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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours ago
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

about 4 hours ago
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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

April 24, 2021 at 12:13am

#1009033

Open Up

So let's talk about actual writing again.

A Close Reading of the Best Opening Paragraph of All Time
From Shirley Jackson's We Have Always Lived in the Castle, of course

I don't effing think so.

One hundred and one years ago today, Shirley Jackson was born.

Article is dated December 15, 2017, so that's the "today" she's talking about. That's fine; that's how time works. Just making it clear.

I'll skip to the actual opening paragraph in question.

My name is Mary Katherine Blackwood. I am eighteen years old, and I live with my sister Constance. I have often thought that with any luck at all I could have been born a werewolf, because the two middle fingers on both my hands are the same length, but I have had to be content with what I had. I dislike washing myself, and dogs, and noise. I like my sister Constance, and Richard Plantagenet, and Amanita phalloides, the death-cup mushroom. Everyone else in my family is dead.

I'll be honest: None of this interests me in the slightest. By which I mean, okay, fine, whatever; don't care. I'm neither tempted to read on nor ready to throw the book against the wall. As opposed to this article, which does evoke emotion in me, all of it negative.

I'm willing to admit that this might be because I lack historical context; the werewolf thing has been done to death now, but was probably somewhat fresh when it was written.

Back to the article:

It almost seems like overkill to explain why this paragraph is so wonderful.

No. No, it doesn't. Explain to me, a lifetime reader of science fiction, fantasy, horror, and supernatural genres, exactly what makes you think this is anything but boring.

You must have a certain sensibility to truly appreciate its charms.

Oh, hell no; you're not going to get away with tricking me into liking it by making me feel inadequate. I could just as easily say "You must have a certain idiocy to think this paragraph is anything but mediocre."

The rhythm is key.

Bovine excrement.

It begins straightforwardly, with our narrator’s name—a somewhat old-fashioned way of opening a book, appropriate for our somewhat old-fashioned, or at least sheltered, perhaps even stunted, narrator.

Sure, it's old-fashioned, but that's not a problem; it's arguably better than waiting until some arbitrary later point to give the reader the protagonist's name. The problem is that nothing else in that paragraph says "sheltered" or "stunted." On the contrary, you're not sheltered if most of your family is dead and you're not stunted if you're familiar with the botanical binomial of a fungus, nor if you feel like you have to explain said fungus by also using its colloquial name.

And what a name it is—a somewhat old-fashioned name, Mary Katherine Blackwood, evocative of witch trials and cultists, dense trees in far-away continents and Nancy Drew mysteries

All I could tell was that the name was so white I could barely make it out against the website's background.

She tells us she is eighteen, but by the very next sentence, she already sounds younger, and she sounds younger still by the third (“I dislike washing myself” is a prim schoolgirl’s complaint—well, prim in tone if not in meaning). This too presages what will we come to discover about Merricat (for that is what she is most often called), who lives by a logic quite disconnected from that of the the adult world, i.e. the world of men, the world exterior to her cherished sororal bond, and who will aggressively reject all encroachments by same.

Oh, now what you're saying is that at least part of the supposed charm of the opening paragraph can only be discerned as it relates to the rest of the story. The story that I'm not going to read if the first paragraph doesn't hook me. Fail. I'm skipping over a bit here, but...

The perfect surreality of her matter-of-fact association—the length of her fingers to her potential as a werewolf—signals that Merricat is a magical thinker, and a confident one.

Again, I read fantasy and horror. This sort of thing neither disturbs me nor intrigues me. A person "confident" in magical thinking is, as far as I'm concerned, an unsympathetic character.

Some readers might be tempted to wonder if I'd say the same thing if the protagonist's name were, say, Mark Aloysius Blackwood. The answer is yes. I don't give two shits about the gender of a protagonist.

Submit to her logic or give up now.

I choose the latter.

As for her likes: first we get a second pointed mention of Constance in twice as many lines, which should alert us to her importance.

I can actually agree with this bit, though again, it's only on reading the actual story that the theory "Constance is important" will play out or not.

So at any rate, now we know we’re dealing with a teenage girl bizarre and erudite enough to name a member of an ancient English dynasty, killed before he could achieve the goal he’d fought his whole life for, as one of her three favorite things.

Thus once again contradicting the "sheltered" and "stunted" descriptions.

The third favorite thing, of course, is a poisonous mushroom. This should set off certain alarm bells, especially when it is immediately followed by the revelation that everyone in her family is dead. I also count this revelation as a third mention of Constance.

That's because you read the story, which, I reiterate, won't happen if the opener fails to hook me. A good opener has to stand by itself, and not be good only in hindsight.

This paragraph is brilliant because of Merricat’s voice, and so is the rest of the book. It immediately teaches us who she is, and what this book is going to be like.

Oh? Well, then HARD pass on the rest of it.

And now that I've forever destroyed any chance of getting published on LitHub, I'll stop ragging on the article and on a story I've never read. Yes, of course I've read other stuff by Shirley Jackson; it was required reading in high school. And I liked it. I'm not criticizing Shirley Jackson here.

So what does Waltz think is the greatest opening paragraph of all time? Well, obviously, to answer that, I'd have to have read every story ever written, and that's an even less attainable goal than my desire to visit All The Breweries.

But the one that has always stuck out in my mind is the opening paragraph of To Reign in Hell, a somewhat obscure novel by one of my favorite authors, Steven Brust:

Snow, tenderly caught by eddying breezes, swirled and spun in to and out of bright, lustrous shapes that gleamed against the emerald-blazoned black drape of sky and sparkled there for a moment, hanging, before settling gently to the soft, green-tufted plain with all the sickly sweetness of an over-written sentence.

What can I say? I prefer my fiction to be self-aware.

How about the rest of you? Any favorite opening sentences/paragraphs? Why not make it a

Merit Badge Mini-Contest!

Give me your favorite opening line(s) (including title and author). Copy them in if you can; or link to them. My favorite will win a Merit Badge. Yes, you can try entering your own effort, or that of other WDC authors. My requirement is that I be able to read it, else I won't be able to tell whether I like it or not. The one I like best will get a Merit Badge sometime on Sunday. As usual, you have until midnight tonight, the end of the day on Saturday, in accordance with WDC time.

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