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

November 18, 2020 at 12:01am

#998616

Original

For as long as I can remember, I liked superhero origin stories.

PROMPT November 18th

Write about origins. You can take this literally as in where your family originated from (your history, culture, traditions), or you can focus on your own origins as a person.

Other people with actual backgrounds in liberal arts have said this better than I ever could, but comic book superheroes are essentially updated mythology. Where the Greeks, for example, wrote of gods and heroes from the point of view of their own culture, our industrial/technological society has produced characters drawn from the fringes of our scientific approach to the world.

Not that there's anything actually scientific about them. Costumed adventurers have always occupied a place between genres, drawing primarily from fantasy and science fiction -- with, of course, several others thrown into the mix. Exposure to gamma rays won't turn you in the Hulk; it will sicken or kill you. Aliens don't leave glowy rings lying around for test pilots to find. Mutated genes are more likely to kill you than give you teleportation powers. Incredibly rich orphans find other ways to make the world a better place than dressing up as a bat to fight crime. Getting bitten by a radioactive spider (which, come on, really?) probably wouldn't do anything that getting bitten by an ordinary spider wouldn't do.

I like to say that my origin story occurred when I got bitten by a radioactive sloth.

Point is, though, these characters and stories occupy a place of speculation and wonder. The radioactive spider thing, or whatever, is incidental; the character's abilities draw a reader (or viewer) in, while their approaches to conflict keeps things interesting. We all know that the origin story, and everything about them, is fanciful. Some dismiss the entire genre for its lack of realism, and that's fair. But they're still stories, and just as with the magical aspects of fantasy, once you accept the more unrealistic parts, the rest can follow.

There's no real difference, then, between stories of the labors of Hercules (born of a god and set up to be an Earthly hero) and stories of the adventures of Superman -- except for the time period in which they're written, and the cultural milieu of their audience. Our modern stories take into account speculation about aliens, scientific discoveries, alternate universes, whatever -- things the Greeks had only the barest inkling of. Even the ancient gods are explained away in the comic books, as advanced, but natural, beings.

Much has been discussed about the differences between, say, Batman and Superman, or the difference in approach to superhero stories taken by different publishing companies, and everyone has their favorites, as well as those they dismiss out of hand. Most of my friends were more into Batman. I was always a Superman fan.

This has little to do with his powers and more to do with his classic origin story. An orphan sent to Earth from a dying planet, adopted by simple farmers who raised him as their own.

As an adopted child raised on a farm, this appealed to me.

Don't get me wrong; apart from the usual childhood fantasies about flying or x-ray vision or whatever, I knew damn well I didn't have superpowers and probably wasn't actually from another planet (unless Indiana counts as another planet). Though after making the entry from a couple of days ago, maybe "avoiding broken bones" counts as a superpower. And then there's my ability to drink different kinds of booze with no additional ill effects... hmm.

No, I'm pretty sure I'm human. 98% sure, at least. Well, maybe 95%.

But -- apart from the flying and the super-strength and tights and whatnot -- I have a significant departure from Clark Kent.

See, Clark was always interested in finding out his biological origins. In every version of the story, he ends up finding out about Krypton and his birth parents and whatnot, and soaking in alien science and lore. Not that he abandons those who raise him, but he has this need to find out where he "actually" came from.

Maybe it's because I know I'm not Kryptonian, but I have no such need. One family was enough for me, and apart from a mild curiosity regarding possible genetic predispositions to certain ailments, I couldn't possibly care less about my biological heritage.

I like to think that this gives me a less tribal worldview. Other people can be -- and in most cases, should be -- proud of their heritage, be it European or Asian or African or American or whatever. But for me, besides looking like I almost certainly have Northern European ancestry, I consider myself a product, not of one place or culture, but of the entire planet.

After all, when it comes to origins, whether you buy into creation stories or follow the science, ultimately, we're all descended from one human -- all one extended family. And I don't know, maybe if more people had that attitude, people might be less interested in bombing each other into oblivion.

But then again, knowing how some families are, probably not.

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