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 15 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 18 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/

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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 12, 2023 at 9:55am

#1054000

Yosemitic

Back in May, I linked an article showing that Yosemite Sam (pronounced Yo-sem-it-ee) is probably Jewish: "Sam: I Am"

Now, Cracked takes a different crack at the character.

Looney Tunes: Yosemite Sam’s Best Nonsense Swear Words, Ranked

"Nonsense?! Why, you dadburn rassa-frassin varmint..."

If Yosemite Sam is remembered for one thing, it would be his massive, marvelous mustache. But if the Bugs Bunny villain is to be honored for a second accolade, it would be his colorful, almost-but-not-quite-off-color vocabulary, which has the whiff of obscenity but is cloaked in euphemisms and old-timey lingo.

Which is somehow even funnier than actual swear words.

Now, these aren't necessarily original to Warner Bros. cartoons. Lots of minced oaths have been around for a long time, possibly in some form since at least ancient Sumeria. They say a lot about cultural taboos. For example, in keeping with my 18+ rating, I can only say "fuck" once in this entry. But I can say "frak" as many times as I want, and everyone knows what the frak I really mean.

You might note there's an issue with the numbering in the article. Doesn't matter; I'm only going to highlight a few.

10. Landlubber

Landlubber has been around since the 1700s, and I love it even though I’m a landlubber myself: i.e., a non-sailor, as I might be disgustingly labeled by a seafarer. Though no sailor, when Sam occasionally found himself in sea mode, he’d modify his folksy lingo accordingly, adopting this term.

Mel Blanc's voice was, as we know, extraordinarily versatile. But Sam's voice and a classic pirate accent always seemed similar to me. As the article points out, this isn't exactly a minced oath, or even fightin' words.

9. Dad-Burn

As in, “Dad-burn it!” Sam’s lingo is one of the few remnants of this sense of dad as a euphemism for God.

This one, however, is a classic minced oath. You can't say goddamn in a kids' cartoon.

6. Varmint

I reckon this is the most characteristic Yosemite Sam word: He is the patron saint of calling critters varmints. The term has referred to vermin since the 1500s. By the late 1600s, a broader meaning emerged relevant to Sam and Bugs: “An animal of a noxious or objectionable kind,” as the Oxford English Dictionary puts it.

It should be obvious, but it may not be, that "varmint" is a linguistic morph of "vermin," while "critter" comes from "creature." All varmints are critters, but not all critters are varmints.

The distinction might be better explained here:

"So basically, a varmint is a sub-set of a critter that's not tasty enough to eat or hunt into oblivion. Sort of like the difference between a plant and a weed"

4. Lily-Livered

There’s an implication of femininity here, which is a heckuva accusation for a man to throw at a rabbit.

Have you met Bugs? That bunny was genderfluid before the term "genderfluid" ever existed. Anyway, as the article points out, "lily-livered" has nothing to do with perceived gender role.

2. Eejit

This variant spelling of idiot signals that the speaker may not have any right to ride a high horse of non-idiocy. A predecessor is found in 1853: eediot.

In addition to being unable to imagine Sam sounding out the word "idiot," it may be that "idiot" was at a point in its linguistic evolution that "retarded" is right now. Both words were originally value-neutral ways of saying that someone has diminished mental capacity. Humans being humans, such words inevitably become slurs, and we have to find new value-neutral terms, which are destined to inevitably become slurs. Don't believe me? Try calling someone "special" these days.

1. Rackin’-Frackin’, Rassa’-Frassin’

Ah, yes, the absolute Platonic ideal of a minced oath.

Words like this are the lexical equivalent of grawlix, a comic strip gimmick consisting of symbols that indicate swearing, like %*!#$.

And despite over half a century of enjoying comic strips, this is the first time I've encountered the word "grawlix." I love learning new stuff. Incidentally, I got in huge trouble for using grawlix (though obviously I didn't know that term then) back in middle school.

I was trying to avoid getting into trouble, you see. Turning over a new leaf, as it were. I don't remember the context after all this time, but I'd written a note containing the "words" @$$ and $#!+. I thought I was being clever, you know, replacing "ass" and "shit" with what would now be perfectly legitimate substrings of internet passwords. The teacher, though, wasn't stupid, and translated them immediately, sending me across the hall to the principal's office, all while probably thinking what a doggone varmint I was.

Well, #@%* me.

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