Is 0 zero?
Um, yes...
Is 0.0 zero?
Yes...
Is 0.00 zero?
Yes. I don't know where you're going with this, but it better be entertaining.
Hang on and it will be.
It better.
Is 0.000 zero?
Yes. Get on with it!
Okay. If I keep placing 0 at the end of the number, is it still zero?
Yes.
So 0.00000... is zero, then?
Assuming you're using only the digit 0, then yes.
Ah! What happens if we take that assumption away?
Then I don't know if the number is zero or not. You could have some digit other than 0 in there.
Yes. Let's play a game.
Okay. What are the rules?
I think of a digit after the decimal point. You can choose between asking me for the digit, or declaring whether or not the number is zero. If you ask for a digit, I get to think of another digit and the process starts again.
And if I declare whether or not the number is zero
I reveal my digit. If you're right, you win. Otherwise I do.
Heh. Seems like you can win every time.
Whatever, let's play. I've thought of my first digit.
Ask.
3, go.
That's easy! The number is not zero.
Right you are, my next digit was 3.
You let me win.
Yes. Let's play again. Go.
Ask.
0, go.
Ask.
0, go.
Ask.
0, go.
...we're not going anywhere are we? I guess the number is zero.
Bzzzzt! My next digit was 8.
You only say that because I guessed it was zero.
Right you are. Let's modify the rules a little bit. I have to write my digit down on this piece of paper, and show it to you every time you ask [etches something on the paper with a blunt pencil].
Ask.
[Reveals a "0.0", then etches some more onto the paper.] Go.
I guess the number is zero.
[Reveals "0.00".] You win!
That was easier. It seems to me like the only viable thing for you to do is to always write down a zero, or I might ask for it and reveal the obvious truth. I should always guess that it's a zero, at random times.
But what if I write a 1 just when you're about to guess?
You'd win, but there's a pretty slim chance of getting it right.
True.
On the other hand, if I keep asking, you might slip and give me something other than a 0.
But if I don't?
We'd be getting nowhere, and I'd be inclined to guess zero.
And so I'd be inclined to put something other than a 0 the longer it goes on, right?
Right... but that means I can just keep asking and I'll win...
Good, you completely grok the game.
It's weird and pointless. Can I go now?
Nope. So, what did you learn? How do the two games compare?
Well, I always lose on the first game, and I usually win on the second game.
But why?
Because you can't change your mind in the second game?
Exactly. Let's try a simple variation on the second game now.
Oh, not again...
I decide what the number is up-front and write it down, or an equation representing it. You can ask for digits, etc...
Okay.
[Pauses for a moment, then scribbles something down.] Go!
Ask.
1, go.
Not zero. Why did you let me win?
To prove a point. Let's play again. [Scribbles something else down.]
Ask.
0, go.
Ask.
0, go.
Ask.
0, go.
...we're at this again. I guess the number is zero.
[Reveals 0.0001]. Oops! I win! Maybe you could have played better?
You mean I just had to keep asking?
Apparently. Wanna try again? [Scribbles some more.] Go!
Ask.
0, go.
Ask.
0, go.
Ask.
0, go.
Ask.
0, go.
Ask.
0, go.
Oh, whatever... I'm guessing not zero.
[Smiles, and reveals a nice round 0.] Geez, I thought you'd have learnt not to trust me by now.
Hey, that hurts! So... how can I tell if it's a 0?
...you can't! That's my point. You can't tell if it's a zero just by asking digits.
So basically we can both win this game. I can win through perseverance, you can win through cunning. Should I go now?
Wait! We haven't even reflected on this.
It's getting dark outside...
Calm down, nothing bad is going to happen if you learn a little bit more.
Just get on with it.
Kids these days, so hasty... Anyway, you can't tell whether or not my number is a 0 just from looking at the digits individually. What if I tell you what all the digits are?
Sounds simple enough.
Oh really? Here: the nth digit is 0, for all n.
Well, that's obviously 0...
Okay. Try: the nth digit is the same as the n minus one -th digit.
Depends on the first digit. You aren't giving me all the information here.
Fair enough. Suppose I did give you all the information. Would you always be able to tell me whether or not the number is zero?
Of course! If I know every digit of the number, I know what the number is!
Really? I wanna see that. What about this: the nth digit is 0 if n converges to the 1-4-2-1 cycle of the Collatz conjecture, and otherwise is 1.
...oh, I don't know that!
...
...
... I haven't figured out a good way to end this yet.
