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.

3 Like

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 28, 2021 at 12:01am

#1018177

To Sleep, Perchance To Demon

Today's science article, courtesy of a dick joke site:

Dealing With The Demon of Sleep Paralysis

Yeah, I know I've written about sleep paralysis in here before. I have to experience it, but the only way I can share the misery is to write about it. You're welcome.

Ever wonder what the sensation of dying in your sleep might be like?

Nope. Don't have to wonder.

Oh, and with an obscure presence by your side? And what if this figure resembled one of your fears?

An eye surgeon? Nope, haven't had that one yet.

A physical feeling of entrapment, sleep paralysis is a troubling sensation that one may experience while waking or falling asleep.

It's a "troubling sensation" the way a hydrogen bomb is an explosion.

Usually accompanied by hallucinations, you may see disquieting images as your mind is active while your body remains asleep.

"Hallucinations," also known as "the same shit that happens during dreaming." Also, seeing is only part of it. You get to hear it, too. Footsteps approaching the bed. Someone calling your name in anguish. A dog barking. Which wouldn't be so bad if I had a dog, which I don't.

What is known is that while dreaming, the mind is at rest and the body's muscles relax as a means of not harming itself in case you dream about snapping your pinkies in half to see what'll happen.

Thanks, I really needed that image.

I think -- but I don't have any data to back this up -- that sleep paralysis is the other side of the sleepwalking coin. In the former case, your mind disengages your body while you sleep, but that can continue into a mental waking state. In the latter case, the clutch mechanism, as it were, engages, so your body moves while your mind's checked out. Either way, shit's not working right.

When the paralysis creeps in, it causes a disruption in our REM cycle and awakens the mind while the body remains confined in a paralyzed state (cue the horror music). You'll want to scream but can barely contort your face, and no one would hear you.

And that might be the worst part of all. The instinct is to call out for help, but you can't even manage a whisper, let alone a scream. You might as well have been shoved out an airlock into the void.

For example, "you might mistake your cat sleeping at the foot of the bed as some sort of goblin," and as ridiculous as it sounds, it's the truth.

Actually, the cat is a comforting presence for me. But I can imagine this happening to someone more ambivalent about our feline overlords.

The scariest of hallucinations is referred to as the "intruder" and is just what it sounds like, a sensation of an intrusive figure at your side. Mainly, it has been reported that this sense manifests itself as "the sounds of doorknobs opening, shuffling footsteps … a shadow man."

It's worse when it's a shadow ex-wife. That's happened. The author also describes tarantulas. I haven't had those, but other bugs have invaded this mental state.

(Yes, I know arachnids aren't technically bugs.)

In my experience with sleep paralysis, it's a lovely thing to sense a presence of danger as you lay there, frozen helplessly in the dark. You don't have a darn clue what to do because you can't do anything. One theoretically cause is unbalanced sleeping patterns, so the best advice to treat it seems to be: Get more damn sleep.

Yeah, I don't know. It's happened to me when I was chronically sleep-deprived, and it still sometimes happens to me when I get enough to be well-rested. In my experience, the only thing that makes it subside is... exercise.

Sometimes that's worse than the nightmares. At least with sleep paralysis, I end up with story ideas.

Mini-Contest Results!

Thanks for all the comments! Some of those I was familiar with, others not so much.

Warped Sanity, that's one I hadn't heard of, but now I want to check it out. The thing about science fiction is that it's not a stand-alone genre; it always has some other angle: often action/adventure, sometimes mystery or, as this one seems to be, horror.

ForeverDreamer, I read A Clockwork Orange at a very impressionable young age, and a lot of it stuck with me. I also enjoyed (well... appreciated) the Kubrick movie.

Kåre เลียม Enga, I never quite got into Pern. Probably I could attempt it again; it's been a very long time and I think a lot of it was beyond my childhood mind (which, as I noted above, was warped by A Clockwork Orange and other stories).

WakeUpAndLive~doingNaNo'24, a lot of people really liked The Matrix. I had just been learning about historical Gnosticism when it came out, though, and through the movie I was like "this is just techno-gnosticism." Beautiful effects, and I do like Reeves as an actor, but especially afterwards, when people started to take seriously the whole "what if we're living in a simulation?" nonsense, I went from "meh" to "bleh" on it. Personal opinion of course, and I definitely understand why people liked it.

Cubby, oddly, I read the novel before I saw the movie, and at the risk of being a cliché, as ForeverDreamer noted, the book was better. In the book, Lastday was at the age of 21, not 30, for starters, and the scope is far grander. Then I made the mistake of reading the sequels, but that's another story...

Prosperous Snow celebrating, I can't argue with Star Trek. It's far from perfect, and the screenplays vary greatly in quality, but overall I think it's been a positive cultural influence. I do also like Star Wars; I see no need to choose between them, either.

Lazy Writer est 4/24/2008, I always meant to read The Handmaid's Tale and never got around to it. I know it's been adapted now, but it's on Hulu, and I won't deal with that site. Eventually, I will read the book.

A lot of these comments were compelling; like I said, I'm not picking the winner based on agreement. This day's Merit Badge will go to Warped Sanity for mentioning a book I haven't even heard of and explaining exactly why it's worth reading.

I'll do this again, so there will be more chances at an MB. Next month.

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