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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October 5, 2022 at 12:01am

#1038635

Fanfiction

Entry #3 for "Journalistic Intentions"

[18+]. Blind quote, no link:

"Sure, when teenage girls write self-insert characters that hang out with their faves it's 'cringey' and 'bad fanfiction', but when Dante does it it's a 'literary classic' and 'redefines the way we view Catholicism'."

Uh huh. In 700 years, we'll see how much of today's fanfiction is taught as "the classics."

Look, teenage writing is generally cringey. Mine certainly was, and I wasn't a girl. You know what Dante wrote before his masterpiece? Love poems. LOVE POEMS. Starting when he was in his teens.

I wrote love poems back then, too. Well, not in Dante's day, but almost. I'm glad they've been lost to the mists of time; Dante had no such luck. One line I specifically remember is "Now my broken heart has mended / But the scars won't let it beat."

You know what else is almost always cringey? Self-insert characters. One of the most successful writers of our time is Stephen King, and you know what it was when he did it, well past his teenage years? Cringey. Like he was writing Stephen King fanfiction.

Oh sure, occasionally you'll get a young writer who's very talented and their early works are things of beauty. Shelley wrote Frankenstein when she was 20, while all of her dudes were stuck in the "love poem" rut (to be fair, those are considered classics too). But not everyone can both create a lasting work of literature and invent a new genre, especially not at that age. No, most of us write cringe in the beginning. And then, if we stick with it, if we really learn our craft well, then maybe—maybe—we write something that stands the test of time. Well, not "me" "we." The general authors' "we."

That said, my advice (for whatever it's worth) to young authors? Do it anyway. Some great authors have built careers starting with fanfiction. Greg Bear comes to mind; he wrote Star Trek novels. So did Diane Duane. Okay, you've probably never heard of either of them, but they both went on to write more than passable SF and/or fantasy.

Doesn't mean I'm going to read it. But just like worldbuilding for a fantasy novel, you're setting up a foundation for the possibility of later greatness.

Speaking of Star Trek, probably the second most famous work of fanfiction (after 50 Shades) in modern history is the story (a work of parody) that coined the term "Mary Sue." I've ranted about that in here in the past, probably for another JI prompt. The canonical "Mary Sue" (insofar as I can use the word "canon" for anything related to fanfiction) is an author self-insert, usually but not always penned by a teenage girl, who's unbearably competent and has no real flaws, and who's universally loved and accepted.

Not fun to read, usually. It's a clear case of wish fulfillment. But in the end, isn't all fiction wish fulfillment? The difference, at least in my mind, between good characters and bad is that the good characters have both external and internal battles to overcome. We're far more satisfied reading about someone who has trouble solving all the puzzles and gets into tough situations—and the Mary Sues or Marty Stus of the writing world are already at the top of their game. It would be like John Wick having the invulnerability, strength, and laser vision of Superman: the stakes would be nonexistent, and the story would be boring.

But the Superman character itself started out as... wish fulfillment. Writing a super-powered character takes some skill; in a compelling story, they have to face challenges that their powers aren't useful for. Like the latest TV incarnation of Superman, who has to deal with having teenage sons, something that, as far as I know, no superhero has ever been able to handle with powers alone.

So, yeah, I don't like teenage fanfiction, and I don't have to or want to read it.

But I'm not going to tell them not to write it. You gotta start somewhere. Artists—by which I mean people who make drawings or paintings—often get their start by copying others; they thereby learn the techniques and only later come into their own style. Or so I heard; I can't draw to save my life. Musicians start out by learning others' chord progressions. And so on. It should be no different with writers.

As a final note, "fanfiction" has somehow acquired the connotation of being sexual in nature, expressing the writer's desire to doink some character someone else created. While it can be erotic (again, 50 Shades), that's not the essential feature of fanfiction. I'm using the term in its broader sense here.

That said, I did once write a scene featuring a liaison between Snape and McGonagall.

"Oh! Yes! Ten points to Slytherin! Twenty points to Slytherin! YES! FIFTY POINTS TO SLYTHERIN!"

...no, I will not be sharing the rest of it. You're welcome.

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