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
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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
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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
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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
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"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
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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
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If the aforementioned exercises don't help; there's still liquor.

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Aiva Raine

Nov 5, 2024 at 3:03pm
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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
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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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August 21, 2023 at 10:19am

#1054398

Meh. Whatever.

Oh hey, an actual writing-related article.

“Whatever!”: In Defense of Anachronism in Ancient Rome
James Hynes on Navigating the Past and the Present in Historical Fiction

It's from LitHub, so it's a book promotion, but it brings up issues I've discussed in the past.

Five times in my historical novel Sparrow, the character Calidus, a young provincial Roman who is the oldest son of a brothel owner uses the late twentieth century idiom, “Whatever.”

Let's be real, here: That dismissive one-word sentence is associated with Generation X, not just a particular time period. But. Y'know. Whatever.

On each occasion one of his free employees is telling him something he doesn’t particularly want to deal with.

I'm unclear as to whether "free employee" means "an employee who is a free person," or "an employee who is not paid," aka a slave.

Not going to read the book to find out. Article says he's a slave-owner, but that meant something different in ancient Rome than it did in 19th century America.

Writing dialogue for characters in historical fiction is always tricky, and the further back in time you go, the trickier it gets.

I can believe it. You know what's even trickier? Going in the other direction and writing future science fiction.

If you’re writing a story set in an American or British city during the mid-twentieth century, you have reams of literature and hours of film from which you can steal idioms, catchphrases, and slang, and even if you go further back—to, say, the reign of Henry VIII—you can still do a fair approximation of how people might actually have talked. But if you’re writing a fictional narrative set in classical or late antiquity, and you’re writing it in English—the market for novels in Latin or Koine Greek having dried up somewhat in recent centuries—you have to resolve the problem of how to, on the one hand, make people sound period appropriate, and, on the other, be psychologically and emotionally understandable to a twenty-first century reader.

And that's as much as I'm going to quote from the piece; there's a lot more there, if you're interested. Here's what I think the author is either missing, or dancing around, though:

It's all translation.

Let me give you an example from the relatively recent past:

The Gettysburg Address famously starts out with "Four score and seven years ago..." This doesn't need translation into English, because it's English, but it might require one to explain to younger audiences that a "score" is an old word for "twenty." I don't know if the construction was considered archaic even in the 1860s, but as written, it's weighty rhetoric that comes across much more profoundly than if he'd said "Eighty-seven years ago..."

But how would one translate it into a different language? Or into a potential version of English 250 years from now? Or a thousand? Kind of like how modern English writers have to translate Beowulf, which is technically written in English, but the language has drastically changed since then.

I suspect it's not difficult in French; their version of "eighty-seven" is already "four twenties seven." But other languages, and future English, might lack context. So a skilled translator would come up with a phrase that captures Lincoln's rhetorical gravity (I'm told he actually wrote his own speeches), in the context of the language to be translated into.

Got me? Basically, an original text or speech has plain meaning, but also it induces certain feelings in the reader/listener. One can translate the plain text easily enough with knowledge or Google, but that won't necessarily convey the same feeling in the audience. The trick in effective translation is to not only get the words and sentence structure right, but also the weight of the words.

That is to say, I could totally see a version of the Old Testament where Moses occasionally says, "Okay, God. Whatever."

I got to thinking about this, not in relation to historical fiction (a genre I generally don't care for, though there are exceptions), but concerning science fiction and fantasy (genres I live and breathe). And I remember touching on it in a long-ago issue of the Fantasy newsletter here.

So, let's take Star Trek as an example, because it's a franchise I'm rather familiar with. Depending on the series, it can be set around 200 to a thousand years from now. We know English has changed quite a bit in 200 years; just read Poe or Shelley (either of them), but it's still intelligible. Shakespeare, whose writing often requires study and interpretation now, was a bit over 400 years ago, as far back in time from the series run of The Next Generation as the setting of TNG is forward. So it's very reasonable to assume that, in the latter part of the 24th century, English will have changed; accents will have changed proportionally; there would be evolution and loan words and all the stuff that make languages different over time.

But TNG actors speak in late-20th century fashion (and their hairstyles, when they exist, are quintessentially 1980s). This isn't just laziness on the writers' part, not wanting to invent 24th century idioms, but also a way to connect with the audience.

It is, in essence, a translation.

It's not just the language, though. Much has been made in fan circles of the changes in alien design over time in that franchise. Especially the Klingons. Early attempts were basically humans with bushy eyebrows. TNG introduced the forehead plates. Discovery, though it was set before the original series, altered them even further, making them more obviously alien. And now in Strange New Words, set between Disco and TOS, the Klingons have gone back to more of a TNG-style design.

Sure, this has been lampshaded in-universe as some sort of genetic virus or deliberate modifications or whatever. But I assert that it didn't need to be. Klingons are whatever the current audience needs them to be, because their appearance is translated.

There are also budget constraints, of course, but that's not what I'm talking about, here.

All of which is to say: Sure, have an ancient Roman say "whatever" or, if you want to play with Latin, "quisquis." It makes just as much sense as a future spaceship captain telling his crew to "Hit it."

Now I just need to remember to adapt this entry into a future Fantasy newsletter, because I'm lazy enough to save myself the work.

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