I’ve been taking buses for a long time. One thing I noticed is that the buses are fullest in the city centre.

Now that’s kind of obvious. After all, most people who take the bus are going either to or from the city centre. It makes sense the buses are packed full of people there.

But that’s not the only effect at play here. There’s another, much simpler reason for the buses being packed full of people in the city centre. It’s the middle of the bus route.

To show how this works, I made a very simple model, shown below:

In my model, there are 10 bus stops. From every bus stop, one person wants to go to each of the other bus stops. So on the first stop, there will be 9 people getting on, while on the 7th stop, there will be 6 people getting off and only 3 people getting on, because there are only 3 stops left.

The blue bars show how many people arrive at each bus stop, whil the red bars show how many people leave each bus stop.

Actually, after I drew the diagram, I was surprised by how flat the distribution seemed around the centre. It looked like only the first two and last two steps were really empty, and otherwise the bus contained pretty much the same amount of people. So just for the kicks, I decided to build the same model with 30 bus stops:

Again, the distribution is very flat around the “city centre”.

Now that I think of it, that’s pretty much what you see in the real world. The bus is more or less constantly full around the city centre, then it slowly starts emptying, and suddenly on the last 3 or so stops, you see a right old exodus!

Hmm, if I spent a bit more time thinking about this, I could probably come up with a lot more examples of this phenomenon. Like cities having a more or less constant population density around the centre, and then quickly dropping the density at the edges. Or mushrooms having a more or less constant height around the centre and dropping off quickly at the edges. This phenomenon would be named Vlad’s Bus Stop Principle, and I would become rich and famous because of its numerous applications…

Ok, maybe not .

Yeeey, we’re back with another edition of throwing cats!

Last week, I showed you what happens when you throw a cat out of an airplane. Today, you will find out what happens when you throw a cat into water!

Please note that this time we’ll be using an idealised mathematical cat. It’s going to stay floating on the surface of the water without moving around, instead of freaking out, shooting out of the water like a bullet and spending years plotting a revenge masterplan against you, like a real cat would. (So don’t try this at home.)

Oh, and in addition to a cat, we’re going to use a dog! Weeee!

Which route will the dog take?

Imagine we’re on a beach. We throw the cat into the water, and then yell “Fetch!”. The dog will start running to fetch the cat. But what’s the best route for him? He can run on the sand a lot faster than he can swim in the water…

If he goes in a straight line, he’s going to be swimming unnecessarily long. Or he could run straight to the point on the sea edge that’s closest to the cat, to minimize the time he spends swimming. Or maybe somewhere in between?

The answer is: somewhere in between. In fact the answer is exactly analogous to light refraction. Light travels faster in air than inside a crystal. So if you shine a light on a crystal, the light will not go in a straight path. Instead, it will go in the path that minimizes the time taken.

This was a big “Ah-ha!” moment for physicists. When they first discovered photons, they thought they travel through the shortest DISTANCE between two points. But they don’t. They take the route that will guarantee the shortest TIME between the two points! It’s almost like photons can THINK!

Some people also propose a theory that dogs can’t really think about the route they’re taking. These people claim the dogs simply run along the beach because they don’t want to get wet, and then jump into the water when staying dry is no longer an option. But there’s a fatal flaw in these people’s claims – we can refute them experimentally! And as a bonus, the experiment shows that dogs are smarter than photons!

How dogs are smarter than photons

The experiment is very simple. Just place the dog in the water. Then throw the cat in the water far away from the dog, and tell the dog to fetch. Which path will he follow?

That’s right, he’ll follow the quickest path! That means getting out of the water, running along the beach, and then getting back in the water! (This was shown by experiments using a stick, because the experimenters couldn’t find an idealized mathematical cat.)

A photon faced with the analogous problem won’t be that smart. If you shine a light inside a crystal and try to find out how it gets to another point inside the same crystal, it will always be in a straight line. The photon isn’t smart enough to get out of the crystal, fly along the edge, and then re-enter when it’s convenient.

And that’s all I have to say about throwing cats into water! It’s probably also the end of my short series about throwing cats. I really don’t think I can frame my next blog post to include throwing cats (well, I could call it “Throwing cats on a bus”, that would be really pushing it).

(Hmmm. Last week I used a cat to illustrate air drag. Today I used a cat and a dog to illustrate refraction. The way this is going, next week I will probably explain the origins of the universe using a giraffe, an elephant, and a duck-billed platypus.)

Ah, division by zero. One of the first things that math teachers start ramming into pupils’ heads. They say “You cannot divide by zero!” with as much vengeance as Gandalf shouting “You cannot pass!”.

But they usually don’t give you any reasons! It’s more like “You cannot never ever divide by zero!” rather than “Here, look at why division by zero wouldn’t work…”

In the better cases, they tell their students “division by zero is undefined”. That’s not strictly always true, but it’s close enough. But quite often, the teachers say “Anything divided by zero is infinity.” No, no, NO. That’s completely, UTTERLY WRONG.

Infinity is not a number. Seriously. It’s an abstract concept. It’s a shorthand for “gets as big as you like”. You can’t treat it as a number, and you most definitely can’t get it as a solution to an equation. In fact, if you start throwing “infinity” into equations, you can get stuff like:

infinity = infinity + 1
infinity + 0 = Infinity + 1
subtract infinity from both sides
0 = 1
Yay!

But I won’t rant about infinity in this post. Instead, I will attempt to explain exactly why division by zero goes wrong – and why the only time it does NOT go wrong, it gives you a completely useless result.

I’m going to start at the very beginning…

The origins of negative numbers

When humans first started counting, they used only natural numbers, and they only used addition. They would ask questions like “I used to have three cows. A salesman came along, and now that he left, I have 5 cows. How many cows did I get from him?”

In other words, “How much do I add to 3 to get 5?”, or “3+x=5″.

That all worked fine, until one day, a farmer asked this question: “I used to have three cows. A salesman came along, and now that he left, I have only two cows. How many cows did I get from him?”

In other words, “3+x=2″. And we run into trouble, because no matter what you try adding to 3, you will never end up with 2.

So the farmer invented the notion of “minus one”, or, as I like to call it, “anti-one”.

Now pay attention, because this is going to be really important. There is no subtraction. In fact, “subtract five” actually means “add anti-five”. We call “minus five” the additive inverse of “five”. But don’t worry about the buzzwords too much. Just try to get wrap head around the concept that what we call “subtracting x” actually means “adding anti-x”.

A quick side-note: There are two elementary operations. Addition and Multiplication. Sutraction is actually Anti-Addition, and Division is actually Anti-Multiplication. (And no, multiplication is NOT just repeated addition. But that’s a topic for another day. And “powering” as in “x squared” also comes into play, but let’s not worry about that for now.)

So subtracting actually means “anti-adding”.

“5 – x = 3″ actually means “What number do I need to add to three to get 5?”

Okay, with that background, let’s tackle division in general, and then specifically division by zero.

Why we can’t divide by zero

When you have 2 * 3 = x, it means “What do I get when I multiply 2 by 3?”

And because division is actually anti-multiplication, 6 / 3 = x means “What number do I need to multiply 3 by to get 6?”

You with me so far?

Therefore, 1 / 0 = x means “What number do I need to multiply 0 by to get 1.” And there is no possible answer to that. We call this “undefined.”

And here comes the next fun bit. What if you have 0 / 0 = x? Obviously, you can multiply 0 by ANY number and get 0. The answer is very well defined… unfortunately, you have no way of telling which answer it is. And no, you can’t just pick one that suits you. Let me explain…

Let’s play a game! I’m thinking of a number, and when I multiply that number by itself, I get 4. What number am I thinking of?

Say it.

Okay, got it? The number I was thinking of was… minus two! If you said “two”, you lose. If you said “either two or minus two”, you were right… but you still lose, because I was thinking of only one number. If you said “minus two”, you win, you tricky bastard.

The point is, if I say “x squared is 4, what is x?”, you have no way of knowing the answer. It could be either 2 or -2, but you can’t tell which.

Analogously, if you try to find 0 / 0 … it could be any real number, and you have no idea which one. So if you get 0 / 0 = x in an equation, all you know is that x actually exists. But you have no idea what it is.

Phew. So much for division by zero. Let’s tackle something easier next time. Like octonion analysis…

Here’s a fun question! What will happen to a cat if you throw it out of an airplane?

Are you ready for the answer? …In fact, nothing will happen to the cat! Because if something WERE to happen to the cat, you would get killed by rain! (Wait, what?)

Okay, okay, I’m getting ahead of myself. Let me backtrack a little bit…

Why rain doesn’t kill you

Rain clouds are anywhere from 2 kilometres up. If you still remember anything from high school classes, the acceleration under gravity is roughly 9.8 m/s^2, and the distance traveled 1/2at^2. Without bothering you with the calculation details, a raindrop from even your lowest 2 km cloud would be falling at 197 m/s by the time it hits your head, according to those equations. In other words, fast enough for the rain to knock you unconscious :p

And yet it doesn’t. In fact, raindrops hit you at just 2 to 9 m/s, depending on the size. Why? Because of air resistance!

The raindrop starts at rest. There’s no air resistance, so it starts falling. And as it falls faster and faster, the air resistance increases, until it exactly balances the acceleration due to gravity. At that point, the raindrop just continues falling at a constant speed, which we call the terminal velocity.

(image courtesy of Institute of Physics)

A human’s terminal velocity is around 200 km/h. That’s fast enough for you to make a nice wet splat noise when you hit the ground (or so I gather from Saturday morning cartoons). And even if you don’t make a splat noise, you still wouldn’t want to hit the ground at that speed. Which is why we use parachutes.

And now back to cats!

So what does all this have to do with throwing cats out of airplanes?

Cats have great skills at landing on all 4 feet. In fact, unless you drop them from 5 cm above the ground with a lead weight chained to their back, they WILL probably land on their feet. And they would have to hit the ground pretty damn hard to get hurt.

But here’s the catch! A cat’s terminal velocity is about 40 km/h. And at that speed, the cat won’t get hurt!

So no matter how high you throw your cat from, it won’t get hurt by hitting the ground. Unless you throw the cat from so high up it chokes or freezes, it will survive!

And that’s all folks! Just a quick disclaimer: I never actually tried throwing cats out of airplanes. I’m not even fully confident about the terminal velocity of a cat – I just read it online. So if you throw a cat out of an airplane and it gets hurt, you’re a dick!

Stay tuned for next week’s installment, where we’ll be shooting hedgehogs into space with a sling!

(no, not really :p)

Update: The next (and probably final) edition of throwing cats is up! You can now find out what happens when you throw a cat into water!

Hi there, I’m Vlad, and I’m one of Leeds Math Geeks.

Some weeks ago, I was browsing the internet looking for math blogs (it’s a long story). And one of the most interesting blogs I came across was The Math 152 weblog. After digging around a bit, I found out it’s a blog maintained by a bunch of Harvard students who are taking the same university course.

That got me thinking. I’ve got lots of ideas about math, and they don’t really fit my main blog‘s theme. Yet I don’t have enough of these ideas to start my own math-themed blog. So I thought… how about getting together a bunch of people from my course, in the spirit of Math 152? I contacted a couple of my friends, and they all thought it was an awesome idea!

So we’re launching this blog! Leeds Math Geeks – a blog about interesting math ideas, written by a bunch of students from the University of Leeds.

Cheers!