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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QPdoll

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I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 18 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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August 18, 2023 at 10:33am

#1054270

Boogie All Night Long

What's worse than a scary story? A scary story that happens to be factual.

11 Terrifying Urban Legends That Turned Out to Be True

Urban legends—those unsubstantiated stories of terror that allow us to use our imaginations to fill in increasingly horrifying details with each retelling—have been with us forever.

Longer than the adjective "urban," even. Or the cities it references.

While the internet has made the dissemination of them easier, humans have been goading one another with spooky anecdotes for centuries.

Yeah, and some of them become actual myths, legends, or religions.

Urban legends often come with a dose of skepticism. (No, a killer with a hook hand has never terrorized necking couples.) But sometimes, these stories turn out to be true.

That's what the hook-hand no-neck conspiracy wants you to think.

As the title suggests, these are supposedly verifiable. No Jersey Devil, no Mothman, no Nessie. I, however, couldn't be arsed to research them, so take it however you want.

1. Rats in the toilet bowl

Never saw a rat there, but I did get attacked by a frightened, wet mouse once. Not where I live now.

And I have seen articles about toilet snakes. And I don't mean the plumbing kind, but the no-legs forked-tongue scaly kind.

2. Cropsey

For years, kids living in and around Staten Island raised goosebumps by relating the tale of “Cropsey,” a boogeyman who lived in the woods and made a nocturnal habit of disemboweling children.

It's important to have a bogeyman to scare your kids into behaving. Well, as much as a kid can behave, anyway. It's generally best if said bogeyman is fictional, though.

Parents no doubt eased their kids’ fears by telling them no such monster existed.

Ha! You overestimate Staten Island parents in the 1970s. "Oh, he totally exists, but he only eats kids that don't do their homework."

Incidentally: "bogeyman" is probably a more proper spelling, but boogeyman, boogieman, bogyman... whatever. The article spells it with two Os. I stick to the official spelling, to help distinguish the kid-scaring villains from the disco-dancing kind. In a nod to equality, the bogeyman can be female, nonbinary, or genderfluid, too.

But he did. In 1987, Andre Rand was put on trial and convicted for a child abduction.

He was probably mad he was named after the worst libertarian ever.

To be sure, this one might be a case of "he killed a couple of kids and the legend grew." Still, the only acceptable number of kids' deaths at the hands of a serial killer is zero—unless, of course, the kids in question don't finish their vegetables.

3. The leaping lawyer

Sooner or later, Toronto residents hear the tale of a lawyer who had a peculiar fondness for running full-bore into his office windows to demonstrate how strong they were.

I'd heard about this one. Maybe from the Darwin Awards? I don't remember.

In a eulogy, managing partner Peter Lauwers called Hoy “one of the best and brightest” at the firm.

Not exactly a persuasive ad for your firm's services. Still, it's at least mildly amusing that there's a lawyer named Lauwers.

4. The body under the bed

Vacationing couples. Newlyweds. Disneyland guests. All have been the subject of an urban legend involving hotel occupants who fall blissfully to sleep, only to wake up to an awful stench coming from either under the bed or inside the mattress.

Gleefully reported by the competing hotels.

While finding a dead person in your room would be legitimately horrifying, disgusting, and weird, things could be worse.

5. The Maine Hermit

For decades, people who vacationed in central Maine’s North Pond area were puzzled by items that would go missing.

The only really horrifying thing about this (dude was certainly strange, and he'd steal shit, but to the best of my knowledge, he never physically harmed anyone) is that Stephen King didn't think of it first.

Skipping a few for brevity's sake.

8. The legend of the bunny man

If you lived in or around Virginia in the 1970s, you were probably exposed to the story of the Bunny Man. In the tale, an escaped mental patient takes to gutting bunnies and hanging them from a bridge underpass.

I'm actually kind of salty that I never heard of this, despite spending my hearing-about-bogeymen years two counties over at about the right time.

Or maybe I did, and I just blocked it out of my head.

9. The legend of Polybius

Vintage video gamers have long traded stories about a coin-operated arcade game circa early 1980s Portland that had strange effects on its players. The game, titled Polybius, was alleged to have prompted feelings of disorientation, amnesia, game addiction, and even suicide.

Here, the article departs from "verified." This isn't even a case of "well, maybe it's true." There's no evidence that this is anything but an actual legend, myth, falsehood, whatever.

Of course, that's what they want you to think.

That one seems to be an outlier here, though. There are a few more at the link. While it's not always true that truth is stranger than fiction, it certainly can hit harder when you realize that the bogeyman is real. Now go clean your room!

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