Is Zero an Even Number? Yes. Five Ways to Prove It

Yes, zero is an even number. It is not odd, it is not both, and it is not some special third category that sits outside the rules of arithmetic. Zero passes exactly the same tests as every other even whole number. Proving it takes seconds; the interesting part is why so many people hesitate when asked. Even adults who know the answer take measurably longer to classify zero as even than other small even numbers, surveys of schoolchildren keep finding uncertainty about it, and in 1977 it even caused problems for the police in Paris. First, though, here are five simple proofs.

Five quick proofs

1. Zero divides by two exactly. The simplest definition is also the strongest: an even number is a whole number that can be divided by 2 with no remainder. Zero divided by 2 is 0, with nothing left over. Every even number can also be written as 2 × n, where n is a whole number, and zero fits perfectly, because 0 = 2 × 0.
2. Zero cannot be odd. Odd numbers take a different form, 2 × n + 1. If zero were odd, then 2 × n + 1 = 0, so 2 × n = −1, and n would have to be −½. That is not a whole number, so zero cannot be odd, and it satisfies the definition of even instead.
3. It fits the pattern on the number line. Write out the whole numbers around zero, from −3 to 3, and they alternate perfectly: odd, even, odd, even, odd, even, odd. Zero sits between −1 and 1, both of which are odd, which makes it the even number in the middle. Children can discover this without any algebra; in one classroom study described on Infogalactic, a group of 15 second-graders convinced each other that zero was even using exactly this alternating pattern.
4. You can split nothing into two equal groups. Imagine sharing zero lollies equally between two children. Each gets zero, and none are left over. It sounds like a silly example, but it captures the basic idea of an even number, one that splits into two equal groups with nothing remaining. Zero is simply the case where both groups are empty.
5. The last-digit rule says so. A whole number is even if its last digit is 0, 2, 4, 6 or 8. That is why 10, 20, 30, 100 and 1,000 are all even. If zero were not even, the rule would need a strange exception for its own most common digit.

There is an even broader reason mathematicians are so firm about it: the ordinary rules of parity only work cleanly if zero is even. Subtracting one even number from another always gives an even number, so 2 − 2 must be even, and that answer is 0. As Wikipedia’s article on the parity of zero puts it, the standard parity rules of arithmetic require zero to be even.

So why does zero feel wrong?

Most of us learn odd and even by counting from one. We learn that 2, 4, 6 and 8 are even and 1, 3, 5, 7 and 9 are odd, while zero is introduced as “nothing” rather than as a number with the same properties as the rest. That seems to leave a surprisingly stubborn mental roadblock. In a survey of nearly 400 seven-year-old children, only about 45% said zero was even, and when they were also given choices such as “neither”, “both” and “don’t know”, even more of them wavered, as Science ABC reports. Adults are not immune: Wikipedia notes that some teachers also believe zero is odd, both or neither, and that some test study guides have told students it is “neither even nor odd”. The children are not necessarily failing at arithmetic. They are applying an everyday intuition, that you cannot split nothing in two, when mathematically you can; both halves are simply empty.

Your brain really does hesitate

The hesitation is not imagined. In reaction-time experiments pioneered by French cognitive scientist Stanislas Dehaene, people shown numbers and asked to sort them into odd and even as fast as possible were measurably slower with 0 than with 2, 4 or 6. Some versions found a delay of as much as 60 milliseconds, roughly a tenth of an average response, as Science ABC explains. Sixty milliseconds is far too short to notice in conversation, but in an experiment like this it shows the brain doing a little extra checking. Zero is special in plenty of ways; it is neither positive nor negative, and it stands for the absence of any quantity. None of that changes its parity, but it seems to make our brains pause before agreeing.

When zero confused the Paris police

The confusion has escaped the classroom at least once. In 1977, Paris brought in an odd-even driving restriction during a period of air pollution, with cars allowed on the road on alternate days depending on the last digit of their number plate. That sounds simple until you ask about a plate ending in 0. According to Science ABC, on an odd-only day, police reportedly let drivers with plates ending in 0 go unfined, because officers were not sure whether zero counted as even, and Wikipedia’s article on odd-even rationing notes the scheme did cause confusion where the rules were unclear. Zero had not changed and neither had the mathematics; the uncertainty was entirely in the people applying the rule. Odd-even schemes have been used in many places since, including Delhi from January 2016, and the lesson from Paris is that the rules need to say plainly which side zero falls on.

The same idea turns up in casinos. On a roulette wheel, bets on “even” lose when the ball lands on 0, but not because zero is odd. The house simply treats 0 (and 00 on American wheels) as belonging to neither side. That is a game rule, not a mathematical one, and it is a large part of how the casino keeps its edge.

Nothing, but definitely even

I find it oddly satisfying that the number for nothing at all has such a firm property. Zero divides by two exactly, sits neatly between two odd numbers and keeps the rules of arithmetic working; the only thing it fails is our intuition, which is a good reminder that “it feels wrong” is not the same as “it is wrong”. The systems we take for granted often hide quirks like this. The alphabet only settled into its final shape a few hundred years ago, as I found looking into the last letter added to the alphabet, and even our units of measurement have surprising stories, such as whether a buttload is a real unit. So next time someone at trivia night insists zero is neither odd nor even, you will have five ways to prove them wrong.

Sources

• Wikipedia: Parity of zero
• Infogalactic: Parity of zero, including the classroom and reaction-time studies
• Wikipedia: Parity (mathematics)
• Wikipedia: Odd-even rationing
• Science ABC: Is zero an even number or an odd number?