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### 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!"

Wrong!

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."

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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