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 2, 2020 at 12:01am

#997445

More Than One Shot

I figured out long ago that everything -- well, almost everything -- that we humans create goes through four phases.

PROMPT November 2nd

Write about something you collect. If you don't collect anything, write about the collection of a friend or family member.

There are exceptions, of course. Perishables, for one; you want to consume those as fast as possible before they go bad. Strawberries, for example. After one day in the fridge, they become a biology project. Avocados have an approximately 15-second window between "too hard" and "rotten;" only once in my life have I seen a perfectly ripe avocado, and by the time I was done eating it, it was already turning black.

Art is another exception. Real art, great art, I mean, which can't be defined but "I know it when I see it." The Sistine Chapel ceiling is probably just as valuable now as it was when Mikey painted the sucker. Great care is taken to preserve works of art, because each is unique.

But for most mundane objects, the first phase is shiny and new and therefore valuable, while the second phase is when it's lost its luster and has become, in the common parlance, junk.

It's at the second phase that things get discarded, thrown away, scrapped, recycled. Few care about such objects, unless they hold some kind of sentimental value. Your mom throws away your comic book collection. You trade in your iPhone for the latest model and the earlier one becomes garbage. Entire neighborhoods are leveled to make way for a shiny new apartment complex or stadium, which, some decades from now, will in turn be demolished for something even newer and shinier.

But some things, should they survive, enter a third phase of their existence. Their contemporaries have mostly been destroyed, and they gradually become valuable again, for their uniqueness. And it doesn't matter what the book value of such a thing is. It could be a house, the last remaining example of its architectural style, a window into a bygone era. A street - when I was in Alexandria the other day, I walked along an alley of original cobblestones, preserved because there weren't many examples left of Colonial-era cobblestones. An antique automobile, all of the other vehicles from that long-scrapped assembly line having been wrecked or junked. Even something as relatively valueless as a beer can; I've seen entire collections of vintage beer cans, ones with pull-rings instead of tabs, even earlier ones that you had to use a pointy opener on.

It's always been my intention to preserve certain things through the second phase until they entered their third phase. My ex-wife put a stop to that. My stamp collection is gone. My book collection is severely diminished. I had a piece of the Berlin Wall that she thought was just another brick (cue Pink Floyd here), and a jar of fine ash from Mount St. Helens. All gone.

I can forgive her for a lot of things, but not for that.

Now, I'm the first to admit that I'm a bit of a hoarder, for that reason, but I've never kept old beer cans. I do have a couple of wine bottles of sentimental value, signed by the winemaker or something of that sort. But I do have a bit of a collection of shot glasses going on.

Shot glasses are easy to collect. They're generally not very expensive, and when I'm traveling, sometimes I'll pick one up at one of those tacky tourist shops. Dice on a glass from Vegas, or a seashell shot glass from Virginia Beach. A crude drawing of Bear Lodge (aka Devil's Tower) from a gift shop in Wyoming. One done in Navajo style that depicts the Four Corners location. Things of that sort.

I rarely use shot glasses, myself, and when I do it's a generic one just so I can properly portion out the ingredients of whatever alchemical concoction I get it in my head to try. Like, tomorrow, I expect to make a few that involve: two shot glasses' worth of vodka, one of Kahlua, one of Rumchata, pour into a glass with ice, top off with heavy cream. It's a riff on the White Russian so beloved of my role model, The Dude, from The Big Lebowski. Because it's an American creation with a Russian influence, I call it "The American Election," which is why I intend to throw a few together on Election Day.

Whether that will be to celebrate or drown my sorrows remains to be seen, but that's one of the many beautiful things about booze: it works for both.

But I digress.

There is, of course, the matter of the fourth phase. No matter how much we try, no matter what techniques we employ, even those items which have enjoyed some time in the third phase of their existence will, eventually, crumble into dust.

Some find that depressing. Not me. It means they're real, and to be celebrated. Things that are real will eventually pass into oblivion; anything that we think is eternal is but an illusion. Sure, some things will last longer than others (including, I would hope, my shot glasses, because glass is remarkably durable unless you break it on purpose), but eventually, entropy always wins. Until one day, far in the future, even entropy will stop and with it, time itself.

But that makes it all the more important to appreciate things while we have them.

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