Is Zero an Even Number?

Apply the definition and it is settled

A whole number is even when it is a multiple of two - when dividing it by two leaves no remainder. Zero passes on both readings. 0 ÷ 2 = 0 exactly, and 0 = 2 × 0, so zero is two multiplied by a whole number. Nothing in the definition asks the number to be positive, or large, or to have anything in it at all.

That is the whole argument, and the thing worth noticing is that no special case was needed. Definitions in mathematics are written so that they keep working at the edges, and zero is the edge here. If zero had to be excluded by hand every time, the definition would be the thing at fault.

• Even: a whole number that is a multiple of 2 - 0, 2, 4, 6, and also -2, -4, -6
• Odd: a whole number that is not - 1, 3, 5, and also -1, -3, -5
• 0 = 2 × 0, so zero is a multiple of 2 and therefore even

The checks that actually convince people

Definitions rarely settle an argument in a classroom. What settles this one is running zero through every test that even numbers are supposed to pass and watching it pass all of them.

The alternation check is the strongest of the five. Odd and even alternate along the number line with no exception anywhere, and for zero to be odd, or to be neither, that pattern would have to break at exactly one point and nowhere else. It would also leave -1 and 1 as two odd numbers with nothing even between them, which happens at no other place on the line.

• Halving gives a whole number: 0 ÷ 2 = 0
• It sits between the two odd numbers -1 and 1
• Even plus even gives even: 0 + 4 = 4
• Even plus odd gives odd: 0 + 3 = 3
• Its last digit is 0, one of the five digits - 0, 2, 4, 6, 8 - that mark a whole number as even

Where the idea that zero is neither comes from

There is a true fact next door, and it gets misfiled. Zero is neither positive nor negative - it is the one number sitting on neither side of the line - and a child who learns that correctly often carries the shape of it across to odd and even. Zero is special once, not twice.

The second source is counting. Even numbers usually get introduced as the ones that can be shared into two equal groups, and sharing nothing feels like not doing anything at all. Zero counters in two piles of zero each is a strange picture, but it is a legal one, and the arithmetic raises no objection.

Adults hesitate here as much as children, which is why the question keeps being asked. The hesitation is almost never about the definition - most people can state it - but about whether a definition is allowed to apply to a number that feels like an absence rather than a quantity.

When it changes an answer

It matters wherever zero is inside the sample space. A spinner numbered 0 to 5 has three even outcomes and not two - 0, 2 and 4 - so the probability of an even score is 3/6 rather than 2/6. A student who quietly leaves zero out has the right method and the wrong answer.

The same thing happens in sequences and in sorting questions. The even numbers between -5 and 5 are -4, -2, 0, 2 and 4, five of them. A rule that says every number ending in 0 is divisible by 2 is general only because zero itself obeys it.

Parity arguments depend on it most of all. Proving that something cannot happen because one side of an equation is odd and the other is even only works if every whole number is one or the other. Allow a number to be neither and the technique fails wherever zero can turn up, which is most places.