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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Brandiwyn🎶

a day ago
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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

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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