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.

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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.

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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.

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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

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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.

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August 31, 2023 at 11:27am

#1054900

Pasta Prime

How about we take a lunch break?

How Spaghetti Squash Squiggled Its Way Onto American Tables
It took a shift in food culture for consumers to embrace the “noodle plant.”

Yes, one reason I'm featuring this article is the glorious alliteration in the headline.

The other reason is that I actually like spaghetti squash, and I even liked it when my mother prepared it way back when. Most of the vegetables she cooked ended up like soggy rubber.

The year was 1974, and Takeo Sakata was holding a dinner party. “Conversation centered around the spaghetti squash,” a guest later recalled, “and the struggle Mr. Sakata had had in introducing it to the West.”

We would have planted it, I guess, sometime in the late 1970s, so that effort was at least partially successful. I guess.

Since the 1930s, Sakata had been the exclusive marketer of spaghetti squash in the United States. According to Dr. Harry Paris of Israel’s Volcani Institute Agricultural Research Organization, perhaps the world’s foremost expert on spaghetti squash, “[the Japanese] were the only ones interested in it for some time.”

The irony here is that squash is an American vegetable (as with tomatoes, it's botanically a fruit, but we're talking about culinary use here). By which I mean, its origin as a cultivated plant was in Central and South America. Cultural appropriation! (The article eventually goes into its cultivation history.)

Americans took a long time to warm to this mild-tasting, oblong yellow squash, set apart by its densely coiled inner fibers, which separate when the squash is cooked into strands that distinctly resemble noodles. “You’d be surprised how many examples there are of things that have been around a while until they are rediscovered by people who appreciate them,” says Dr. Paris. He compares the spaghetti squash with the zucchini, a crop that was local to Milan, Italy for decades before it became the world’s most popular summer squash.

To me, the greatest thing about spaghetti squash isn't the spaghetti-like feature (though that's pretty cool). It's that the stuff doesn't taste even a little bit like zucchini or yellow squash. Not that I hate those squashes, but I never choose them on purpose, either.

It would take a massive shift in Western food culture before consumers finally took notice of spaghetti squash: this time, as a health food.

Well, shit. If I'd known it was a health food, I'd have hated it. Like with kale.

In the late 20th century, the marketing of Sakata’s spaghetti squash began to center around its appeal to an increasingly calorie-conscious public. The flesh of the squash had long been compared with noodles, but now, for the first time, it was marketed as a healthier alternative to them, with one-fifth the calories, one-quarter the carbs, and a negligible amount of fat.

Let's be real: it's not exactly like spaghetti, or any other noodle. But it's similar enough that one can do the same things to it as with spaghetti, like loading it up with butter, bolognese sauce and Parmesan cheese, thus negating most of its health benefits.

Although recipes of the 1980s and 1990s still suggested boiling or steaming the squash, one of the first recipes to present it explicitly as a healthier pasta alternative, published by Frieda’s Branded Produce in 1975, was developed using a microwave oven.

I'm sure there are several ways to cook it, as with most vegetables, but roasting works well and produces less mess. I'd keep it far away from the microwave, though.

Spaghetti squash was finally in the limelight. “But you have to be a pretty good cook to know what to do with it,” says Dr. Paris.

Or just look for instructions on the internet, scroll past all the pages and pages of superfluous background information food bloggers seem to be compelled to preface their recipes with, and do what it says. It's really not that difficult, and I say that as a dedicated slacker. It's like hard-boiling eggs: if you know the right procedure and time involved, it's easy to get good results. It's not like making a soufflé, or even getting your steak temperature just right.

In any event, I didn't know the whole history, hence the article. And now I'm hungry...

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