Parity of zero

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.

[3] The following explanations make sense of the idea that zero is even in terms of fundamental number concepts.

[9] 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.

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.

This apparently trivial observation can provide a convenient and revealing proof explaining why an odd number is nonzero.

[17] Finally, the even number of odd vertices is naturally explained by the degree sum formula.

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.

[18] A stronger statement of the lemma then explains why this number is odd: it naturally breaks down as (n + 1) + n when one considers the two possible orientations of a simplex.

[1] One consequence of this fact appears in the bit-reversed ordering of integer data types used by some computer algorithms, such as the Cooley–Tukey fast Fourier transform.

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.

[32] The subject of the parity of zero is often treated within the first two or three years of primary education, as the concept of even and odd numbers is introduced and developed.

[43] 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.

[44] Deborah Loewenberg Ball 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.

[45] 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.

In another study, David Dickerson and Damien Pitman examined the use of definitions by five advanced undergraduate mathematics majors.

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.

[48] Researchers of mathematics education at the University of Michigan 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 does not vary between traditional and reform mathematics.

Betty Lichtenberg, an associate professor of mathematics education at the University of South Florida, 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".

[51] Mathematically, proving that zero is even is a simple matter of applying a definition, but more explanation is needed in the context of education.

[52] 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.

Some variations of the experiment found delays as long as 60 milliseconds or about 10% of the average reaction time—a small difference but a significant one.

[55] Dehaene's experiments were not designed specifically to investigate 0 but to compare competing models of how parity information is processed and extracted.

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.

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".

[58] The effect also suggests that it is inappropriate to include zero in experiments where even and odd numbers are compared as a group.

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.

[67] To avoid such confusion, the relevant legislation sometimes stipulates that zero is even; such laws have been passed in New South Wales[68] and Maryland.

[70] In the game of roulette, the number 0 does not count as even or odd, giving the casino an advantage on such bets.