A history of mechanical calculating devices

Today I gave a talk to a group of Cambridge maths students on the history of mathematics. It was an exhilarating and exhausting experience. I've been asked to put up a video of this, unfortunately the best I can do is a rather poor camera-phone video, which may make understanding me difficult. But there's a lot in the slides so I've uploaded them too. Thanks to everyone there, especially the person to my left who keeps interrupting with corrections and clarifications that I really needed. He is of course the legendary Prof Piers Bursill-Hall whose lecture series this was.

Sadly due to battery life limitations I can only give you the 1 hour 10 min talk and not the much more interesting 40 min question and answer session afterwards where I really showed up how shallow my research was. Abraham Izrael Stern especially, I make the claim that he's doing something really important, but I've done almost no research into the guy's life and impact, that needs serious work. But this is such a vast topic that in the Q&A afterwards a phd thesis and a whole seminar of talks were identified of extra study that could be done. Which sounds great to me ... you know, after exams and stuff.

Video: http://www.sendspace.com/file/hgf52a
My slides: http://www.sendspace.com/file/6btn0q

Jupiter and beyond the infinite - Part 5

Solving Zeno's paradox

Adam: Ok, we have seen that different sizes of infinite sets can exist by thinking about size as pairing things up. We've seen that adding one to an infinite set makes it the same size and today I'd like to try and deal today with some of the problems raised by Zeno. Would you like to explain the paradox of the arrow so I can see how best to respond?

Plato: Ok, suppose an arrow is fired at a target. Before it can hit the target it has to get half way there, before it can get half way there it has to get a quarter of the way there and so on. So it needs to do an infinite number of things before it can hit the target. Doing an infinite number of things takes an infinite amount of time. So the arrow will never hit the target.

Jupiter and beyond the infinite - Part 4

Different infinities.

Adam: So last time we worked out that we measure size by pairing things up. And from that we worked out that the set of evens is the same size as the set of all whole numbers, that the set of square numbers is the same size as the set of whole numbers, etc.

Jupiter and beyond the infinite - Part 3

Counting the infinite.

Adam: Hello Plato, sorry I was busy in Friday, care for another chat about infinity?

Plato: Definitely, let's see. Last time we decided that a set is a collection of some number of objects, that can be thought of as a box. We also said it makes sense to say that an infinite set cannot be given a number as its size. But I said that we had missed something and wanted to know how we could say the size of something infinite.

Jupiter and beyond the infinite - Part 2

defining infinity, sets and counting

Plato: So lets recap what we worked out last time. If I remember right we realised that if our definitions aren't sorted out we can end up confusing and contradicting ourselves. And as we dont find definitions set in stone, we have to make them to suit our needs. (Though I'm not sure I agree with you about that).

Adam: Yeh that's right, basically today I'm going to suggest that the idea of a set is a useful one for discussing infinity. … hang on a second … why am I talking to you?

Plato: Oh it's called a Socratic dialogue. You talk to fictional student,in this case me, then in arguing with me you find a solution to the problem.

Jupiter and beyond the infinite

Infinity is a very interesting subject, and one that I need to understand better. I'm going to make this post a regular (weekly?) thing. I hope that by going slowly over what infinity is and what we can say about it that I can a) ensure my understanding of it is clear b) interest people who might want to know about it and c) show how mathematical ideas are generated and created. I hope you enjoy this. Feedback, as ever, is appreciated.

Brainteasers

Some interesting problems at various levels of impossibility:

• Think of a chessboard, can you cover the board in dominoes so that each domino straddles two squares exactly?
• Can you do it for other sized boards? Not just 8x8 but nxn?
• Can you do it if you take out the bottom corners and just try to cover the rest of the board?
• Can you do it if you take out diagonally opposite corners?
• Think of a chessboard which has a number of squares on each side which is a power of two ie 2x2x2x2x.... some amount of times. Take out any square at random. Can you cover the rest of the board in trionimos, L shaped blocks made of three squares?
• Two trains are on the same track, each is moving at a constant 30mph and they are a mile apart on a collision course. A super-fly starts out at one train and rushes towards the other one at 100mph, once there it turns round instantly and rushes back to the other at 100mph. How far does the fly travel before it gets squashed between the trains?
  

RESPOST Why I love maths.

Stop! Please dont run away, I promise this wont be scary or hard. A lot of people are scared by maths because what you have to do in school is awful and boring and hard and crap. I know it's awful, I hated it as much as you all did. But I still love maths, because there's something more to maths, there's beauty. And if you'll give me just 5 minuets of your time, I'd like to show you something that I think is more profound than a lot of poems and more lovely than a lot of art. And I promise no big words and no sums, not one.

REPOST Derren Brown's "Deep Maths" is easy

If you didn't watch the lottery prediction or the event program you're crazy, go watch it online right now. But if you did you may have been interested by the "deep maths" of coin tossing that Derren claimed would take an hour to explain... not so, it's fairly simple really. So I'm going to try and explain it quickly and without anything more than common sense.

The game:

You take a coin and flip it a lot of times in a row, and you note every time that two different sequences of 3 results comes up.

The assumption:

Your initial reaction (and mine) was that all of the 2x2x2=8 results are equally likely to come up. Because if we flip a coin 3 times only this is true. You try and flip a coin 3 times and record weather your sequence came up or not and then start again. If you do this enough times (you'll need a few hours and nothing better to do) and you can easily show that all 8 results are just as likely. We are lead to believe this is true for the game above.

But:

This isn't what happens, in a long string of results HTHHHTHTHTTHTHTHTHTTTHTHTH

THTTHHTH etc it is possible for results to overlap. For instance HTHTH contains HTH twice, and HHHHH contains HHH 3 times. How can you use this overlapping to your advantage? You start by thinking about what happens if you nearly "win" but fall down at the end. What can you do if you predict HHH and it comes up HHT? You've got a result you dont want, and you have to start again from scratch. but what you can do is get a head start. If the last coin in your sequence is different from the first one something interesting happens. You pick THH, the first coin is a tail, you're happy, the second is a head, you're very happy, the third one is a tail ... you haven't won this time, but you're not upset, because you've already got the first coin of the sequence in place. To put it simply: with THH every time you predict a coin wrong you are already a third of the way through the next attempt, but if you pick HHH then every time you go wrong you have to start again at the beginning.

I predict THH: T win! 1/3 done, H win! 2/3 done, T ignore what happened before, start again with the first one right , 1/3 done

I predict HHH: H win! 1/3 done, H win! 2/3 done, T ignore what happened before, start again with the first one wrong, 0/3 done

Conclusion.

Pick a sequence where failing to get the end right means you are automatically in with a second chance. This means you can win far more often than you would think. The reason is just common sense, no deep maths, no psychology, no team thinking, no magic, just common sense.

As for how he predicted the lottery, there are dozens of equally plausible solutions, but however he did it I'll stab myself in the foot if a group of 24 people in a trance had the slightest involvement.