Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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May 28, 2022 at 12:03am

#1032941

Trouble

There weren't a whole lot of interesting things that happened on this day in history, and I picked this one at random.

On This Day
28-May-1974
Collapse of the Sunningdale Agreement

Here's the thing, though. Well, two things.

1. What little I know about Irish issues can be stuffed into a thimble; and
2. The linked article is largely unsourced and suspect.

So I won't talk directly about the Sunningdale Agreement. You can go to that page and read it yourself if you want. I'm not saying it's not interesting, but anything I say I'd be talking out my ass.

Not that I've never done that, but I'm trying to go into recovery.

What I am aware of is that a big part of the problem has to do with religion. Catholics, Protestants, whatever. Blame Henry VIII, I guess. The whole thing is way above my pay grade.

A friend of mine, a Wiccan, took a trip to Ireland once. Someone asked her something like "So, are you Catholic or Protestant?" (This is apparently less volatile a question than which football club you support).

"Actually, I'm Pagan," she said.

"We're all Pagan, dear. Are you a Catholic Pagan or a Protestant Pagan?"

I have no real reason to believe that actually happened, but it's how she told it. I've seen it elsewhere, so it might have been a joke. Either way, it adheres to Waltz's First Rule of Comedy: Never let the facts get in the way of a good joke. Or a bad one. Especially a bad one.

Speaking of bad jokes, I'm also reminded of this old one:

An Englishman, a Scotsman, and an Irishman are all drinking in a pub, getting loud. Problem is, they're all stuttering terribly. The other patrons are starting to get annoyed. This gorgeous woman comes up to them and says, "Tell you what, boys. If any of you can tell me where you were born, without stuttering, I'll show you a night you will never forget."

The Englishman is the first to speak up. He braces himself, prepares, and says, "Liverp-p-p-p-pool. Shite."

The Scotsman is next. He goes, "Edinb-b-b-b-burgh. Ach, dammit."

The Irishman looks calmly at the woman and says, "London."

Surprised, she takes him by the hand and leads him upstairs.

Half hour later they come back down. Irish guy's got a big grin on his face, he sits down, quaffs the rest of his ale, and goes, "...d-d-d-d-derry."

I suppose that the most unbelievable part of the joke is where those three were drinking together amicably, but I suppose it can happen.

Point is, I guess, that wherever you are, someone's not going to like the government, and that's why shit like this is relevant. Because, as always, if you don't learn from history, you're doomed to repeat it. Well, not necessarily "you." Like I said, I'm not well-versed in this aspect of history, but then again, I'm not in charge of anything but a couple of cats and a bit of property.

And this is really why I'm doing these entries: to learn.

There's no perfect government, and some people will never be pleased with whatever government they end up with. Ideally, that's what democracy is for, but things rarely turn out ideally, so you get demonstrations, riots, insurrections, civil wars, not-so-civil wars. These only get worse when other nations get involved, too. Again, I can't speak to the Irish situation, but there's another country right now that recently annexed part of a different country, and still wants more. Not exactly parallel, I suppose, but another echo.

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