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').

1 Like

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

1 Like

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

September 28, 2022 at 12:02am

#1038262

Purrometer

And now for a truly impurrtant article.

Purring Is a Love Language No Human Can Speak
Some cats do it, but others can’t—and researchers still don’t fully understand why.

My love language is avoiding people who use the phrase "love language." Stop it. Still, as it's an article about cats, I read it anyway.

On the not-so-infrequent nights when I’m plagued by insomnia, no combination of melatonin, weighted blankets, and white noise will do. Just one cure for my affliction exists: my cat Calvin, lying atop my shoulder, lulling me to sleep with his purrs.

I get it. When my cat is tired, I'm tired. Since she's a cat, that's pretty much most of the day.

But purrs—one of the most recognizable sounds in the animal kingdom—are also one of the most mysterious. “No one, still, knows how purring is actually done,” says Robert Eklund, a phonetician and linguist at Linköping University, in Sweden.

And I hope it never is. Life is better when some things are mysteries, especially regarding cats.

Cats purr when they’re happy—but also sometimes when they’re anxious or afraid, when they’re in labor, even when they’re about to die. Cats are perhaps the most inscrutable creatures humans welcome into our homes, and purring might be the most inscrutable sound they make.

As I noted in a recent Comedy newsletter (yes, really), I've witnessed the distress purr. It can be disconcerting if you aren't expecting it.

There is, at least, some consensus on what purring is. In the strictest sense, the sound is a rhythmic, rumbly percolation that’s produced during both exhales—as is the case with most typical animal vocalizations—and inhales, with no interruptions between.

And that's really weird, when you think about it.

Purring isn’t easy to study: Felines aren’t usually keen on producing the sound around researchers in labs.

That's because the phrase "curiosity killed the cat" isn't referring to the feline's curiosity.

Carney told me that in some animals, purring could be a sort of vocal tic, like nervous laughter; cats might also be trying to send out pleas for help or warning messages to anyone who might dare approach. Or maybe bad-times purrs are self-soothing, says Jill Caviness, a veterinarian and cat expert at the University of Wisconsin at Madison, and parent to a feline named Electron. They could even be a cat’s attempt to dupe its pain-racked body into a less stressed state.

I happen to favor this latter hypothesis myself, in a very unscientific way. The frequency of sound that a cat emits with a purr might have some healing properties.

I had a cat for a long time who wasn't the most affectionate creature. She was what some people think of as the archetypal cat: nervous, aloof, sometimes even mean. Not a lap cat at all, and I still bear some scars to prove it. But when I was lying in bed, immobile due to some kind of pulled muscle or whatever, this cat curled up right on top of where the pain was and started purring. Like she knew, and was trying to help.

In the early aughts...

Noughties, dammit.

...a researcher proposed that purring might even have palliative properties for cats—pinging out vibratory frequencies that could, for instance, speed the healing of wounds or broken bones. The thought isn’t totally bonkers, Eklund told me.

Like I said.

Carney has had plenty of clients who “swear that the cats lying in bed, purring beside them while they were ill, kept them from passing away,” she told me. But alas: Although cats can purr at frequencies that overlap with those used in vibration therapy, none of the research on these treatments has actually involved felines. “I don’t think we have any studies that are like, I sat with a purring cat on my broken leg for 15 minutes a day; I healed more rapidly than someone else,” Caviness told me; the same goes for the effects of purrs on the purrer.

So my story remains anecdote, but it seems like it'd be worthy of further study.

If people feel better around their cats, that might be less about purring’s direct mechanical effects on human tissues, and more about the entire companion animal being a psychological balm.

Which I'd buy if my cats weren't so damn annoying sometimes.

And unlike many other cat noises, purrs stubbornly elude human imitation (though some people on YouTube might beg to differ). Humans can easily meow back at their cats; “it’s like a very rudimentary pidgin language,” Eklund said.

I used to meow back at my cats. Their reactions indicated that I should stop, like, right now. So I did.

Purring is a language barrier we have yet to surmount. Which, in some ways, is so, so cat. Humans have spent generations breeding dogs to emote in very people-esque ways, using their soulful eyes and slobbery, smiley mouths. Cats, though, continue to thrive on subtlety; their mugs aren’t evolutionarily set up for obvious expressions, defaulting instead to “resting cat face.”

Don't draw the conclusion from any of this that I don't like dogs. That's what most people leap to, like they think I hate children because I never wanted to own one. But that doesn't follow. I just never wanted the extra work, in either case.

Plus, I want to bust the stereotype of the crazy cat lady by being a crazy cat dude. But I can't bring myself to have more than three at the same time, so I'll probably never qualify. Right now, I have two, and so does my housemate. The threshold for crazy cat person status is N+3, where N is the number of humans in the house. Neither of us is there. But she also has guinea pigs, so if one of us is crazy, it ain't me.

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