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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 9 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 11 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.

June 6, 2021 at 12:12am

#1011375

Fun & Games

Today's article may have more complicated implications than it would seem at first glance. So, first, the article:

How Video Games Satisfy Basic Human Needs

The publication date is more than four years ago, but I don't think that changes much.

Grand Theft Auto, that most lavish and notorious of all modern videogames, offers countless ways for players to behave. Much of this conduct, if acted out in our reality, would be considered somewhere between impolite and morally reprehensible. Want to pull a driver from her car, take the wheel, and motor along a sidewalk? Go for it. Eager to steal a bicycle from a 10-year-old boy? Get pedaling. Want to stave off boredom by standing on a clifftop to take pot shots at the screaming gulls? You’re doing the local tourism board a favor. For a tabloid journalist in search of a hysteric headline, the game offers a trove of misdemeanors certain to outrage any non-player.

I've never played GTA, but from what I understand of it, it's a single-player game; that is, you control an avatar, and every other object in the game (including the ones that resemble people) is programmed. In that way, it's much like any other single-player video game. GTA, though, explicitly focuses on playing as a criminal, or "bad guy."

For the British artificial intelligence researcher and computer game designer Richard Bartle, the kaleidoscopic variety of human personality and interest is reflected in the video game arena. In his 1996 article “Hearts, Clubs, Diamonds, Spades: Players Who Suit MUDs,” he identified four primary types of video game player (the Killers, Achievers, Explorers, and Socializers).

And then suddenly the article switches to MUDs - originally Multe-User Dungeons, in which you might have NPCs, but other avatars are controlled by other human players. These were essentially online role-playing games, in the vein of D&D. They're mostly niche, now, as people prefer multiplayer games with all the graphics and whatnot, like World of Warcraft or whatever.

Personally, I prefer single-player games, because gamers are assholes and generally treat other player characters as if they were NPCs.

Bartle’s research showed that, in general, people were consistent in these preferred ways of being in online video game worlds.

Well, as usual, I violate the general case.

While players sometimes experiment by, for example, playing an evil character just to see what it’s like, Bartle found that such experiments usually lead to affirmation rather than transformation. “Basically,” he said, “if you’re a jerk in real life, you’re going to be a jerk in any kind of social setting, and if you’re not, you’re not.”

Yeah, that hasn't been my experience. People who are kind in real life can blow off steam by being a jerk in an online game. People who are assholes in real life, though, are also assholes in games. I guess this says something about me, or humanity in general. Maybe. Or it says something about how we approach fantasy.

I like to think that on the occasions where I've played in multiplayer games, I wasn't a jerk to other players. But maybe they didn't see it that way.

They found that video games that allowed players to play out their “ideal selves” (embodying roles that allow them to be, for example, braver, fairer, more generous, or more glorious) were not only the most intrinsically rewarding, but also had the greatest influence on our emotions. “Humans are drawn to video and computer games because such games provide players with access to ideal aspects of themselves,” the authors concluded. Video games are at their most alluring, in other words, when they allow a person to close the distance between how they are, and how they wish to be.

This makes a lot of sense; however, I've taken great joy in playing (in single-player games) as a despicable assassin, and that's hardly my ideal self. I've also played as a paragon of virtue, and find it too limiting. LIke, when playing Skyrim, a "bad guy" has a whole lot of options for quests and fun things to do, whereas a white-knight type gets bored quickly.

I solve this, in Skyrim, by being a good guy in public while sneaking around doing evil deeds where the guards won't catch me.

But, again... it's a game. It's fiction, not real life. Does me playing as an assassin in a game mean I'd want to kill people in real life? Of course not, any more than writing a story about murder would. And people love stories about murders, and bad guys doing bad guy stuff.

“It’s the very reason that people play online RPGs,” Bartle said. “In this world we are subject to all kinds of pressures to behave in a certain way and think a certain way and interact a certain way. In video games, those pressures aren’t there.” In video games, we are free to be who we really are—or at least find out who we really are if we don’t already know.

Yeah, I'm not buying it. I play games to be something other than who I really am, be it virtuous hero of despicable villain -- neither of which describes my actual personality -- or anything in between.

That's why they call it "role-playing." Did I play a druid in D&D because I love nature and the outdoors? Shudder, hell no. Did I play a Drunken Master monk because I like to drink? Well, okay, that one may have been typecasting. The point is, regardless of what this guy claims, I don't play aspirationally; I play characters that may or may not reflect my own real-life preferences, exploring different personalty types that may or may not reflect who I am in real life.

I guess that's an occupational hazard for a writer.

Not every game, however, allows us to act in the way that we might want to. The designer, that omniscient being who sets the rules and boundaries of a game reality, and the ways in which we players can interact with it, plays their own role in the dance. Through the designer’s choices, interactions that we might wish to make if we were to fully and bodily enter the fiction are entirely closed off.

Yes, and that's annoying, but for single-player games, sometimes mods can fix that. Let me give you an example.

Fallout 4 was produced by the same people who did Skyrim, and it's another game I keep going back to because there are so many character-based ways to approach it. It's harder to go the "white knight" route in it, because the game setting doesn't really fit that archetype. Which is fine, because it mimics the real world in that sometimes you have to make tough "moral" choices. I put moral in quotes because, again, it's a single-player game and every other object in it is programmed into the software, including NPCs with their actions and reactions. So nothing you do in the game has any bearing whatsoever on the "real world." Want to blow a guard's head off? You can do that. Set off a nuke in a marketplace? You can do that. Take on a companion and use them as a pack mule? You can do that. There may be repercussions in the game, but those are programmed, too.

What you can't do, in the base game, is kill "children." By which I mean, there are certain NPCs who resemble young teenagers, and, unlike the NPCs who look like grown-ass adults, the program simply will not let you do anything to permanently remove them from the game world.

And there seems to be a general consensus that, while it's perfectly okay to sneak up behind an "adult" innocent NPC and "kill" them, somehow it's different when it's a "kid." But the only difference between the two is the programming of their size and maybe voice.

So I run a mod that lets you kill any NPC that isn't essential to the storyline. Including the annoying little brats.

This, apparently, makes me a horrible person. But it's okay, because I also run a mod that makes it impossible to kill the game's cats.

Priorities, people.

Anyway, the full article is worth a read, even if I quibble with the general conclusions. I guess being able to slip into different character types is mostly a writer thing, and most gamers aren't writers; if they were, they'd be writing instead of playing video games. There, too, I'm an exception.

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