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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February 2, 2022 at 12:01am

#1025849

Time and Time and Time and Time Again

I've noted before that, sometimes, these articles that I'm picking at random come up on appropriate occasions. This is one such occasion.

A Defense of the Reality of Time
Time isn’t just another dimension, argues Tim Maudlin. To make his case, he’s had to reinvent geometry.

Why is it appropriate today? Well, because it's Groundhog Day, of course.

The meaning of Groundhog Day has changed over the course of my life. It used to be about giant rodents seeing their shadow (or not) and thus making a prediction for the remainder of Northern Hemisphere winter. In 1993, that definition changed forever; it is now about reliving a time period over and over and over again, thanks to the movie of that name that came out that year.

I've noted before that GHD (the movie) wasn't the first pop culture story to feature a time loop. ST:TNG did it the previous year. But that's okay. Nowadays, every TV show that is even slightly science-fiction-ish has to feature at least one time loop episode. And almost invariably, one of the characters, sensing they're in a time loop, will use it as a verb. "We're being Groundhog Dayed!" Even Doctor Who referenced it in the last special, which came out at New Year's.

Now, the concept of a time loop of that sort is completely unscientific. If time were to loop, we'd never know it, because we wouldn't be able to keep our memories from the previous loop. You might be in one now and not even know it and not even know it and not even know it and not even know it.

Okay, that was a cheap joke. Anyway, unscientific or not, it can make for good storytelling if you can explain or handwave why someone knows they're in a loop. And it's still got way more scientific justification than squirrel meteorology. I mean, when I first heard about the mythology as a kid, I was like, "So if the woodchuck sees his shadow, it's six more weeks of winter, but if he doesn't, it's an early spring? How does that make sense? You can only see your shadow if it's sunny. If it's sunny, things warm up faster. That's just logic." And then I went back to eating paste or whatever it is that kids do.

I was going to put this rant in this week's Comedy newsletter, by the way, but decided instead to talk about walking in the freezing damn cold. Because if I waited until for my next Comedy newsletter in March, that one would have lost a lot of its timeliness. So it's a good thing the article about time came up because it gave me an opportunity to write in here about ImbolcGroundhog Day on Groundhog Day.

You'll note I haven't even started to talk about the article itself. So how about I start now?

Physicists and philosophers seem to like nothing more than telling us that everything we thought about the world is wrong.

And now you're going to tell us that that's wrong.

They take a peculiar pleasure in exposing common sense as nonsense.

That's because it generally is. If there's anything I've learned in life, it's that "common sense" is neither. Anyone who insists on leading from "common sense" is automatically a bad leader.

But Tim Maudlin thinks our direct impressions of the world are a better guide to reality than we have been led to believe.

I've been railing for years against the numinous "time is but an illusion" bullshit promoted by mystics and charlatans and even some serious scientists. Sometimes even here in this blog. Therefore, I'm favorably inclined toward any scientific take on the subject that tickles my confirmation bias. So I'm not sharing this article to rag on it, but because it makes me think I'm right about this particular philosophy. I might not be, and this guy could have it all wrong too, but it's nice to know I have some backup.

The rest of the article, honestly, gets into stuff that's beyond most of us, and there's not a lot of point in quoting it out of context. I still think it's worth a read, even if you don't grok it in fullness. I can't say that I did.

So take some time and read it. I promise you won't get Groundhog Dayed into reading it again unless you want to.

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