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

October 12, 2021 at 12:03am

#1019158

The End

It's almost like my random number generator has become sentient.

The Body’s Most Embarrassing Organ Is an Evolutionary Marvel
And yet we have very little idea where anuses come from.

I'm still kind of drunk from my journey to the drafthouse cinema today (I hope I remember to do a movie review at the bottom), so it's Confession Time.

Confession #1: This article has been hanging around in my queue for a while, but it popped up this week. Why is this significant? Because, being old, I have a colonoscopy scheduled this week, so it's relevant.

Go look at the article. Look at it. The top picture is a cat butt, something I'm way more familiar with than I'd prefer. You might even call it... a cat-ass-trophy.

You know, if you've been drinking, that is.

To peer into the soul of a sea cucumber, don’t look to its face; it doesn’t have one. Gently turn that blobby body around, and gaze deep into its marvelous, multifunctional anus.

No, thanks.

Confession #2: I don't like butts. I just don't get the sexualization of them I see everywhere. "Wow, nice ass." "So?" Butts are vehicles for shit, and that's all they are. Okay, they're also moderately useful for sitting on -- what would I do if I couldn't sit on my ass all day, every day? -- but that's ancillary to their only real purpose, which is waste elimination.

I'm not trying to kink-shame anybody here. If that's what you're into, hey, you do you. But for me, I don't care -- male, female, trans, whatever, I don't want to see your ass.

I have what I consider to be a perfectly reasonable aversion to shit, no matter whose ass it comes out of. One of the main reasons I don't have a dog is I'd have to walk the little bastard and carry plastic bags with me. Cats? I can scoop that shit up without touching it. Yes, even through a plastic bag I'm still touching it. Nope.

This also applies to life forms that are as different from human as they can be; e.g., the sea cucumber.

The sea cucumber's posterior is so much more than an exit hole for digestive waste. It is also a makeshift mouth that gobbles up bits of algae; a faux lung, latticed with tubes that exchange gas with the surrounding water; and a weapon that, in the presence of danger, can launch a sticky, stringy web of internal organs to entangle predators. It can even, on occasion, be a home for shimmering pearlfish, which wriggle inside the bum when it billows open to breathe.

I could have gone my entire life without knowing this, but since I do, I have to inflict it on you readers, too.

It would not be inaccurate to describe a sea cucumber as an extraordinary anus that just so happens to have a body around it.

Oh, so, kind of like a politician or a talk radio host.

Bodily taboos have turned anuses across the tree of life into cultural underdogs, and scientific ones too: Not many researchers vocally count themselves among the world’s anus enthusiasts, which, according to the proud few, creates a bit of a blind spot—one that keeps us from understanding a fundamental aspect of our own biology.

And look, I'm not ignorant of biology. Hell, I take my username from a carrion-eater. I'm aware of the functionality and even appreciate it. I just don't need to go anywhere near it.

The appearance of the anus was momentous in animal evolution, turning a one-hole digestive sac into an open-ended tunnel.

I'm also peripherally aware of the mathematical discipline of topology. In topology, as I understand it, they only care about things like how many perforations a shape has. You have a sphere, for example, which is topologically equivalent to a cube, a dodecahedron, or even a cylinder -- because it doesn't have any holes. Then you have shapes with one hole: a toroid, shaped like a bagel (or I could say donut, but my people invented bagels so I'm going with that), for example, which is topologically equivalent to... a teacup (or I could say coffee cup, but I dislike coffee too). The point being that both a bagel and a teacup only have one perforation.

And when you think about it, humans (and most other animals) are topologically equivalent to bagels.

So I try not to think about it.

But anuses are also shrouded in scientific intrigue, and a fair bit of squabbling. Researchers still hotly debate how and when exactly the anus first arose, and the number of times the orifice was acquired or lost across different species. To tap into our origins, we’ll need to take a squarer look at our ends.

Confession #3: The only reason I put up with this article at all was in hope of butt puns, and I wasn't disappointed.

One of the oldest hypotheses holds that the anus and the mouth originated from the same solo opening, which elongated, then caved in at the center and split itself in two.

It's rather important here to understand that this is evolution-speak shorthand for changes that occurred in animal species over a long, long evolutionary time, and not something that literally happened to a single organism over its lifespan.

Cloacae are fixtures among birds, reptiles, and amphibians, and although they tend to get a bad rap, their internal architecture is actually quite sophisticated, Patricia Brennan, a cloaca expert at Mount Holyoke College, in Massachusetts, told me.

"Glad you could join me for dinner. What do you do for a living?"

"I'm a cloaca expert."

"Check, please!"

Whatever the reason behind it, the partitioning that did away with the cloaca made human anuses, as Manafzadeh said, “completely boring.” As far as exit holes go, ours are standard-issue, capable of little more than extruding waste from the gut, with no frills to speak of.

See? I told you.

The only redeeming quality of humans’ humdrum posterior hole is the feature we evolved to cushion it: our infamous buttocks, the most voluminous one documented to date, thanks to our bizarre tendency to strut around on our two primate legs.

In the Before Time, when I was still going to the germ-infested hellhole known as a "fitness center" or "gym," I'd see videos on screens. It's about the only time I ever saw ads, because I avoid them with the same single-minded focus that I avoid assholes, and for generally the same reasons. Unfortunately, one time, I happened to see an ad for... I don't know what it was for, because the volume was off (you can select your own channel for audio; I chose to ignore all of them and instead watch videos about physics, math, and sometimes writing on my mobile). This ad featured butts. Not just any butts. Female butts. And for some reason, they featured expanded, unnatural-looking, enormous butts. I don't mean fat, either; I'm not body-shaming here any more than I'm kink-shaming. But it gave me the impression that it's somehow fashionable to make your buttocks look bigger, which, all my life, I'd been told was the polar opposite of what's attractive.

"Does this dress make my ass look big?" "Why, no, honey." "Oh, good."

But now? Apparently now, big butts are back in style, and I'm in hell.

It's been a couple of weeks, but there was finally a movie I wanted to see. Because of my upcoming medical procedure, though, I was unable to munch on popcorn, as is my habit while watching a movie in a theater. They didn't tell me I couldn't have beer, though. Or a shaken-not-stirred martini. Okay, two martinis.

One-Sentence Movie Review: No Time to Die

While previous Daniel Craig Bond movies didn't do much for me because of their impenetrable plots, this one is notable not just for the acting, stunts, and more straightforward story, but because it features some of the best camera work I've ever seen -- almost enough to distract from the actual content.

Rating: 4.5/5

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