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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Adherennium Plotting something

about 2 hours ago
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For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

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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.

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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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Nov 6, 2024 at 10:22am
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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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May 7, 2024 at 9:47am

#1070653

Fort Knocks

Just an interesting Atlas Obscura article today...

Star Forts Are Military History, and the Base of Some Strange Conspiracy Theories
Examining why these ubiquitous, obsolete fortifications carry an air of mystery.

Anything with even the slightest hint of mystery becomes, in the minds of humans, a conspiracy theory. Or at least something supernatural. It's like:

"I wonder why the sun moves across the sky."

"Oh, the gods must be pulling it on a chariot."

"Oh! Yeah, that makes sense!"

Or:

"Who left these big footprints in the mud by the edge of the forest?"

"Not sure. Must be aliens."

"What? Come on. It's gotta be something perfectly natural, like Bigfoot."

Star forts, like all forts, having been (most likely) built by actual humans from actual Earth, have nevertheless been subject to these same impeccable lines of reasoning.

As usual, I recommend going to the link for the pretty pictures. I've only got a few comments here.

I first began thinking seriously about bastion forts after reading W.G. Sebald’s final novel, Austerlitz. The book centers around the eponymous architecture critic, Jacques Austerlitz, who explains to the protagonist early on that “it is often our mightiest projects that most obviously betray the degree of our insecurity.”

For once, I don't think this is a book ad in disguise. Though the author's own books are linked at the bottom of the article.

“No one today,” he continues, “has the faintest idea of the boundless amount of theoretical writings on the building of fortifications, of the fantastical nature of the geometric, trigonometric, and logistical calculations they record, of the inflated excesses of the professional vocabulary of fortification and siegecraft, no one now understands its simplest terms, escarpe and courtine, faussebraie, réduit, and glacis, yet even from our present standpoint we can see that towards the end of the seventeenth century the star-shaped dodecagon behind trenches had finally crystallized, out of the various available systems, as the preferred ground plan.”

For some reason I can't quite put my finger on, no "mystery" question is ever answered by conspiracy theorists with "Because humans can be really clever."

They were first developed after the fall of Constantinople in 1453, and became widespread throughout the 16th century, until their gradual obsolescence became apparent in the 1800s.

Technology advances, in part, because of an arms race. You design a better weapon, I design a better defense. I design a better defense, you design a better weapon. And so on, until you design a weapon good enough to wipe out the planet, and do so, ending the arms race... probably.

Meanwhile, a lot of those older weapons and defenses look cool, but aren't useful for their original purpose.

I returned to Sebald’s novel again as I learned that there is an entire conspiracy theory built around these fortifications. According to the stranger corners of the Internet, these forts are not elaborate remnants of long-obsolete defense strategies, but rather structures built by some lost, hyper-advanced civilization.

"Because I'm not clever enough to have designed them, I guarantee you no human was clever enough to design them."

Littered all through Youtube, TikTok, and Reddit are videos and threads with names like “Star Forts: Tesla Technology, Vibration Healing, Ancient Electricity, Sound Frequency Technology,” in which bastion forts are recast as key players in the New Age gobbledygook that makes up so much of social media these days. (Among the regular posters in the starforts.org thread is a man named John A. Warner IV, the son of former U.S. Senator and U.S. Secretary of the Navy John Warner III, who’s self-published a novel that involves star forts, among its many other plots, and argued, for example, that they were not built but rather “grown” through some combination of frequency resonance and “water cymatics,” whatever that is.)

John Warner was a senator from my state. He's dead now. I didn't usually agree with him, but I respected him (which I hardly ever say about a politician of any sort). He apparently neglected his kid, though.

The star pattern, these conspiracists allege, mimics certain frequency waves as they appear on oscilloscopes, suggesting that these structures’ design reflects not military technology but an organic shape that somehow harmonizes with the natural world.

I'm sorry... I can't... excuse me a moment, would you?

BWAAAAAAAAHAHAHAHAHAHAHAHAHAHAcoughcoughHAHAHAHAHAcough

Ahem. Okay, I'm back.

I won't quote further. But I will say that the article left me with a sense of the inevitability of rampant speculation and conspiracy "theories" in the wake of our construction of a post-truth society. No evidence in the world—and there is plenty of said evidence—will turn these speculations around, because, in the end, truth is what they believe it is.

And that's an arms race we've already lost.

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