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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 5 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 7 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

April 11, 2024 at 10:38am

#1068505

Things Go Better With Cahokia

By coincidence, this article from BBC, which I've had saved for a while, is almost exactly three years old.

The US' lost, ancient megacity

Well, it can't be entirely lost, as we know about it.

Pity the event planners tasked with managing Cahokia's wildest parties. A thousand years ago, the Mississippian settlement – on a site near the modern US city of St Louis, Missouri – was renowned for bashes that went on for days.

Well, at least they didn't have the internet and its trolling, grifting denizens to deal with.

Throngs jostled for space on massive plazas. Buzzy, caffeinated drinks passed from hand to hand. Crowds shouted bets as athletes hurled spears and stones. And Cahokians feasted with abandon: burrowing into their ancient waste pits, archaeologists have counted 2,000 deer carcasses from a single, blowout event.

Pretty sure most of that still happens when the Cardinals win the World Series.

A cosmopolitan whir of language, art and spiritual ferment, Cahokia's population may have swelled to 30,000 people at its 1050 BCE peak, making it larger, at the time, than Paris.

Math aficionados might note that this would have been about 3,000 years ago, not "a thousand" like in the sentence I quoted up there. This is because, apparently, the BBC made a (gasp!) mistake here, at least according to this Wikipedia entry,

which appears to be thoroughly sourced. The Beeb should have written 1050 CE, not BCE.

I point this out only to show that everyone makes mistakes. I don't think that means we can't trust the rest of the article.

It's what Cahokia didn't have that's startling, writes Annalee Newitz in their recent book Four Lost Cities: A Secret History of the Urban Age. The massive city lacked a permanent marketplace, confounding old assumptions that trade is the organising principle behind all urbanisation.

Oh, look, a book promotion. Startling.

"Cahokia was really a cultural centre rather than a trade centre. It still boggles my mind. I keep wondering 'Where were they trading? Who was making money?'," Newitz said. "The answer is they weren't. That wasn't why they built the space."

Archaeology, or really any scientific discipline, is subject to human cultural biases. One reason for science's existence is to minimize, at least over time, the effects of bias. It helps to have people from diverse cultures studying things. We in the English-speaking world tend to assume everything's always been about money and trade, but this is not necessarily the case. (In fairness, it appears that math and written language in Eurasia both began as means to record business transactions, but even that appearance may be the result of some bias.)

When excavating cities in Mesopotamia, researchers found evidence that trade was the organising principle behind their development, then turned the same lens on ancient cities across the globe. "People thought that this must be the basis for all early cities. It's led to generations of looking for that kind of thing everywhere," Pauketat said.

Like I said.

They didn't find it in Cahokia, which Pauketat believes may instead have been conceived as a place to bridge the worlds of the living and the dead.

Yeah, let's not immediately leap to that conclusion, either. "May... have" doesn't fill me with great confidence. Nevertheless, it indeed seems to fit as a hypothesis.

Eventually, Cahokians simply chose to leave their city behind, seemingly impelled by a mix of environmental and human factors such a changing climate that crippled agriculture, roiling violence or disastrous flooding. By 1400, the plazas and mounds lay quiet.

Great, more ammunition for the anti-climate-change crowd to seize on. "See? Climate always changes." I mean, sure, it does, but usually on a much longer timescale than what we're experiencing now. Plus, recall that this was in the Mississippi flood plain, and that river has been known to shift, even during a single human lifespan.

But it's more than that. Cahokians loved to kick back over good barbecue and sporting events, a combination that, Newitz noted, is conspicuously familiar to nearly all modern-day Americans. "We party that way all across the United States," they said. "They fit right into American history."

The only thing missing is beer. And I'm not going to wade into the cultural quagmire on that subject.

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