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.

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').

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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.

3 Like

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

October 21, 2024 at 10:16am

#1078678

Mything the Point

From the BBC, an article at the intersection of science and mythology.

The myths that hint at past disasters
Myths and fables passed down over thousands of years are full of fantastic creatures and warring gods. But they also might contain evidence of environmental disasters of the past.

It's long been assumed that many, if not all, myths contain some elements of fact. (Like the article, I'm using "myth" in the sense of a passed-down story, not the more modern meaning of falsehood.) The tough part is teasing out the fact from the fiction.

The article opens with a bit about expected sea level rise, comparing it to past sea level rise. This might add fuel to the "climate's changed before and we're still here" campfire of climate change deniers, but so be it.

With the possibility of a catastrophic global sea level rise of 3ft (1m) by 2050 which could force millions of people to leave their homes, researchers have now started to look at ancient stories about land lost to the sea and downed cities in a new way.

Hell, it might have looked like the whole world was flooding, leading to one extraordinarily popular myth (though that one might have been due to its Mesopotamian origins).

Some researchers argue that tales of hot boulders thrown into the sea or the building of sea walls comprise factual information, albeit exaggerated and distorted to some extent.

Well, okay, argue all you want, but how about some science to back it up? I have this hypothesis that the "nephilim" mentioned exactly once in the Bible, in Genesis, are actually a passed-down memory of Neanderthals. There's no definitive translation to it, and the KJV muddled the waters by rendering the word as "giants," with even less evidence (we didn't know about Neanderthals at the time). Like I said, it's my hypothesis. I have literally no way of supporting or disproving it.

These researchers are geomythologists.

As good a name as any, and better than most.

“Geomyths represent the earliest inklings of the scientific impulse,” says Adrienne Mayor, folklorist, historian of ancient science and research scholar at Stanford University, California, and author of the important The First Fossil Hunters, “showing that people of antiquity were keen observers and applied the best rational, cohesive thinking of their place and time to explain remarkable natural forces they experienced.”

I suppose, for various definitions of "rational" and "cohesive," if such words can be applied to deciding that floods are due to gods yeeting boulders and whatnot.

There's a lot more at the article, including specifics; I won't quote much more.

Back when the glaciers retreated, humanity was already infesting the European lands (I'm including Neanderthals in that group). In that era, Britain wasn't an island; it was connected to the mainland by a low-lying chunk of land now called Doggerland. This wasn't particularly ancient, in geological terms; apparently, it flooded out about 6,000 years ago, around the time civilization was flourishing in Mesopotamia, thanks to beer.

I'd often wondered about undersea archaeology in that region, and what it might find in terms of artifacts. It's often called a land bridge, but that implies that humans mostly lived in Britain or Nederland, only crossing the "bridge," but there's no reason why people wouldn't have lived there.

Turns out, of course, that actual scientists were way ahead of me.

The whole thing's fascinating to me, though I doubt the article's thesis that there's much practical use in terms of dealing with catastrophic modern climate change in an era when there are billions of people, not hundreds or thousands, living near a coast. But finding more truth about ancient humans is practical enough for me.

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