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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 14 hours ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

about 17 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

about 20 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

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.

6 Like • Comment

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