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

about 13 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 16 hours ago
In response to "Landmarks"

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

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

Bienvenue en France.

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WakeUpAndLive️~Happiness

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

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
In response to "Paris, I Guess"

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/

3 Like

WakeUpAndLive️~Happiness

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

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

2 Like

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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September 10, 2023 at 9:13am

#1055524

Revisited: "Annoying Things"

We're not reaching very far into the bowels of the past in today's Revisited. Barely even penetrating the rectum, really.

Anything within the last 12 months is ineligible for revisiting, but this one's from May of last year: "Annoying Things"

Sometimes, to mix things up a little, I like to participate in blogging activities or contests. That entry was prompted by "Invalid Item"

. Another perennial favorite of mine is "Journalistic Intentions"

[18+], which I mention because there's a new round of that next month.

Plugs aside, the entry was, as per the title, about things that annoy me. Specifically, they're annoyances about living in Charlottesville, apart from the blindingly obvious annoyance of my town being internationally famous for a deadly protest/riot that mostly involved out-of-towners.

As I've noted before, when it's not freezing (defined by me as 55F or below) outside, I like to sit on my deck. I paid a lot of money for that thing and by Marduk's manhood, I'm going to use it.

I'd forgotten that particular oath. Along with "Thor's balls" and "Ishtar's tits," I'm going to need to remember it and use it more.

This is normally quite pleasant. Songbirds are tweeting (and not complaining about Elon Musk)...

You know what else is annoying? Every gods-damned news outlet still insists on reporting Xits, and always with a line like "...posted on X (the platform formerly known as Twitter." We get it. It's X now. Anyone who doesn't know this didn't know what Twatter was to begin with. So just call it X. Or Xitter (where the X is pronounced like sh). And call the blathering a Xit, like I just did. Also pronounced like sh, or maybe like z; either way works for me.

...bees are buzzing, owls are hooting (yes, yes, I did my Duolingo lessons; shut up)...

Over four years of daily Duolingo now. They keep changing the lesson plan. It's annoying. Mais je crois que j'apprends quelque chose.

Yes, very pleasant, except during the day and at night.

Sometimes, I forget what I wrote, and when I find it again, I crack me up. This was one of those times. That's another reason I like to do these retrospectives.

During the day, everyone gets their yard work done. This wouldn't be so bad if everyone did it at the same time. But no; that would be too kind. The neighbor mows his lawn. Then, when that's finally done, the other neighbors mow their lawn. After that one's finished, one of the people behind me mows their lawn. And then another. And then another.

For whatever reason, this hasn't been nearly as bad, this summer. It's been pointed out to me that this particular noise problem may have been the result of everyone using one lawn care company. But it's not. There are like four dozen different lawn care companies around here, and besides, most of the mowing in my neighborhood is done by the resident. Not mine, though. Too lazy.

And also the nightly muscle car race. For some reason, that's still a thing.

That hasn't stopped or slowed down. But the other night, I heard one of those 8-cylinder engines rev up and gear on down the residential streets. Then, I heard CRUNCH... followed by blessed silence.

For about 2 minutes, at which point the sirens started.

I'd like to say I hoped no one was injured, but I'd be lying. Let's leave it at "I hope no one died."

Oh, and as that entry was posted in late spring, I'd forgotten one of the massive annoyances of fall: every other damn Saturday, on average, UVA has a sportsball game. I live a bit over a mile from the stadium, just close enough to hear the announcer, the bands, and the roar of the crowd, but just far enough away so I can rarely understand what they're actually saying. They also use the stadium for other events, so this isn't limited to football.

I can usually ignore that, though. But last year, some rocket surgeon decided it would be a good idea to have some Air Force jets fly low overhead at great speed as part of the halftime show. That sucked, especially since they flew directly over my neighborhood, and you don't hear them until they're right on top of you (speed of sound and all that).

Anyway, it helps me sometimes to count my annoyances. Once I'm done with that, I can go back to being content with life.

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