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.

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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.

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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

June 23, 2023 at 11:03am

#1051484

The Cat's Meow

Yes, this article is basically an ad for books. But it was like catnip for me.

The big idea: Are cats really domesticated?

Guardian link, so UK spellings ahead.

Few people would mistake a wolf for a dog. But if you saw the ancestor of the domestic cat in your backyard, your first thought would likely be “What a cool-looking housecat!” rather than “What’s an African wildcat doing in Manchester?”

I'd be like "Hey, when did I get a house in Manchester? Cool! Where's the nearest pub?"

What about behaviour, then? Which of the traits we commonly associate with our furry friends are the result of domestication, and which do they share with their wild relatives?

This is why the article is interesting to me, though much about cat behavio(u)r remains a mystery—as it should.

Let’s start with the classic cat sound.

Yes, let's. What I find fascinating is not so much the meow itself, but the different interpretations of the sound across cultures. Oddly, this article spells it meow, when proper British cats go miaow. French cats say miaou—not so different (I suspect, but I'm not sure, that the plural is miaoux). Japanese cats say something like "nyan," which is the source of a popular meme from a few years ago. Most meows in other languages are at least close to onomatopoeic, though.

I had always assumed that cats talked to each other by meowing, and that they were just including us in their social circle. However, detailed behavioural observations of unowned groups of cats living in southern England have revealed that they rarely meow among themselves.

This appears to be borne out by other observations. I have a cat who almost never vocalizes. She'll purr when you expect her to, but I think I've heard her meow twice in eight years. I got her as an adult, and apparently she wasn't well-socialized with humans as a kitten. You'd never know it now, though, because she doesn't act feral at all.

Domestic cats are thought to be like other felines: solitary, aloof and asocial. But that is not always the case. When unowned cats occur in large, dense populations – as happens when people provide a lot of food – they do live in groups, composed mainly of related females.

Whoever thinks that doesn't know many cats. No, they're not always up in your face like those other animal companions are; to me, this is a feature, not a bug. But when they do want to hang with you, they won't leave you alone.

Why, then, do we consider domestic cats to be loners? Remember that the key aspect of lion and domestic cat groups is that they are made up of female relatives. But when multiple cats are brought together in the same house, they often arrive at different times, from different families. Not surprisingly, they frequently don’t get along.

My two (female) cats come from different places, and they have, at best, a wary armistice; at worst, a howling fight. Well, one of them howls. The other is silent, as usual. But they both get along well with my housemate's cats.

And as for the disturbing claim that your cat would eat you if you died at home and your body weren’t discovered: don’t believe it. Research shows that dogs are the culprit much more frequently.

I did want to address this libelous slander. Even if it were true... so what? What do you care? You're dead. And at that point, there is only one way for you to continue to fulfill your contractual obligation to your feline overlord.

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