h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Could we please discuss the accuracy and lack of it in my conversation starter, A6338586. I'd also like to figure out surreal numbers and their relationship to infinitesimals, and what on earth the axiom of choice is.

I'm afraid I'm going to take issue with your 'playing with infinity'. By 'point 0.[0-repitand]1' I'm guessing you mean a decimal point followed by an infinite number of zeros then a one. Well, that's not a real number and there's no way you can list all the real numbers so as to take two 'adjacent' ones.

Your last sentence is correct.

I figured most of it was wrong.
However, I was hoping you'd go into more detail.

Hummm. Well, for a start 179.9(recurring) is the same as 180. The fact that they are two different decimal expansions doesn't mean that they are two different numbers.

It's a basic property of the real numbers that for any two different ones you choose, there is another real number (in fact lots of numbers) that lie between the two of them. Attempts to find two that are next to each other are therefore doomed.

Oh and, if I can complain (sort of) about the book you're using, it doesn't make allowance for those of us who've studied topology or some non-Euclidean geometries - it says a polygon must have at least three sides, well I've read papers that refer to monogons (one vertex, so one side) and null-gons (no vertices, but still one side) and I referred to digons (two vertices, two sides) in my thesis.

Don't judge a geometry textbook too harshly.
Could you please explain, then, the rest of the question (infinitesimals, axiom of choice, and surreal numbers)?

Yes, poor textbook.

I can't really help you with the others - you'd be better off looking them up. If you want to know about knot invaraints I'm your man, though.

just to say that I have read it, but it seems a bit beyond me. sorry not to be any help

Knot theory? Fascinating! I almost moved into braid groups once but eventually I didn't. I like the conferences about these though, they're always full of fancy pictures

I have a little amusing problem here. Imagine a wall with two nails. You have a picture and a sufficient length of string attached to it, one end of the string on one corner of the picture and the other end on the next corner. The question is as follows.

Can you hang the picture on the wall in such a way that:
1) It won't fall down;
2) Removing any of the two nails will make it fall down?

Hello!

I would drive the nails close together, loop the string from the picture between the nails and then tie a knot in the end of the loop of string, so that it wont slip back through them. Is this permitted?

Brilliant solution, Tony. Not the sort that toy box was looking for though. For this sort of mathematical problem you have to assume infinitely thin frictionless string. It's not too difficult a problem.

Doug, taking the traditional formalist approach to mathematics, here's what's wrong with your "proof":

1. "My math textbook has no definition of zero."

For reference, zero is that value which when added to something does not change its value.

2. "The angles of a circle, of which there are infinite, are all 179.9 reccuring degrees."

You can't just state this at this point, as you have not yet proved it. Try not to state things and prove them later, as you get confused about which things are proved and which aren't. If you have already proved something, you can then prove it. If you must state something before proving it, label it very clearly "Reuqired to Prove:" or something like that.

3. "All the elements are numbered as real numbers," - you can not do this, unless you provide a system for doing it. Starting at a particular point, you can call it the first point, but there is no second point - the set of points in a line is not countable - there is no way of making it correspond to the set of numbers 1, 2, 3 etc. You can read more about this concept of uncountability in the entry on Cantor A593552. No matter which point you pick as the second point, there are always other points between the 1st and the 2nd.

4. Also, that means that if you add zero (the size of a point) to itself infinitely many times, you'll get a larger-than-zero length. No you won't.

5. "THESE POINTS ARE ADJACENT" There are no adjacent points on a line.

6. "You also probably thought that 179.9 reccuring was the same as 180. It's not" These two values differ by an infinitely small non-zero number. Such a concept does not exist in the traditional analysis framework of modern mathematics. 179.9 recurring = 180. The recent re-invention of infinitesimals may change this a bit though.

As I said before, this is intuitively right. A circle is a polygon with infinitely many sides. But applying full mathematical rigour, the phrase "with infinitely many sides" is meaningless. Mathematicians don't talk about infinity much because you can't pin down what it means. Instead they talk about limits, where things get closer to some value as some other value increases. A regular polygon gets closer to a circle as the number of sides increases. The circle is the limit because you can make the polygon as close as you like to the circle by picking an appropriate number of sides. But the number of sides is always finite.

Hope that makes things a bit clearer. If you understand all this, you'll sail through calculus!

Thank you Mr Ng

1. If the nails are placed the width of the string apart, then it has to work surely? Maybe the knot is infinitely small, but the nails are closer.

2. I thought of an alternative, which is known in pure maths circles as the "Sod the string" approach. Just drive two nails in the wall and balance the picture on top of them. (I bet he doesn't like this, either)

I've signed up for the OU double unit in Pure Maths next year, so maybe I'll discover the real answer then.

Tony (aka Icy)

I love your "sod the string" approach too. Here's a clue: let's think about beer. Not just any beer, but one particular type of beer:

http://www.liv.ac.uk/~spmr02/rings/beer.html

Look at their logo. Note how the three rings are linked. If you cut any link all three fall apart. No ring is connected to any other ring, but all three stick together.

We're looking for something similar with the string.

Oh that's clever! I can see it now.

Does this configuration have any practical applications (I guess it must have)?

Hmm, my solution doesn't bear that much resemblance to the Borromean rings. I hope I'm not giving too much away if I say that the string passes over each nail twice.

My solution doesn't bear much resemblance to the borromean rings either. But it gets you into thinking the right way about the puzzle - it's not a trick solution, it's just the way the string is looped over the nails.

Ah, good. I must confess that my solution was actually inspired by some of the braid theory I studied for my thesis. Looking at it afterwards I realise that it is no coincidence that this worked and that this problem links in very well with the Gassner representation.

The solution is simple: commutator.

And that's all I have to say.

Now there's a clue that will really help your average layman.