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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Write_Mikey_Write!

about 8 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 10 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.

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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.

July 29, 2020 at 12:01am

#989410

Humanity's Last Invention

As a lifelong consumer (and sometime writer) of science fiction, and having lived through the last half-century, I can attest that technology can certainly make one's life better, for various definitions of "better," but as with a monkey's paw or a genie's wishes, everything comes at a cost.

PROMPT July 29th

Write about an invention or technology that you wish existed that would make your life better.

This prompt came in at the same time as my friend was texting me her idea that they should remake some of the original episodes of Star Trek, with a new cast and better budgets. And she wasn't even stoned. So of course we're going to talk about Star Trek today.

The first thing that came to mind, of course, was the holodeck (holosuite in DS9 because it wasn't set on a starship). That would certainly make life better in so many ways, but as someone (Dave Barry, maybe?) pointed out, the holodeck would be humankind's last invention.

There are a lot of inventions speculated about in the various incarnations of the show. Some of them have even been realized in some form (at one point, William Shatner's character in Boston Legal opened a flip phone and it made a communicator chirp). Others probably can't be, at least not in the foreseeable future. All of them have their benefits... as well as their downsides. I'm going to go the overachiever route and talk about not just one, but many of them today. And hell, maybe this will be the first draft of my next Fantasy newsletter editorial because I'm lazy and out of ideas. I mean, come on, I was so starved for inspiration that I talked about tea in this week's newsletter. Tea. Earl Grey. Hot.

So let's start with tea. Specifically, let's talk about how Picard was able to get his tea just the way he liked it in his ready room. The replicator is, like, a 3D printer on steroids. Upside: Using transporter technology (which I'll discuss in a bit), it can craft almost any arrangement of matter. Want a sandwich? Order it at the replicator. Want a thick, juicy steak? Order it at the replicator (no cows were harmed in the making of this episode). Want some Romulan ale? Ooops, sorry, that's not going to happen. Downside: some people actually enjoy cooking and preparing food, not to mention yet another profession lost to tech.

Speaking of the transporter, this idea was introduced as a cost and time saving measure so they wouldn't have to show planet landings and such all the time, but it's become just as intrinsic to Trek as warp drive. Your atoms are scrambled, converted to energy, and then reconstituted at some other point. There's an ongoing philosophical debate as to whether this in fact kills and resurrects you. Upside: near-instant travel. Downside: Transporter accidents are usually gruesome and fatal. But then again, so are many plane crashes.

While we're talking about warp drive, I mean, come on, how would that not make life better? As with the transporter, though, there's little chance it'll happen anytime soon. Upside: explore strange new worlds, seek out new life, etc. Downside: Klingons.

Then there's artificial gravity. Oddly, every other system on a starship could be fried, and the gravity still works. Okay, not that oddly, because it's still just a TV show produced on a planet. Point is, though, if you can create gravity at will, you can also negate it and use the tech to dampen acceleration, so... Upside: easy transportation, loft stuff into orbit; also, not suffering bone loss and other effects known to occur in microgravity. Downside: Well, who knows how much power such a thing would have to use? And it would suck for it to glitch out while you're accelerating at 20 Gs.

Another thing of great personal benefit would be the Universal Translator. Whatever language they're speaking, it becomes English, like magic! It even makes their lips move in English. Technology, I'm telling you! Upside: Understand everyone who doesn't speak in metaphor (watch the TNG episode Darmok if you don't know what I'm talking about). Downside: There are a lot of benefits to actually doing the work to learn a different language, including a better understanding of the culture. Besides, I'm not even sure such a tech would even be possible; there are just too many complexities to language.

Want to make life better? Invent the phaser. Upside: stun setting eliminates threats without those messy ethical issues. Downside: you know we'd abuse the fuck out of it just like cops are abusing pepper spray now.

Going back to drinks for a minute, there's synthehol, which is the obvious portmanteau of "synthetic" and "alcohol." Upside: Drink all the scotch you want without getting drunk! Downside: Drink all the scotch you want without getting drunk.

Want to know everything about a particular, say, rock? Then you need a tricorder. This handheld device can analyze the molecular and crystalline structure of, well, pretty much everything. Upside: know what everything is. Downside: where's the joy of discovery?

And so we finally come to the holodeck, the ultimate Virtual Reality experience. Upside: Any scenario you can imagine, it can create, much like the replicator but for more than just food. Rest on a beach. Go camping in the mountains. Experience the pleasure domes of Risa. All without leaving Deck 38 or whatever. Downside: spend the rest of your now-shortened life getting foot massages from Uhura. "Nichols or Saldana?" "Pourquoi pas les deux?"

But you can't tell me it wouldn't be worth it.

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