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.

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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.

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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.

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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.

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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.

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June 6, 2024 at 9:51am

#1072234

Elevator to Heaven

From Wired, some panic about a thing that might not even happen.

What Happens If a Space Elevator Breaks
These structures are a sci-fi solution to the problem of getting objects into orbit without a rocket—but you don’t want to be under one if the cable snaps.

Yes, they're a staple of science fiction, but early concepts for them were thought up way back in the 19th century, before even controlled powered atmospheric flight and way before achievement of space travel.

They're also called Tsiolkovsky towers after the Russian who apparently first conceptualized them. According to Wiki,

other names include "space bridge, star ladder, and orbital lift." Personally, I find all those names descriptive but boring. No, I think it should be called a space lift. Because it rhymes with "face lift," and I find that amusing. Everything should be named in accordance with what I find amusing.

In the first episode of the Foundation series on Apple TV, we see a terrorist try to destroy the space elevator used by the Galactic Empire. This seems like a great chance to talk about the physics of space elevators and to consider what would happen if one exploded.

Not even the prospect of a Foundation series is enough to get me to pay any attention to Apple TV. But the idea, as I said, isn't new, nor is thinking about the security issues surrounding them. If you think about it, the idea of a tower connecting Earth to space will inherently offend a large portion of the population, who will inevitably call it the Tower of Babel, decide that it's against God's will, and actively try to blow it up.

This is why we can't have nice things.

Well, actually, we can't have a space lift because, so far, no one's been able to come up with a material that with the appropriate requirements, or address numerous other issues including powering the thing or shielding any passengers from space radiation. As far as I know.

Anyway, the article mentions that, sort of, but starts out by explaining, in great detail, how a space lift would be an improvement over rockets. This takes up most of the article, and I'm not quoting that part. Only after that does it get into what the headline promised.

In the first episode of Foundation, some people decide to set off explosives that separate the space elevator’s top station from the rest of the cable. The cable falls to the surface of the planet and does some real damage down there.

Again, not the first time this has been considered. I don't remember a scene like that from the Foundation books (by Asimov, whose entire output I devoured as a teen), but it's been a very long time. I do remember Kim Stanley Robinson addressing the issue in his Mars series.

The (over-simplistic, which the author acknowledges) video model in the article aligns with what I recall of Robinson's description.

So not only is building a space elevator very difficult, but you really don’t want the cable to snap and fall.

At this point, though, it's firmly in the realm of speculation, both in science fiction and actual science. The problem may be so tough that we'll actually develop antigravity first, which would negate the need for a space lift. Which is why you don't see many space lifts in Star Trek (budget considerations notwithstanding).

Of course, all of that assumes that a small remnant of humanity doesn't get plunged back into subsistence living before either happens, which I'm not prepared to rule out.

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