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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 29, 2021 at 12:03am

#1018245

Han Shot First

I have to get up at an ungodly cow-milking hour (8) for a routine tooth torture session, so it's just as well that I don't have much commentary on this one.

The '90s 'Star Wars' Special Editions Kicked Off The Never-Finished Movie Era

I saw the original Star Wars in a theater in 1977 (and of course its direct sequels later). When I went to see some other movie in the 90s -- pretty sure it was a science fiction flick, but I can't remember what it was -- they showed the preview of the re-release Episode IV. I remember that the theater was completely packed full of people primed for space movies.

At the end of the trailer, when (spoiler alert) the Death Star blows up, there was a moment of awed silence. Then someone said, "I think I just saw God," and the whole theater erupted in whoops and cheers.

If only we'd known...

Rolling Stone interviewed Lucas later that summer, and he bemoaned how the movie was insufficient and only "about 25%" of what he wanted it to be, adding: "There is nothing that I would like to do more than go back and redo all the special effects." And that was way back in August 1977. So when the film's 20th anniversary rolled around, and Lucas wanted to update the movie, it shouldn't have come as a shock. Lucas convinced Fox to pony up $10 million in what was advertised as a restoration of George Lucas' artistic vision, which had been tragically ruined by 1977's conspicuous lack of desktop computers.

I took a cinema class as a humanities elective in college one semester for the easy A and an excuse to watch movies every Wednesday night. This would have been nearly 10 years after the original Star Wars. I don't remember much about the class, but I do remember the otherwise very stodgy professor proclaiming that there were two movie eras: pre-Star Wars, and post-Star Wars. Whatever you think of the movie itself, the way it changed the way movies were made clearly made an impression on stodgy film class professors.

It's 35 years later now, so maybe there's another era. Post-Matrix, maybe. I don't know; I'm not a professor or a critic; I just like movies (but not The Matrix).

No, Lucas trumpeted that he had "gone back and fixed the trilogy." To him, this was Star Wars, and the public had merely gotten accustomed to his rough drafts along the way. And it's not as though we didn't get a heads up about this; the '95 VHS release advertised that it was everybody's "last chance to own the original Star Wars."

I actually own those. Of course, I can't fucking watch them.

While many of the new effects blew away audiences at the time, other changes immediately rubbed some fans the wrong way. Most notoriously, the awkward shot of Greedo ineptly firing his blaster milliseconds before Han drew fan outrage even before the film hit theaters, thanks to primordial internet communities.

Goddammit, HAN SHOT FIRST. I have spoken.

After all, in hindsight, why would a trilogy that ends with a tribe of woodland teddy bears begin with a dude murdering someone in cold blood?

"In hindsight," I should have taken the Ewoks as a sign that Star Wars had jumped the shark.

The article argues that this is something different than Director's Cuts, which I can kinda see. But I'd counterargue that the idea of re-releasing new versions of old movies owes its proliferation to the success of the Director's Cut of Blade Runner, which was the first movie I was aware of being re-released as something substantially different from its original theatrical form. The DC was, by every measure, a far superior movie, though it was not without its flaws as well, leading to the eventual release of the "Final Cut" (which I don't believe for a second that it actually is).

Since it was so much better, I think audiences were primed to believe that updating would improve the movies.

I insist that it's entirely a coincidence that Harrison Ford was in both films.

Anyway, this was supposed to be short, dammit. I didn't quote much of the article; it's there for you to read if you're interested.

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