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.

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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.

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

November 15, 2022 at 12:01am

#1040668

Captain's Log

Look, sometimes the Random Number Generator (peace be unto it) gives me the same source twice in a row. But I promise you this one's worth it, because it resolves a question I've had since I first learned what a question was.

Where All The Poop On The Enterprise Goes -- According To 'Star Trek' Experts

I know I've commented on this before. For a while, I was methodically going through every single episode of every Star Trek show, plus the movies, in chronological order. I think the first time I noticed any reference to a toilet (or that would be head, since it's a ship) was sometime in the early 90s, some 25 years or so after the show's beginning.

But, apparently, I'd missed some. We'll get to that.

Without its fantastical future technology, Star Trek would just be a series about people who love sitting in comfortable chairs.

People watch shows like that all the time.

What has wowed audiences for decades are inventions such as the faster-than-light warp drive, the matter transportation system, and the Starfleet human resources nightmare that is the Holodeck.

Someone, I think it was Dave Barry, noted that the holodeck would be humankind's last invention. Personally, I don't think we'll ever invent it. Not because we won't be able to, but because, well, look at the Metaverse. Any real-life implementation of a holodeck will be absolutely loaded with safeguards, to the point where no one will be able to do anything fun with it, it will be a joke, and everyone will laugh at the inventor (who will consequently become a supervillain: "Laugh at ME, will you? Let's see who laughs now muhuahahaha!!!")

But anyone who’s ever watched any Star Trek TV shows, movies, or adult movies probably has some serious questions about how this fictional universe really works – perhaps the biggest being: where the hell does everyone go to the bathroom?

This is, indeed, one of the biggest questions in the universe.

I should note that, at about the same time DS9 was airing, and also around the same time I saw someone in the Trek universe finally acknowledge the existence of a toilet, Babylon 5 (another SF show about another space station that stayed in one place) not only acknowledged it, but set scenes in it.

Anyway, the next bit points out that there was a door labeled HEAD on the bridge of the Enterprise-D. That was TNG, before DS9.

And while the production “did not design or build the inside of that bathroom,” it was still there, just in case Number One had to take a … well, you know. Also, in the crew members' “various living quarters,” there is an unopened door that “we assume led to a bathroom.”

I should also note that this was around the time when, in attempting to answer my own version of the question (and the Trek novels were no help in that regard, either), I finally decided the key had to be transporter technology. Think about it. A transporter, under normal use, records the position and connections of every molecule, atom, proton, electron, and so on in your body (they have "Heisenberg compensators" to handle the Uncertainty Principle) and your clothes and equipment, right? And when you arrive at your destination, your clothes aren't somehow bonded to your body, etc.

Well, part of your body is the shit in your intestines and piss in your bladder. Sorry, but it's true: transporting someone necessarily requires transporting whatever waste products are awaiting exit. And with the transporter able to easily distinguish between different molecules, certainly it can tell shit from shinola. So. All you have to do is program the transporter to image you, then pull out the waste. No need to even strain on the throne; just push a button and boom, it's gone.

Where it goes is... well, let's keep reading. My theory turns out to be wrong, anyway. But it could have been right.

So we know that the Enterprise is equipped with toilets, but do we know what happens to the human waste? Do they make Chief O’Brien beam it out? More sensibly, one would think that the Enterprise crew could simply fire their poop into space from time to time, like a smellier version of Spock’s funeral.

Send it all to the Klingons like Scotty did with the tribbles?

But it turns out that this might be a terrible, terrible idea.

It is, for many reasons, not the least of which is the next starship zipping through the area will come into spacedock with skid marks.

Dr. Siegel speculated that, while none of the Trek shows go into much detail about toilet-based issues, “every bit of eliminated human waste is useful matter that you can reconstitute into something else.” Meaning that any waste created could potentially be used to “power the Replicators” – you know, the device that people of the future use to create small objects, cups of piping hot tea, and food.

Now, look. Some people are going to be grossed out by this idea. But come on. How do you think it works right here on Earth, and mostly without technological interference?

Everything you eat contains atoms that were once part of poo. Everything you drink contains atoms that were once pee. Every breath you take inhales atoms that were in a fart. Hell, worse, they've all been part of dead things. I don't mean fresh dead things like the kind you eat (even if you're vegan), but yummy, delicious carrion... oh, sorry, channeling my spirit animal again.

A replicator would just speed up that process.

Part of civil engineering, though not a part I ever participated in directly, is sewage treatment. You take all that waste and process it, and (in theory anyway) the water becomes clean enough to dump into a river. It's then either evaporated, falling back as rain that you might eventually drink; or it's processed by fish that you might eventually eat. Solid waste is sterilized and becomes fertilizer. Trek would just use the technological evolution of the same sort of processes.

According to Dr. Siegel, this is a pretty solid plan, although admittedly, you have to get over the “gross factor.” Like if you were to replicate a clarinet, à la Harry Kim in Voyager, you’d have to look past how “the atoms making up the clarinet that I'm putting in my mouth and playing right now were defecated out by me and a thousand other crew members onboard the ship.”

Again, the only difference between that and the way things occur naturally here on Earth is scale.

Still, seems to me that in Trek it would be simpler to cut out the whole "plumbing" bit and use the transporter like I said. But I guess that might set things up for some really gross practical jokes.

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