This post was inspired by a training session I attended on Saturday, where I watched in awe as a video was played of a lady teaching an entire lesson to Year 2 students (aged 6) about zero. And only about zero. Very impressive.

So where to begin… the beginning of zero I suppose. Zero began a lot later than all the other numbers. An afterthought of convenience that changed our numerical system forever. Some 5000 years ago, traders in Mesopotamia began using double dashes like this // to mean “nothing goes here” when writing out large numbers.

Consider this for a moment, before zero, the typical structure of numerical symbolism was to use new symbols or symbol repetition as you ascended the number line. Roman numerals are a great example. X is ten, L is 50 etc. The downside to doing things this way around, is that you either have infinite symbols, or ridiculously long combinations of symbols for relatively small numbers.

For example: LXXXVIII is 88, and MCMLXXXVII for 1987. So it was a bit of a stroke of genius when someone started using a ‘gap’ symbol. It allowed you to write 101 as ‘one gap one’, or ‘1 // 1’. Now you can use a mere 9 symbols, and just show everyone that sometimes there’s nothing in your ‘hundreds’ column, or ‘tens’ column using your new // symbol. Handy. Eventually this evolved into our special zero. Although not before various disappearances and banishings, and it wasn’t considered a NUMBER until much later. But we’ll save those for another day.

Zero evolved further to become the most important gate keeper since Zuul the gatekeeper of Gozer.

Zero is the gatekeeper between the positives… and the negatives (“Booooo, hissssss”).

Zero annoys a lot of mathematicians because it doesn’t play by all of the rules. It needs special care and gets a timeout pass for certain operations.

First and foremost, we don’t divide by zero. No no no no no. Dividing by zero is like the end of the Matrix Trilogy. No-one knows what just happened, it doesn’t make sense, you’re a bit scared and you wish you stayed at home. You can’t even divide zero by zero. Your calculator will likely display NaN (Not a Number … or perhaps Now Annihilation Nears) How annoying.

And look at this crazy mess:

0 x 6 = 0, 0 x 7 = 0, but 6 doesn’t equal 7 ! But we get the same answer. Zero is really messing with things. No other numbers get that kind of nutty result.

The craziness is easier to see if we switch to algebra

ab = d, ac = d, but b ≠ c.

Bloody mind bending zero!

Here are some more properties that may ease your mind, or make you scream uncontrollably.

Zero is the only number that is neither positive or negative

Zero is even

Zero is neither prime nor composite.

Anything multiplied by zero is zero… or is it…?

Anything divided by zero is undefined… and ends the universe.

x ± 0 = x

There are infinite zeroes hidden in front of every number. (e.g. 3 = 03 = 003 = 00000003)

Zero factorial, written as 0! is 1. If you’re unfamiliar with ‘factorial’, 4! = 4 x 3 x 2 x 1 = 24. But 0! = 1… but anything multiplied by 0 = 0…

WHAT?!?!?!

Anything to the power of zero is also 1.

e.g. 250^{0} = 1

WHAAAAT?!?!

(I need an adult!!)

Zero is melting my brain. Let’s look a little closer at those two counter intuitive examples.

Why is 0! = 1 ?

If you’re impatient, the key reasoning is highlighted pretty well below:

And as for 250^{0} = 1, well that’s linked to the laws of indices.

I’ll leave you with this final mind bender…

-3^{0} = -1 but (-3)^{0} = 1

I’ll let you ponder that one on your own 🙂

I haven’t seen the one about 0! before. Neat, but it is not a proof. The existence of 0! does not follow from the definition of factorial, so really to say 0! = 1 is a definition.

No Bramagupta? Oh – on the email, that last equation showed as (-3⁰) = 0 – had me worried my brain was atrophying at exponential rate since retirement 🙂

yes that was a typo. I got them the wrong way around! Brain freeze.