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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

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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