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.

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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

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.

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February 24, 2023 at 12:08pm

#1045458

Food, Glorious Food

In contrast to yesterday's ramblings about exoplanets, today's article is both humorous (because it's from Cracked) and slightly more down-to-earth.

There’s No Such Thing as ‘English’ Food

I think that's a matter of definition, but we'll see what the article says about it. After all, when you think of Italian food, what comes to mind? Pasta with tomato sauce? Noodles came from east Asia, and tomatoes came from the Americas.

In a nice break, this isn't a countdown list.

English food has a reputation as bland at best and outright objectionable at worst, full of vegetables boiled to mush, dubious “puddings” that bear no resemblance to the Snack Packs we love overseas, and jellied eels, possibly the least appetizing combination of sounds in the English language.

While that reputation may have some basis in reality, having been to England, I can say it has a lot more than that.

All of that is beside the point, though, because it turns out there really is no such thing as English food, or at least not what we think of as English food. Fish and chips, afternoon tea, beans on toast, all manner of meat pies or non-pies that are still confusingly called pies, and yes, even sausage rolls — they or their components all come from somewhere else in the world. The Brits have created an entire cuisine out of little more than theft and colonialism.

And? Lots of nations' cuisines came from somewhere else. Japanese tempura? That has the same source as British fish & chips: Sephardic Jews from Portugal. Hell, even the name has a Latin root.

Let’s start with what many bewilderingly consider England’s national dish: curry. Obviously, curry wasn’t invented by some dude who looks and sounds like Colin Firth, coming to England via its colonization of India.

One thing England did do well on its own: beer. You might have seen the proliferation of the beer style known as IPA, or India Pale Ale. That's not an Indian beer. That's a British beer, hopped-up (hops are a preservative) to survive the long trip around Africa (in the days before the Suez Canal), because apparently that was easier than setting up breweries in India.

Incidentally, speaking of curry, Japan has its own take on curry. You'd think they'd have gotten it from India, too. But it turns out that they got it from England, who got it from India, which is about as convoluted as it's possible to get. Whether India or Thailand invented it, I don't know and can't be arsed to find out right now; it doesn't matter. Point is, delicious food gets around.

Tea, of course, also came from India, which got it from China, though it’s as entwined with English identity as large clocks and the wrong kind of football.

You mean the kind that doesn't give its players brain damage?

For their part, scones originated from Scotland. That might sound close enough, but tell some burly Scottish dude that he’s basically English and see what happens.

I wouldn't brag about scones.

The article doesn't mention that other quintessential British food: crumpets. Depending on who you talk to, they might have actually been an English creation... or, perhaps, Welsh.

Another country that England didn’t colonize, despite their best efforts, but still cribbed much of their cuisine from is France.

I'm not going to rag on English food, but I don't think anyone except the most committed Anglophile would argue that the French are, in general, better cooks. There's an old joke to the effect of: the difference between Heaven and Hell is that, in Heaven, the Germans are the engineers, the British are the police, and the French are the cooks; in Hell, the French are the engineers, the British are the cooks, and the Germans are the police.

Basically, the most traditionally English meals, broken down to their constituent parts, have roots outside England, sometimes literally. For example, the only English part of fish and chips is the newspaper it’s served on, and that might be suspect as well. The fish came from Jewish immigrants, who fried it in flour for religious reasons (it turns out God wants us to live deliciously), and was initially advertised in England as fish “in the Jewish manner.” The chips, which we — probably correctly, it turns out — call French fries, actually do come from either France or Belgium. (We got something right, guys!)

Worse, it's not like potatoes are native to Europe. Or even Africa or Asia.

The mince pie, an English Christmas staple, came from the Middle East, and anything with custard owes a debt to Ancient Rome.

Okay, but let's be fair, here: the British conquered a good chunk of the world and appropriated various cultural foods, but not Ancient Rome. Rome instead invaded the British Isles. "We learned it from YOU, Dad!"

Anyway, a fun article if, like me, you enjoy knowing where your food came from.

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