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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Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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

October 4, 2018 at 1:10am

#942615

Nightmares Are Real

https://www.theatlantic.com/health/archive/2018/10/hallucinations-hearing-voices...

There’s a good chance you’ve hallucinated before.

If you’ve ever felt the buzz of your phone against your thigh only to realize the sensation was entirely in your head, you’ve had a sensory perception of something that isn’t real. And that, according to the psychologist Philip Corlett, is what makes a hallucination.

The more you try to define "reality" and "illusion" as they relate to each other, the more slippery it gets. I find it easier to stop trying to do so; I'm an engineer, not a philosopher, and therefore my worldview is largely practical. Can I sense it? Yes? Then it's real.

But hang on - I can sense these hallucinations as well. I swiftly know them for what they are, but at the time they seem real.

And what about dreams? I'm a bit disappointed that the article doesn't mention that phenomenon, which is known to happen to the vast majority of people (and other animals as well). I got to wondering if maybe the biochemical mechanism that prompts hallucinations is somehow related to the one producing dreams.

Most of the time when I get sensory hallucinations as described in the linked article, it happens when I'm about to fall asleep. Maybe I'll hear my name called, as plain as if someone were speaking directly to me. Sometimes it's related to other senses, such as the shaking of a minor earthquake, or a sudden, unbidden visualization. And dreams themselves are sensory - I don't know about other people, but sometimes I'll get smell or taste or touch in a dream. During a dream, things seem real enough, until the logical mind takes over and you realize that you wouldn't really have gone to class without pants. Or whatever. (I've rarely had that one, but I keep getting dreams about being back in college and there's this one class that I've spent most of the semester having forgotten that I signed up for it.)

Dreams are real, in a sense. That is, they can affect a person emotionally. Sometimes they can even present a solution to a problem you've been thinking of, and so they spill over into everyday reality.

Confession time: I've experienced sleep paralysis way more than I'd like. Of course, once is way more than I would have liked, but it's been significantly more than once. The way it presents with me is a very vivid sequence that, at the time, I think is actually happening. Usually there's someone else in the room, someone who's not supposed to be there (these days, no one is supposed to be there, but this started when I was married). The presence is never benign. At some point I think I might be dreaming, which for a regular dream is about when I wake up, go "okay, that was weird," and get on with my day - but in an episode of sleep paralysis, I desperately try to wake up and can't for a subjectively long time; meanwhile, the presence becomes more and more inimical. This is the classic "night mare" of historic description.

But even that isn't the worst part about it. The worst part is that when I do wake up, the feelings are still with me - and I. Can't. Move. I can breathe, and my heart is still beating (fast), but no voluntary muscles will so much as twitch, not even my eyelids. I know, logically, what is happening: when we go into dream/REM sleep, the brain shuts off motor functions so that, in the dream, one can walk or fly or swim, or pick something up, or bite into a cheeseburger, or whatever, without these things translating to physical body actions. Other people have the opposite problem, in that the brain doesn't fully shut off those pathways, and so they get their legs all tangled up in the sheets or whatever. I've had that too, but it's not even close to as scary as a sleep paralysis episode.

Anyway, like I said, I know this logically, but the back of my mind - especially active at the edge of sleep - still thinks there's someone or something else in the room and I'm utterly unable to kick in the basic animal flight-or-fight reflexes. I'm just as stuck as if someone's wrapped me in duct tape (don't even think about it).

I'm pretty sure that many of the stories of alien abduction (those that aren't hoaxes, anyway) are the result of sleep paralysis, or something very similar. Their descriptions match. Now, I've never had Grays probing my ass, but the stories are otherwise consistent - being unable to do anything, inimical presence, sometimes physical displacement, extreme limbic reactions. In the old days before we really conceived of aliens as such, people would describe demons, or a demonic presence. People in other cultures probably experienced whatever monsters their particular cultural zeitgeist (pun intended) presented them with.

Eventually, it wears off, and once I concentrate on moving just one muscle, and do so successfully, it breaks me out of the paralysis.

Nightmares suck at the time, but despite the scary description, once it's over, I find the experience intellectually fascinating. In other words, it makes me think and be curious about the process.

So in that sense, they're real.

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