Parity of zero

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 0 × 2. 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 y is even then y + x has the same parity as x—indeed, 0 + x and x always have the same parity. Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, 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. 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 0 × 2 = 0 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 5 × 2. In the same way, zero is an integer multiple of 2, namely 0 × 2, so zero is even. 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 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. 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. 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. With the introduction of multiplication, parity can be approached in a more formal way using arithmetic expressions. Every integer is either of the form (2 × ▢) + 0 or (2 × ▢) + 1; the former numbers are even and the latter are odd. For example, 1 is odd because 1 = (2 × 0) + 1, and 0 is even because 0 = (2 × 0) + 0. Making a table of these facts then reinforces the number line picture above.

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. 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:

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