In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer multiple of 2, specifically . 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 is even then has the same parity as —indeed, and always have the same parity.
Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as , require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all. "By the same test zero surpasses all numbers in 'evenness.'"; "Thus, the integer is the most 'even.'
Among the general public, the parity of zero can be a source of confusion. In reaction time 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 mathematics education propose that these misconceptions can become learning opportunities. Studying equalities like can address students' doubts about calling 0 a number and using it in arithmetic. 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 abstraction of a familiar concept to an unfamiliar setting.
Why zero is even The standard definition of "even number" can be used to directly prove 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 . In the same way, zero is an integer multiple of 2, namely so zero is even.: Lemma B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that and this follows from the equality ."
It is also possible to explain why zero is even without referring to formal definitions. 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. 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 zero.
Basic explanations
Given a set of objects, one uses a number to describe how many objects are in the set. Zero is the count of no objects; in more formal terms, it is the number of objects in the empty set. 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. "...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."
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 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. "Zero groups of two stars are circled. No stars are left. Therefore, zero is an even number."
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.
Numbers can also be visualized as points on a number line. When even and odd numbers are distinguished from each other, their pattern becomes obvious, especially if negative numbers are included:
The even and odd numbers alternate. Starting at any even number, counting up or down by twos reaches the other even numbers, and there is no reason to skip over zero.; 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."
With the introduction of multiplication, parity can be approached in a more formal way using arithmetic expressions. Every integer is either of the form or the former numbers are even and the latter are odd. For example, 1 is odd because and 0 is even because Making a table of these facts then reinforces the number line picture above. "At a more advanced level ... numbers expressed as are even numbers ... zero fits nicely into this pattern."
Defining parity The precise definition of a mathematical term, such as "even" meaning "integer multiple of two", is ultimately a convention. Unlike "even", some mathematical terms are purposefully constructed to exclude trivial or degenerate cases. Prime numbers are a famous example. Before the 20th century, definitions of primality were inconsistent, and significant mathematicians such as Goldbach, Lambert, Legendre, Cayley, and Kronecker wrote that 1 was prime. The modern definition of "prime number" is "positive integer with exactly 2 factors", 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 fundamental theorem of arithmetic is easier to state when 1 is not considered prime. "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 .
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 algebraic rules governing even and odd numbers. The most relevant rules concern addition, subtraction, and multiplication: even ± even = even odd ± odd = even even × integer = even Inserting 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. 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 exceptions: even ± even = even (or zero) odd ± odd = even (or zero) even × nonzero integer = even 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 integers forces the usual definition and the evenness of zero.
Mathematical contexts 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 positive 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 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 , which is necessary for it to be a multiplicative function and for the Möbius inversion formula to work.
Not being odd A number is odd if there is an integer such that . One way to prove that zero is not odd is by contradiction: if then , 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.
A classic result of graph theory states that a graph of odd order (having an odd number of vertices) always has at least one vertex of even degree. (The statement itself requires zero to be even: the empty graph has an even order, and an isolated vertex has an even degree.) 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 degree vertices. The appearance of this odd number is explained by a still more general result, known as the handshaking lemma: any graph has an even number of vertices of odd degree. 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. The lemma states that a certain kind of coloring on a triangulation of a simplex 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 induction argument. A stronger statement of the lemma then explains why this number is odd: it naturally breaks down as when one considers the two possible orientations of a simplex.
Even-odd alternation
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 idea 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 LF and the Isabelle theorem prover.; ; 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. 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.