Is Zero Odd or Even?

It's been a while since I posted something mathsy, but I've seen a few people being wrong about this one lately, so here goes...

Let's start with the popular-but-wrong response; the one that would get the QI alarm awooga-ing: "it's neither!"


Now I've lit the blue touchpaper for a few ranty comments or tweets, let's look at the facts. A good place to start is to take a step back and ask...

What exactly is an even number?

The first port of call for almost all questions must surely be the great* Wikipedia, which says that "an even number is an integer that can be divided by two and remain an integer or has no remainder."

Wikipedia isn't the best source for dissuading ranty comments, though.

Wolfram|Alpha, possibly a more reliable source of mathematical information in many eyes, says "an even number is a number in the form $n = 2k$ where $k$ is an integer."

Does zero fit here?

Wikipedia first:
  • An integer is a number with no fractional part (i.e. no decimals; a whole number): ✓
  • If we divide zero by 2 we get zero (with no remainder): ✓
And Wolfram|Alpha:
  • If we can find an integer $k$ such that $0=2k$ that means $0$ is even:
    $k = \frac{0}{2}$, so $k = 0$. ✓
Both Wikipedia and Wolfram|Alpha agree:
$0$ is even.

Need some more?

  • Back at primary school you may have learnt a quick way of identifying whether a number of any size is even by glancing at its units digit: any whole number that ends in $0, 2, 4, 6$ or $8$ is even, $0$ ends in... $0$. So it's even.
  • Additionally, you might also recall the rule that if you add two even numbers together the result is always even, for example: \[2+2=4\] \[120 + 346  = 466\]
    Well, $-2 + 2 = 0$, so $0$ is even.
  • It's also true that two odd numbers added together ($3 + 7$, for example) give an even number ($10$).
    What's $-3 + 3$? Oh, it's $0$, so that's even.
  • Multiplicatively, any even number multipled by any integer results in another even number, e.g.:
    \[2 \times 3 = 6\] \[84 \times -37 = -3,108 \]
    $28 \times 0 = 0$, so $0$ is even.
  • We're all familiar with the number line, stretching from $ -\infty $ to $ + \infty$ . Let's look at a section of that line with $0$ in it:
    ... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
    And let's colour in the even ones:
    ... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
    Without $0$ included there's a big hole and the pattern is broken. Why would we do that to a perfectly decent pattern?

Still not sure?

Many a maths fan's bible, the On-line Encyclopedia of Integer Sequences (OEIS) naturally includes an entry on the nonnegative even numbers and this begins with $0$. Are you going to argue with the OEIS?

* And often, I admit, fallible

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