Unplanned.net

So, the mathematicians out there can already explain why the number “0.999…” (i.e.: 0.9 recurring) is equal to one.  Unfortunately, despite there being countless proofs of this — proof by fractions, proof by series, proof by Dedekind cuts — a lot of people still find it a hard concept to grasp. In my aim to make the world an infinitessimally better place, I therefore present…

Proof by cake.

There is no missing cake!

I think there are a good chunk of people who can see the mathematical arguments behind the assertion that 0.999… equals 1. They can read (and understand) the mathematical formulae. And yet, for them it just doesn’t feel right.

I also think that part of this may be due to the way it’s written. I mean, it starts with a zero. A zero! Right there, at the start! That must mean it’s less than 1, surely? Well, no. But how can we get past that mental hurdle… How about cake? Everyone loves cake.

1. Lets take a cake. One cake, to be exact. One real cake. This cake shall represent the number 1 (within the set of Real numbers).
2. Now, take a knife and cut the cake into 10 equal pieces. We’re going to assume that the knife is completely frictionless here, so no bits of cake get stuck to the knife — that would ruin the maths, and mess up your floor with cake crumbs.
3. Okay, take 9 of the 10 pieces of cake and leave them alone. That’s 0.9 of a cake. Then grab the remaining slice and cut that into 10 equal pieces too. Ignore 9 out of those 10 new pieces of cake (0.09 of our total cake), and move on to that tenth slice.
4. Keep doing this process with every tenth slice of the cake bits. Then do it some more. We shall assume that our frictionless knife is also really sharp. No, seriously, reeeaaaally sharp! We’ll soon be slicing away at subatomic bits of cake. But still, we keep going…
5. As we slice to infinity, the cake gets chopped up into 0.999… of a cake, if you look at it as “a cake with lots of slices made into it”. And yet nothing’s ever going to be thrown away, it’s still all there, one cake, the same as we started with.

What about the bit of the cake that we keep slicing? The more slices we take, the smaller the chunk becomes.  If we slice to infinity then this chunk becomes infinitessimally small. But hold on, this is cake we’re talking about, and we can’t have an infinitesimally small piece of cake! We can only have no cake. Well, so it is with the Real numbers.

The Real numbers are like cake y’see. There are no infinitessimally small numbers, just zero. If you take 0.9, and 0.09, and 0.009, all the way to infinity, then you’re just slicing into the number 1 — but never actually losing any of it. In the end it’s still 1, it just looks like someone’s hacked at it with a knife.

Just like with our cake.

Wednesday morning, November 12th.

A short story.

It’s a good morning.

I need to get the train into work, from Ealing to Reading. Today is a good morning because I’ve woken up early enough to go the quick route: Walk to West Ealing station, catch the 08:06 to London Paddington, arrive there in plenty of time to catch the 08:30 high-speed train to Reading. I’m gonna get into work on time. Woo.

Looks simple, huh? You'd be amazed...

Now, I could in theory catch the 08:12 instead. This would, in theory, get into Paddington at 08:25, so still plenty of time to jump on board that 08:30 to Reading. Of course, it’s rush hour and (perhaps more importantly) it’s First Great Western, so I try not to play that game of chance. The 08:06 it is then.

Bad news though: Today the 08:06 has been cancelled. Not such a good day after all. Still, these things happen, life goes on, etc. (Actually, quite why the 08:06 is so problematic is a bit of a mystery — it’s the Greenford train, which has a monopoly over the tiny Greenford branch line for most of it’s journey. Hmmm. Anyway, that’s not the point of this little story…)

So, I take my chances with the 08:12. Naturally, as this train is now carrying two sets of passengers — the usual ones plus the 08:06-wannabes — it’s rammed. No surprises there, huh, it’s just the way these things work. Besides, the 08:12 is running roughly on time, just a minute late so far. Guess I’ll still make that prized 08:30 to Reading. Woo.

Off we go. We stop at Ealing Broadway, and the train gets even more rammed. It’s amazing how many busy commuters you can cram into a train at rush hour, it’s almost (though not quite) Tardis-like. Anyway, apart from a few unlucky ones, we’re all in. Next stop, Paddington.

Ah. Guess not. Next stop is…erm…Acton Main Line. To pick up the handful of passengers who are still waiting for that cancelled 08:06 from Greenford. Except, of course, that this train is totally rammed. Rammed! As in, no more room for those extra people, even by London rush-hour standards. Which should be no surprise, given that…(wait for it)…it’s carrying all those additional passengers who would’ve got on the (cancelled) Greenford train, but couldn’t! I mean, c’mon, it’s not rocket science is it.

Seriously, why make the 08:12 do the extra stop? Why?! It’s rush hour, on one of London’s busiest lines. What exactly was the mindset of the planning geniuses at First Great Western on this one? I can only think of two possible explanations:

1. First Great Western hates its passengers.
2. First Great Western is not managed by geniuses. Their management are, in fact, morons. The really stupid kind of moron.

I would love there to be some third option, really, I would.

“Ah, but wait,” I hear you cry. “You can’t leave those poor passengers stranded at Acton Main Line! Their next scheduled train isn’t for another half an hour!”

Well, yes. But let’s take a quick look at the busy rush-hour morning schedule… Ooh, look, it’s the 08:19 from Ealing Broadway to Paddington! And could it be that this train isn’t burdened with two sets of passengers? Could it be that this train is merely ‘rush-hour full’, rather than ‘ridiculously-rammed full’? I mean, I wouldn’t claim to sport the kind of genius that the planners at First Great Western possess, but it seems to my simple brain that this train *might* have been a more sensible choice to pick up those poor passengers from Acton Main Line.

Anyway, fast forward to 08:29-and-thirty-seconds. I step off the 08:12 to London Paddington. I run but, inevitably, I miss my train. That wonderful 08:30 to Reading. I get into work late. No biggie, but I could’ve stayed in bed an extra 15 minutes and got the slow 08:22 train from Ealing Broadway instead. And an extra 15 minutes in bed is a big deal on a workday morning.

To catch my 08:30 to Reading, how much quicker would my train to Paddington have needed to be? I dunno, about two minutes, maybe a bit less. How much time did the train waste on that extra pointless stop at Acton Main Line?

About two minutes.