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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March 15, 2022 at 12:11am

#1028954

Audience

And now, a few words about movies.

The Movies Are Gonna Be All Right

The article talks about the recent Spider-Man movie. It's still showing in my local theater, so I deem this piece from December still relevant.

You’d think the mind-blowing box-office performance of Sony’s Spider-Man: No Way Home this past weekend would be good news for just about everybody.

And like I said, it's still going, like a certain battery-powered rabbit. I gotta say, though, I was starting to get a little pessimistic about the article's thesis, when there was hardly anything worth watching (besides that, and I saw it twice) in the theater in January and February.

Now The Batman has joined it, so there's the fairly rare occurrence of both major comics publishers having a big movie in the theater at the same time. No, I haven't seen the Batmovie yet. Maybe today. More likely, Thursday.

Superhero-movie fans, of course, were happy. Theater owners, who’ve had a rough couple of years, were definitely happy. “Hey prophets of doom,” Adam Aron, CEO of the much-beleaguered AMC Theatres chain, tweeted Monday to those who’d claimed streaming would kill movie theaters. “#CHOKEonTHAT.”

Another reason to hate AMC Theatres: The CEO twoots and uses pound signs (I refuse to call it the H-word).

For those who’ve watched the slow death of movies made for adults over the past decade-plus, the problem wasn’t so much that a superhero flick was making so much money but that other big releases were crashing and burning all around it.

If by "made for adults," you mean "mind-stunningly boring," sure.

Here, in seemingly stark and undeniable terms, was the pop-cultural dystopia many had feared: comic-book tentpoles obliterating anything that isn’t a comic-book tentpole out of the marketplace.

If by "many," you mean "culture snobs," sure.

I'm an adult, I like many different kinds of movies, and I'll watch the hell out of superhero flicks.

I gave Spider-Man a lousy review, but even I’m happy to see it doing so well.

This is my shocked face.

I've said it before: you like what you like. It's also good to have different opinions to look at. At least this writer isn't all "stop liking what I don't like."

And on a micro level, there are glimmers of hope all over the cinematic landscape. Throughout the summer and fall of 2021, I found myself floating from theater to theater in New York, surprised at the level of turnout.

Sure, it's nice to have streaming services. But there's still something compelling about being in the theater, engaged, unable to put it on pause, among other people who wanted (for whatever reason) to see the same movie. And I say this as an introvert.

I think the author here and I can agree on the attraction of the cinema:

When screenings of Casablanca are selling out (as they did this year at Film Forum during its Humphrey Bogart festival, another big hit), it’s about more than people wanting to see Casablanca; it’s about people wanting to be around other people as they watch Casablanca together. Believe it or not, this is somewhat analogous to the phenomenon of people coming out in droves to see Spider-Man and a nostalgic assortment of old, recognizable villains.

But I really wish people would stop ragging on "nostalgia." Admit you like it. Admit it's a draw.

Besides, I've been hearing rumblings about streaming services wanting to go to an ad-supported model. This would make me cancel a few subscriptions, because if I'm paying for something, I don't want to see ads (movie and show trailers get a pass). This is why I've never had cable.

Meanwhile, the new-release landscape has been more uneven. But even on that front, just two weeks ago, you could come across any number of sold-out screenings in New York of Licorice Pizza or Drive My Car.

I'm still mad I missed Licorice Pizza in the theater. I have no desire to stream it. But I wanted to see it at the drafthouse. Anyway, you can't rag on nostalgia and wax poetic about that movie, at least from what I saw in the trailer.

In the early months of the pandemic, sizable groups of allegedly smart people thought that movie theaters were not only doomed but that they maybe even deserved to die.

Who were these people? I'm thinking Netflix, Disney, Paramount, and Amazon shills.

On the other hand, it's true I wouldn't be as keen on going to the movie theater if the local one was, say, AMC or Regal (are they even different?) instead of a cinemaphile's drafthouse. The whole overpriced-popcorn-and-coke thing doesn't do it for me.

If we can’t be around each other anymore, we are no longer a society, a country, a civilization. We’re just pod people jacked into the Matrix.

Oh, look, a lowbrow movie reference.

And cinema cannot survive without theaters. Without them, it becomes TV, a completely different art form.

I'd like to believe that, but that's an assertion without evidence. Rather than movies becoming TV, I've seen TV shows becoming more like movies. Disney is excellent at this, what with Loki, The Mandalorian, etc. Of course, you could argue (and you'd be right) that those shows are based on theater movies, but my point is that the six-episode mini-series format is more of a six-hour movie now than a traditional TV show.

Even Star Trek has gotten into that act, and while it had movies, it was always a TV show first and foremost.

Well, we’ll be hitting two years of pandemic in less than three months, and almost nobody is spewing this streaming-über-alles nonsense anymore — at least not in public. Maybe because after trying it out for a while, we’ve come to realize that a lifetime spent on our couches is neither feasible nor desirable. Maybe because losing things sometimes makes you appreciate them. Maybe because 2020 and early 2021 were an inadvertent test run for our theaterless, benevolent-cocoon future … and it was fucking horrible.

I can agree with this, too. Back in 2020, I decided, in a rare moment of making a New Year's Resolution, to see one movie a week at the Alamo.

That lasted until mid-March, when they closed for a few months.

Nothing made me happier than coming back to watch a movie, even if it did suck (Tenet).

Lesson learned: don't make New Year's resolutions.

And families have yet to return to theaters at the rates they once attended. That’s why a film like Disney’s Encanto, which in the Before Times would likely have been a big hit even if it were terrible (it’s not — it’s delightful), has made less money than freaking Free Guy.

The theater I go to isn't exactly family-friendly. I mean, they don't exclude brats, not really, but their focus is on adult beer-swilling movie lovers: in other words, I'm their demographic, even if I'm older than some. There are strict rules about talking and texting. No one under 18 is allowed without an adult, regardless of the movie's rating (there are some exceptions).

As an aside, I'm glad to see more people adopting "the Before Time(s)." I'm under no illusion that I started this trend, but shortly after the pandemic started I was early on in my quest to re-watch all of Star Trek (I'm up to Disco now), and that phrase comes directly from an Original Series episode. So I started using it.

Now if I could only get more people to refer to the first decade of this century as the Noughties.

Also, Free Guy was awesome; shut up.

Anyway, there's a lot more in the article that I didn't quote here, and if you like movies, it's worth a read, even if the author does come across as a highbrow snob. Whatever brow you have, low, medium, high, uni -- I think it's good that theaters seem to be bouncing back.

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