge |isbn=0-415-31265-5 |pages=p. 299] while a work of historical fiction suggests that al-Khwārizmī became the first to call 0 even during his arguments to the Caliph that "sifr" was a number. [cite book |title=Marvels of Math: Fascinating Reads and Awesome Activities |year=1998 |first=Kendall F. |last=Haven |publisher=Libraries Unlimited |pages=p. 13 |isbn=1563085852 Haven references himself in a later work making the same claim, cite book |title=100 Greatest Science Inventions of All Time |year=2005 |first=Kendall F. |last=Haven |publisher=Libraries Unlimited |pages=p. 28 |isbn=1591582644 ] These authors do not cite sources for their claims, so they are difficult to corroborate. In more modern times, a claim that zero is even appears in Stephen Chase's 1849 "A Treatise on Algebra". [cite book |first=Stephen |last=Chase |year=1849 |title=A Treatise on Algebra |publisher=G. S. Appleton |pages=p.65 |url=http://books.google.com/books?id=Tu82AAAAMAAJ |accessdate=2007-10-01]

Motivating modern definitions

The precise definition of any mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. And unlike "even", some mathematical terms are purposefully constructed to exclude especially trivial or degenerate cases. Prime numbers are a famous example. The definition of "prime number" has historically shifted from "positive integer with at most 2 factors" to "positive integer with exactly 2 factors", with the effect that 1 is no longer considered prime. Most authors rationalize this shift by observing that the modern definition more naturally suits mathematical theorems that concern the primes. For example, the fundamental theorem of arithmetic is easier to state when 1 is not considered prime.

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 the the algebraic rules governing even and odd numbers.cite book |title=Fundamentals of Mathematics for Linguistics |first=Barbara Hall |last=Partee |year=1978 |location=Dordrecht |publisher=D. Reidel |isbn=90-277-0809-6 |pages=p. xxi] The most relevant rules concern addition, subtraction, and multiplication:
*even ± even = even
*odd ± odd = even
*even × integer = evenInserting appropriate values into the left sides of these rules, one can produce 0 on the right sides:
*2 − 2 = 0
*−3 + 3 = 0
*4 × 0 = 0

The above rules would therefore be incorrect if zero were not even; at best they would have to be modified in some way. 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 numbers contain some exceptions:
*even ± even = even (or zero)
*odd ± odd = even (or zero)
*even × nonzero integer = even [cite book |title=30 Days to the GMAT CAT |first=Mark Alan |last=Stewart |year=2001 |location=Stamford |publisher=Thomson |isbn=0-7689-0635-0 |pages=p. 54 These rules are given, but they are not quoted verbatim.]

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 all integers, forces the usual definition and the evenness of zero.

Countless results in number theory 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 numbers have unique factorizations 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 any other prime factors, it is a product of 0 distinct primes; since 0 is an even number, 1 has an even number of distinct prime factors. This implies that the Möbius function takes the value nowrap|1=μ(1) = 1, which is necessary for it to be a multiplicative function and for the Möbius inversion formula to work, and affects the exact value of the Mertens function everywhere. [cite journal |first=Keith |last=Devlin |title=The golden age of mathematics |pages=30–33 |journal=New Scientist |date=1985-04-18 |volume=106 |issue=1452] Some other mathematical contexts, where the presence of 0 in the even numbers can be felt, follow.

Mathematical contexts

Most of the intuitive reasons why zero is even fall under a few themes:

*Zero is not odd.
*Zero must be even to preserve the alternation between even and odd numbers.
*Zero must be even to preserve algebraic relations among even numbers.
*Zero is even because the empty set can be split into corresponding halves.
*Zero is even because it is divisible by 2, indeed any number.

These themes then reappear in many broader, more abstract mathematical structures. Even and odd numbers have countless applications and generalizations in mathematics, in which the evenness of zero often has identifiable consequences and analogies. Some of these follow.

Not being odd

The observation that zero is not odd is sometimes directly applied in a mathematical argument. If an unknown number is proven to be odd, then it cannot be zero. This apparently trivial observation occasionally provides a convenient and revealing proof that a number is nonzero. In the following examples, a problem lends itself to determining that a number of interest is odd, and an investigation of its parity helps identify the underlying mechanism that forces the number to be nonzero.

A classic result of graph theory states that a graph of odd order always has at least one even vertex. (Already this statement requires zero to be even and not odd in two places: the empty graph cannot have odd order, and an isolated vertex must be even.cite book |title=A Mathematics Sampler: Topics for the Liberal Arts |author=Berlinghoff, William P., Kerry E. Grant, and Dale Skrien |year=2001 |edition=5th rev. ed. |publisher=Rowman & Littlefield |isbn=0-7425-0202-3 |pages=149 For isolated vertices see p.149; for groups see p.311.] ) In order to prove the statement, it is actually easier to prove a stronger result: any odd-order graph has an "odd number" of even vertices. The appearance of this odd number is explained by a still more general result: any graph has an even number of odd vertices. [cite book |title=Discrete Mathematics: Elementary and Beyond |author=Lovász, László, József Pelikán, and Katalin L. Vesztergombi |pages=127-128 |year=2003 |publisher=Springer |isbn=0387955852] Finally, the even number of odd vertices is naturally explained by the degree sum formula.

Sperner's lemma is a more advanced application of the same strategy. Rather than prove that there exists at least one completely labeled subsimplex by directly constructing it, it is more convenient to prove that there exist an odd number of such subsimplices through an induction argument. [cite book |title=General Equilibrium Theory: An Introduction |first=Ross M. |last=Starr |pages=58-62 |year=1997 |publisher=Cambridge University Press |isbn=0521564735] A still stronger statement of the lemma then explains why this number is odd: it naturally breaks down as nowrap|("n" + 1) + "n" when one segregates colorings by orientation. [cite book |title=Fixed Point Theorems with Applications to Economics and Game Theory |first=Kim C. |last=Border |pages=pp.23-25 |year=1985 |publisher=Cambridge University Press |isbn=0521388082]

Even-odd alternation

Zero is the starting point of the even natural numbers. 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 natural number. This property can be formalized into a recursive definition of the set of even natural numbers:
*0 is even.
*("n" + 1) is even if and only if "n" is not even.This definition has the conceptual advantage of relying only on the minimal foundations of the natural numbers: the existence of 0 and of successors. As such, it is useful for computer logic systems such as the Isabelle theorem prover. [cite book |first=Richard J. |last=Lorentz |title=Recursive Algorithms |year=1994 |publisher=Intellect Books |isbn=1567500374 |pages=5-6] [cite book |title=Isabelle/Hol: A Proof Assistant for Higher-Order Logic |author=Nipkow, Tobias, Lawrence C. Paulson, and Markus Wenzel |year=2002 |publisher=Springer |isbn=3540433767 |pages=127] 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 Peano axioms, of which the even natural numbers are a model. [cite book |title=Mathematical Fallacies and Paradoxes |first=Bryan H. |last=Bunch |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5] A similar construction extends the definition of parity to transfinite ordinal numbers: every limit ordinal is even, including zero, and successors of even ordinals are odd.cite book |author=Salzmann, H., T. Grundhöfer, H. Hähl, and R. Löwen |title=The Classical Fields: Structural Features of the Real and Rational Numbers |pages=p. 168 |year=2007 |publisher=Cambridge University Press |isbn=0521865166]

The classic point in polygon test from computational geometry applies the above ideas. To determine if a point lies within a polygon, one casts a ray 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 algorithm 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 switches between even and odd, and the point at its tip switches between inside and outside. [cite book |first=Stephen |last=Wise |title=GIS Basics |year=2002 |publisher=CRC Press |isbn=0415246512 |pages=pp.66-67]

Another application comes from the field of graph theory. A graph whose vertices are split into two groups, such that two vertices from the same group are never adjacent, is called a bipartite graph. If a (connected) graph has no odd cycles, then an explicit bipartition can be constructed by choosing a base vertex "v" and coloring every vertex black or white, depending on whether its distance from "v" is even or odd. Since the distance between "v" and itself is 0, and 0 is even, the base vertex is colored the opposite color as its neighbors, which lie at a distance of 1. [cite book |title=A First Course in Discrete Mathematics |first=Ian |last=Anderson |year=2001 |location=London |publisher=Springer |isbn=1-85233-236-0 |pages=p. 53] [cite book |title=Pearls in Graph Theory: A Comprehensive Introduction |author=Hartsfield, Nora and Gerhard Ringel |year=2003 |location=Mineola |publisher=Courier Dover |isbn=0-486-43232-7 |pages=p. 28 ]

Algebra

The evenness of zero appears in the structured context of abstract algebra. The fact that the additive identity (zero) is even, together with the evenness of sums and additive inverses of even numbers and the associativity of addition, means that the even integers form a group. Moreover, the group of even integers under addition is a subgroup 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 nonempty subset of an additive group that is closed under subtraction must be a subgroup, and in particular, must contain the identity. [cite book |title=Abstract Algebra |last=Dummit|first= David S.|coauthor= Richard M. Foote |edition=2e |year=1999 |location=New York |publisher=Wiley |isbn=0-471-36857-1 |pages=p. 48]

Since the even integers form a subgroup of the integers, they partition the integers into cosets. These cosets may be described as the equivalence classes of the following equivalence relation: nowrap|"x" ~ "y" if nowrap|("x" − "y") is even. Here, the evenness of zero is directly manifested as the reflexivity of the binary relation ~. [cite book |title=Markedness Theory: the union of asymmetry and semiosis in language |first=Edna |last=Andrews |pages=p. 100 |year=1990 |location=Durham |publisher=Duke University Press |isbn=0-8223-0959-9] There are only two cosets of this subgroup — the even and odd numbers — and it can be used as a template for subgroups with index 2 in other groups as well. A well-known example is the alternating group as a subgroup of the symmetric group on "n" letters. The elements of the alternating group, called even permutations, can be completely characterized as products of even numbers of transpositions. The identity map, an empty product of no transpositions, is an even permutation since zero is even; it is the identity element of the group. [cite book |title=Topics in Group Theory |last=Tabachnikova |first= Olga M. |coauthors= Geoff C. Smith |year=2000 |location=London|publisher=Springer |isbn=1-85233-235-2 |pages=p. 99] [cite book |last=Anderson |first=Marlow |coauthors=Todd Feil |title=A First Course in Abstract Algebra: Rings, Groups, And Fields |year=2005 |location=London |publisher=CRC Press |isbn=1-58488-515-7 |pages=pp. 437-438]

Adding in the rule "even × integer = even" means that the even numbers form an ideal in the ring of integers, and the above equivalence relation can be described as equivalence modulo this ideal. In particular, even integers are exactly those integers "k" where nowrap|"k" ≡ 0 (mod 2). This formulation is useful for investigating integer zeroes of polynomials. [cite book |first=Edward Joseph |last=Barbeau |title=Polynomials |year=2003 |publisher=Springer |isbn=0387406271 |pages=98]

The empty set

One way of interpreting the evenness of zero is to say that a set with 0 elements can be partitioned into two subsets of equal size. The cardinality concept of size requires that there exists a bijection between these two subsets. In general, a set "A" has even cardinality if a partition of "A" into disjoint sets "B" and "C" exists where nowrap|1="B" = "C", and thus nowrap|1="A" = 2"B" = 2"C". The empty set can be partitioned trivially as nowrap|1=Ø = Ø ∪ Ø, which immediately shows that nowrap|1=0 = Ø is even. This can also show that 0 is divisible by any integer "n", since nowrap|1=Ø = Ø ∪ Ø ∪ · · · ∪ Ø ("n" copies).

The essentials of the above structure can be specified in more compact language: a finite set has even cardinality iff it supports an involution without fixed points or, equivalently, a free action by Z/2. The empty set, having zero elements, does support such an involution, namely the empty function. [cite conference |first=J.C.E. |last=Dekker |year=1993 |title=A Bird's-Eye View of Twilight Combinatorics |booktitle=Logical Methods: In Honor of Anil Nerode's Sixtieth Birthday |publisher=Birkhäuser |pages=p. 298] Fixed-point-free involutions are mostly studied not on finite sets but on topological spaces, where the most important examples are the antipodal maps on "n"-dimensional spheres "Sn". The involution on the empty set is a base example: it is the antipodal map on "S" -1, the sphere of dimension negative one. [cite journal |author=Conner, P. E. and E. E. Floyd |title=Fixed point free involutions and equivariant maps |year=1960 |journal=Bulletin of the American Mathematical Society |volume=60 |issue=6 |pages=pp. 416–441 |doi=10.1090/S0002-9904-1960-10492-2] [cite journal |first=G. R. |last=Livesay |year=1960 |title=Fixed point free involutions on the 3-sphere |journal=Annals of Mathematics |volume=72 |issue=3 |month=November |pages=pp. 603–611 |doi=10.2307/1970232] In general, for a closed manifold to support a free involution, it must have an even Euler characteristic. [Much more can be said: if a closed manifold has an odd Euler characteristic, then one can put a lower bound on the dimension of a fixed set of an involution; this 1964 result is due to Conner and Floyd. See cite journal |last=Stong |first=R. E. |title=Semi-characteristics and free group actions |journal=Compositio Mathematica |volume=29 |issue=3 |year=1974 |pages=pp. 223–248 |url=http://www.numdam.org/item?id=CM_1974__29_3_223_0] The Euler characteristic of the empty set "S" -1 is the same as the Euler characteristic of any other odd-dimensional sphere, and it is an even number: zero. [In fact, any oriented, odd-dimensional, closed manifold, not just spheres; see cite book |author=Guillemin, Victor and Alan Pollack |title=Differential Topology |publisher=Prentice-Hall |year=1974 |isbn=0-13-212605-2 |pages=p. 116]

Degrees of evenness

In a proof by Solomon W. Golomb that a nowrap|10 × 10 torus cannot be covered with nowrap|1 × 4 tiles, the author reminds the reader that zero is an even number. The proof relies on the fact that a sum of many even numbers, which may include 0, is always even. [cite book |first=Solomon Wolf |last=Golomb |title=Polyominoes: Puzzles, Patterns, Problems, and Packings |year=1994 |location=Princeton |publisher=Princeton University Press |pages=p. 119 |isbn=0-691-02444-8] Golomb's argument does not end with even numbers. It also generalizes to divisors other than 2, resulting in a weak relative of de Bruijn's theorem on box packing. The key is that 0 is not just a multiple of 2; it is a multiple of every other number.

Some multiples of 2 are more even than others. The ancient Greeks already categorized the even numbers as singly and doubly even; 0 is doubly even because it is a multiple of 4, so it can be divided by 2 twice. More generally, 0 is divisible by any number, including any power of two, and this unique property of 0 has some interesting consequences.

One consequence appears in computer algorithms such as the Cooley-Tukey FFT, in which numbers appear in bit-reversed order. This ordering has the property that the farther to the left the first 1 occurs in a number's binary expansion, 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. [cite book |first=Samuel Shaw Ming |last=Wong |title=Computational Methods in Physics and Engineering |year=1997 |publisher=World Scientific |isbn=9810230435] The illustration on the right depicts the evenness of the integers from +256 to −256.

It is clear that 0 is divisible by 2 more times than any other number, but one runs into trouble when trying to quantify exactly how many times that is. For any nonzero integer (or even rational number) "n", one may define the 2-adic order of "n", an integer which can be described as the number of times "n" is divisible by 2, or the exponent of the largest power of 2 that divides "n", or the multiplicity of 2 in the prime factorization of "n". But none of these descriptions works for 0; no matter how many times 0 is halved, it can still be halved again. Rather, the usual convention is to set the 2-order of 0 to be infinity as a special case. This convention is not peculiar to the 2-order; it is one of the axioms of an additive valuation in higher algebra.

The powers of two — 1, 2, 4, 8, … — form a simple sequence of increasingly even numbers. There are mathematically interesting ways to force such sequences to actually converge to zero, including the construction of the 2-adic numbers. [cite book |author=Salzmann, H., T. Grundhöfer, H. Hähl, and R. Löwen |year=2007 |title=The Classical Fields: Structural Features of the Real and Rational Numbers |pages=p.224 |publisher=Cambridge University Press |isbn=0521865166]

References

;In-depth sources
*cite conference |first=Deborah Loewenberg |last=Ball |year=2003 |title=Using Content Knowledge in Teaching: What Do Teachers Have to Do, and Therefore Have to Learn? |booktitle=Archive of the Third Annual Conference on Sustainability of Systemic Reform |url=http://sustainability2003.terc.edu/go.cfm/keynote |accessdate=2007-10-01
*cite journal |author=Dehaene, Stanislas, Serge Bossini, and Pascal Giraux |title=The mental representation of parity and numerical magnitude |journal=Journal of Experimental Psychology: General |volume=122 |issue=3 |pages=pp. 371–396 |year=1993 |url=http://www.unicog.org/publications/Dehaene_ParitySNARCeffect_JEPGeneral1993.pdf |accessdate=2007-09-13 |doi=10.1037/0096-3445.122.3.371|format=PDF
*cite conference |first=Len |last=Frobisher |title=Primary School Children's Knowledge of Odd and Even Numbers |editor=Anthony Orton (ed.) |booktitle=Pattern in the Teaching and Learning of Mathematics |location=London |publisher=Cassell |pages=31-48 |year=1999
*cite journal |author=Levenson, Esther, Pessia Tsamir, and Dina Tirosh |title=Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero |volume=26 |issue=2 |year=2007 |pages=pp. 83–95 |doi=10.1016/j.jmathb.2007.05.004 |journal=The Journal of Mathematical Behavior
*cite journal |last=Lichtenberg |first=Betty Plunkett |title=Zero is an even number |journal=The Arithmetic Teacher |volume=19 |issue=7 |year=1972 |month=November |pages=pp. 535–538
*cite journal |author=Nuerk, Hans-Christoph, Wiebke Iversen, and Klaus Willmes |title=Notational modulation of the SNARC and the MARC (linguistic markedness of response codes) effect |journal=The Quarterly Journal of Experimental Psychology A |volume=57 |issue=5 |month=July |year=2004 |pages=pp. 835–863 |doi=10.1080/02724980343000512 |url=http://math.nmi.jyu.fi/numbra/Mater/NMaterial/NMaterial_txt/NMaterial_Aachen/Nuerk&al_2004_Qjep_marc.pdf |accessdate=2007-09-19|format=PDF

;Footnotes


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