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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June 5, 2023 at 9:21am

#1050592

Driving a Semicolon

I've written on the virtues of the semicolon before. Notably in this entry, in which Kurt Vonnegut railed against them and then used one: "The Sage of Indianapolis"

This one, though, is far more positive than good ol' Kurt on the use of this important punctuation mark, one which I suspect I overuse, but can't help myself.

The Virtues of the Semicolon; or, Rebellious Punctuation
It Cares Not for Your Rules

In 1906, Dutch writer Maarten Maartens—acclaimed in his lifetime but now mostly forgotten—published a surreal, satirical novel called The Healers.

He should not have been forgotten. He had the most Dutch name since... well, no, he had the most Dutch name that ever Dutched. The Dutch equivalent of Vladimir Vladimirovich, or Jon Jonson.

The book centers on one Professor Lisse, who has conjured up a potential bioweapon: the Semicolon Bacillus, an “especial variety of the Comma.” The doctor has killed hundreds of rabbits demonstrating the Semicolon’s toxicity, but, at the beginning of the novel, he hasn’t yet succeeded in getting his punctuation past the human immune system, which destroys Semicolons instantly as soon as they enter the mouth.

Maartens wrote before antibiotics.

Maartens wrote at a time when the semicolon was still an exceptionally popular punctuation mark—so popular that grammarians forecast the extinction of the colon, which 19th-century writers had abandoned in favor of semicolons.

Apparently, the thought that both had their use never crossed their minds.

Nowadays, however, it’s the semicolon that is no longer a la mode; and judging by the number of writers who have something like an allergic reaction to it, plenty of people might find a glimmer of truth in Maartens’s vision of the semicolon as disease vector. “Let me be plain,” wrote the novelist Donald Barthelme: “the semi-colon is ugly, ugly as a tick on a dog’s belly, I pinch them out of my prose.”

Well, that's one author whose novels I won't read.

Semicolons are “idiocy,” Cormac McCarthy scoffed in a Vanity Fair interview.

And that's up there among the most Irish names to ever Irish (but McCarthy is American).

The semicolon’s fall from favor was determined in part by the contradictions, complications, overcompleteness, and incompleteness of rule books. But if we look past those books to the work of some of our best writers, we can see semicolons used to create music and meaning in language that no other punctuation mark could accomplish.

Damn straight.

Raymond Chandler is famous for his noir detective novels featuring Philip Marlowe, but he was a brilliant essayist as well, and the semicolon seems to have been one of his favorite punctuation marks. He used it to create expressive rhythm in his writing, in ways that leapfrog rules.

I should note that using a semicolon in fiction is a completely different thing from using it in essays (or blog posts). I'm not even sure how to articulate the difference; there's a formality to semicolon usage that only flies in a certain style of storytelling.

These moments of rule-chucking in his writing are not accidents: “When I split an infinitive,” he railed to the editor of his film essay, “God damn it, I split it so it will stay split.”

While I support semicolon usage, the "rule" about not splitting infinitives is one of the stupidest restrictions to ever be imposed upon the English language. (See what I did there? I crack myself up.)

What Chandler and other excellent writers knew is that a book of grammar rules is incapable of answering the question of how, and how often, to use a semicolon, because the answer is a matter of what you, the writer, are trying to accomplish.

In other words, it's more a question of style than of prescriptive rules.

No matter what you’ve read in a book or been told by an English teacher, for instance, a sentence can’t be too short or too long in an absolute sense. A sentence can be too short or too long to suit its purpose.

Yeah, no, that one in Ulysses is entirely too long.

Hell, Ulysses is entirely too long.

Perhaps the best use of the semicolon I’ve ever read is in a sentence that’s 318 words long—just under a third of the length of this entire essay. That sentence is in Martin Luther King’s “Letter from Birmingham Jail,” and if you look up the letter, you’ll know exactly which sentence I mean when you get to it. King is describing why he is unwilling to continue to wait for change without taking action, and to make his point strongly felt, he makes you wait in pauseless discomfort while he recounts injustice after injustice, all semicoloned together one after another, never letting you rest on a full stop. Read it out loud and you’ll be exhausted and breathless: justly so.

King's genius wasn't just in his ideas; it was also in his rhetorical prowess.

Chandler and King were so deft with their punctuation because they invested time and attention in reading other good writers, and they poured just as much time and attention into crafting their own sentences.

Meanwhile, I write these posts in less than an hour and then promptly forget about them.

Alas, to punctuate well requires deliberation.

“I was working on the proof of one of my poems all the morning, and took out a comma. In the afternoon I put it back again.”
—Oscar Wilde

But perhaps that’s a good thing. According to the CDC, mindfulness-based practices like yoga and meditation rose significantly between 2012 and 2017. Venture capital firms have poured millions into apps like Calm and Headspace. If we all scanned our emails, texts, Tweets, reviews, and essays, looking for opportunities to use something a little more interesting than the now ubiquitous, catchall em-dash, we might find that thoughtful punctuating is an occasion to stop, reflect, and immerse ourselves in the contemplative art of the well-judged pause, with no monthly subscription required.

And, unlike yoga, meditation, and mindfulness apps, proper punctuation isn't a derivative of copium.

In Maartens’s The Healers, Professor Lisse eventually discovers that the semicolon isn’t a toxin as he initially believed; instead, when formulated properly, it’s a source of vitality.

And apparently for this excerpt's author, a source of income.

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