Stunned

I had a bit of a shock. I was chatting to a generally very sensible fellow, when he came out with the "atheism is a faith" bit, accompanied by a bit of mild invective.

Since he was normally given to being very sensible, I asked what made him think that. Turns that we had different definitions of atheism - he understood it to be an absolute, categorical denial of the existence of any gods; in that situation, his claim has at least some basis, and of course some dictionaries do support that definition.

We then had a brief but very reasonable discussion.

It turns out that, under the definition I gave (a-theism, absence of belief in gods), he is an atheist. Indeed, our positions are extremely close (we're both agnostic on knowledge of existence, but both lack belief).

I pointed to some online definitions of atheism to make it clear my definition was not esoteric.

I was really glad I didn't over-react to that old chestnut. He was being quite reasonable, within the scope of his definition.

Not everyone who brings up the "atheism is a faith" thing is a fundie.

A little tidbit I should have already known

I was reading online about a game I bought while in the US and which I have only just had time to take a peek at. Someone made a point about many parts of the game being based on the fact that a right triangle with sides of 7 and 4 units has a hypotenuse with length very close to 8.

Well, 7² + 4² = 8² + 1

so the hypotenuse is close to 8, as suggested: √(7² + 4²) ≅ 8.

In fact, I knew √65 to be very close to 8 ¹/₁₆

(if x is not too small, √(x² + 1) ≅ x + ¹/_2x).

(Note that (x + ¹/_2x)² = x² + 1 + ¹/_4x² , and if x >> 1, the final term is quite small )

So that's an error of around 1/128, or about 0.8%; pretty good, since the game aims for much less accuracy than that in general.

But then I thought about the fact that 16 in the denominator was a bit too small, and I wondered about how much. I realized straight away that it was in fact about a sixteenth too small. That is, it occurred to me that √65 is very close to 8 ¹/_{(16 + ¹/16}).

A little light went off in my head, so I hauled out my calculator.

Try this with me, if you have a calculator handy:
Take the square root of 65. (You should see 8.06225...)

Now subtract 8 (the bit we know).
Take the reciprocal (¹/x). You get 16 and a bit.
Subtract 16 and take the reciprocal. Looks like you get the same number back...

What is this number? A tiny bit of algebra shows it's 8 + √65.

So far, that may seem like a trivial curiosity. But this happens all over.
For example, you get the same thing with any positive integer, x;
√(x² + 1) + x is a number like that "16 and a bit", where
you can keep subtracting that integer part and taking the reciprocal.

That is, expressions like 8 ¹/_{(16 + ¹/(16+ ...))} come up lots of times (and recognizing that I'd hit one of these objects was what made the light go off).

Take √10 for example - it's 3 ¹/_{(6 + ¹/(6+...))}

And you don't just get it with roots of 1 more than a perfect square. As I said before, it happens all over.

We've hit continued fractions. They come up a fair bit in mathematics, and they appear in numerous places where rational approximation comes in - I remember playing with them when dealing with asymptotic approximations in statistics, for example. There's a much nicer notation (see the wikipedia article), so if you're playing with them you're not stuck with endless layers of fraction running down the page.

So, for example, the sequence 8, 8 ¹/₁₆, 8 ¹/_{(16 + ¹/16)}, ... 8 ¹/_{(16 + ¹/(16+ 1/16...))} would be rendered as:
8, [8; 16], [8; 16, 16], ... [8; 16, 16, ...]

Similarly, √10 is [3; 6, 6, 6, ...].

The well known continued fraction for √2 falls into this class: [1; 2, 2, 2...].

Compute a few terms in that sequence with me:
1, 1.5, 1.4, 1 ⁵/₁₂ = 1.416666... , ...

already we're quite close - and it continues to jump about either side of √2, getting closer and closer.

For larger numbers, the convergence is much faster. The general continued fraction for √(x² + 1) is [x; 2x, 2x, 2x, ...].

Try seeing if you can work out what is going on with square roots with different offsets from a perfect square.

So anyway not only is there a handy way of computing square roots that are close to perfect squares, there's a handy way to improve the calculation if it wasn't as accurate as you needed.

There are many beautiful things related to continued fractions. Take a look over at MathWorld if you've a mind for some boggling factoids.

What fun.

(Two posts in one day! OMFFSM)

Unconsciously annoying

Here's a peeve that I've been seeing all over the place the last couple of weeks:

"I'll leave that to your conscious"

. . . That's conscience, dammit.

You can't quite see them waving, yet.

Images of planets orbiting other stars.

58 down, two to go.

Stevens is behind Begich by 814 votes. With mostly Begich-heavy count left, Stevens is not going to pass him.

The only question remaining: whether Begich can (roughly) double that lead and avoid a petition for a recount (not that a recount would flip it).

A year (and a month)

With all the stuff that happened just after I got back from the US and then the hurried preparations for lecturing this subject I am still busy with, I entirely missed my blogging anniversary [for this blog, at least - I also have a long running, if recently neglected, personal blog that's been going for about 5 years].

Yep, I started this blog in October 2007, a year and a month ago.

Many thanks to my modest band of readers. Hi!

This is also my 150th post, so that's averaging about 3 posts a week.

Volume is down (and traffic with it) since with all the lack of time for much of anything but work lately (hmm... I think I have some kids around here somewhere), blogging is the thing that has had to drop off a bit.

The other thing is the fire mostly isn't there right now. There's plenty to get worked up about, but I just haven't have enough anger to go around the last few months, nor the time to deal with a more reasoned argument. I can't even keep up with science news (at last glance, my science news aggregator had about 800 unread articles).

[I have lots of ideas for things to write about, but by the time I find an hour to write a decent post, it has become out of date. I also have a number of topics that won't go out of date in a hurry, but they would take much longer to write.]

So volume is down and will probably stay that way for some weeks yet. But I am still here.

The joy and relief around here is palpable

I'm sitting in a large building at the moment, and echoing from different parts of the building I hear cheers, laughter, loud conversation as the news filters through the building.

I've never heard people so bubbly, excited, and at the same time, relieved at a US election. Obama's election victory seems to have energized almost everyone.

One colleague said to me "What I have now is hope".

Which about sums it up, I guess.