Alternative … History

Browsing around, I stumbled across this picture, which got me thinking. Could you write a consistent, complete alternative history of the United States (modern history, perhaps), whose common theme was that every president was taken out of some movie? What kind of effects would the events of Air Force One, Escape from New York, Fail-Safe, and Armageddon have on the history of this nation, just to name a few?

(One of those things is not like the other ones…)

Idle thoughts. Maths later ^.^ Primes and polynomials.

Multiplication: You’re Doing It Wrong!

It is of interest to me the extent to which the traditional long multiplication algorithm is of so little use when multiplying irrational numbers. I mean, consider the below with respect to our traditional place value system.

1.41421356237309504880168872421…

Clearly it trails off infinitely to the right. But the point is, in traditional multiplication, you start at the rightmost digit because you can be guaranteed that no lesser places will effect the multiplication of that place through carrying. For example, multiplying 432*599, looking at the last two digits of 2 and 9, it’s clear to see that the last digit of the product must be an 8, since 9*2 = 18. The next digit can then be worked out, knowing you carry that 1. However, starting from the opposite side, 4*5 = 20, so you ‘know’ that the leftmost side must be around (imagine airquotes around that) 20. But you can’t say for sure, because prior terms in the product will effect the final result. Indeed, a more astute observation would be to say that 599 is almost 600, and 4*6 = 24, so the leftmost side of the product will more closely resemble 24. Of course, we’re already using prior terms to adjust our prediction/calculation about the leftmost side of the product. And indeed, if you calculate the actual product out, 258768, we see that the leftmost part is actually 25 – we were close, but some amount of carrying pushed us off slightly. By starting from the lowest place value, we avoid all this.

So, as it applies to irrational numbers, because they have an infinite decimal representation, there is no lowest place value to start at. There is an obvious starting point, the leftmost place, the 1 before the decimal, but the problem then is that the place values decrease from left to right, so any kind of carrying would just fail, and multiplication would just fall apart.

The point though is that I don’t really care, so what happens if we apply the multiplication algorithm to infinite sequences of digits?

Logic: Abstraction and the Theory of Lumps

To begin with, I would like to start by saying that today’s winners are people that ennunciate the ‘t’ in ‘often’.

I was thinking today, about abstract math. Higher level math is largely about abstractions. What this means is we consider mathematical objects with certain properties, and prove things about those objects from those properties. What’s nice about this is that whenever we find something new, if we can verify that it obeys a certain set of properties, anything we’ve proven about things with those properties automatically applies. That’s why abstraction is nice, we can spread results far and wide.

There are different levels of abstraction. Take groups, for example. A group is a set of objects that interact with each other through some operation called ‘addition’. Notice I put that in quotes, because what the operation actually is depends on what you want it to be. Different groups have different operations, but even though the operations are different, what you can prove about sets with those operations, in general, is broadly applicable. It’s nice.

Movie Break: The Wolf Man

I’d just like to take a quick break from the maths (though more will be coming, soon I hope), to draw your attention to the coming remake of The Wolf Man, due in 2009. As someone who will sit through almost any werewolf movie, no matter how bad, I can’t begin to describe how legitimately excited I am about this.

The pictures have to be seen to be believed.

Lon Chaney Jr., 1966

Benicio del Toro, 2008

Grab Bag: Inexplicable

This is just a brief set of things that I’ve stumbled across in my travels, mostly through the random math script at Mathworld. The common theme here is interesting maths that I can’t quite explain or justify.

Primes, Probabilities, Products

Over at Pro-Science, Kris noticed a news item, that researchers were reporting the discovery of a new mathematical object, a ‘third degree transcendental L-function’. And he was wondering just what this means, and why it is important.

The best answer is actually given in the article, that

“It’s a big step towards our understanding the ‘world of L’, which is where most of the secrets of number theory are kept.”

But I think I can elaborate a little further.

Quantum Computers: Math and Time Travel

I’m currently half way through my Spring Break, so the math is slow/non-existent. I’ve got an interesting math problem half written up, but I’m trying to decide if there’s a better way to reach my final conclusion. It’s still nice to get a break from the insanity, though.

In the mean time, in this months Scientific American, there was an article on the Limits of Quantum Computers. You’ll need to register/pay to read the online version, so that’s sort of a bum deal. But it’s a very interesting read. But it draws together two of my favoritest things, math and time travel. And a big heap of quantum computing to boot.

Puzzler #2: Cube Dissection

The idea of interest today is, as the title states, cube dissection.

The question is, given a cube, can it be cut into a finite number of smaller cubes – all of different size?

Clearly, thinking in terms of a rubik’s cube for example, it is easy to cut a cube into a finite number of smaller cubes of the same size. One could imagine too, cutting a cube into an infinite number of infinitely tiny, though dissimilar, cubes. But can it be done in a finite number of different sized cubes? It’s like a Jigsaw Puzzle, where you can pick the shape of the pieces. You just don’t know if they actually form anything when put together.

Solution below the fold!