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 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 18, 2021 at 12:08am

#1012072

Superman

The Random Number Gods have had a nice chuckle at my expense. Over fifty possible links to select from, and I get the one about Superman the day after I made a Souperman pun.

4 Reasons Anyone Who Says 'Superman Is A Boring Superhero' Is Full Of It

Article is from Cracked and dated March 1, but that doesn't change the basic message.

Between the debuts of Zack Snyder's Justice League and Superman & Lois, this month is shaping up to be unusually Supermanly.

I have yet to see the Snyder cut (I know, I know... bad nerd) but I have been watching S&L, and I find it to be very well-written with competent acting -- a bit too heavy on the teen drama aspects for my taste, but that's on me.

A common complaint about Superman is that he's just too damn powerful, which is kind of like saying that the Grand Canyon is too damn grand, or that your Baconator came with too much damn bacon. Still, a lot of people seem to think it's impossible to tell interesting stories with a character who's practically invincible ... including some of the writers and studio execs whose job it is to make those stories.

It is, indeed, difficult to write meaningful stories about a character who verges on the omnipotent. As something of a writer myself, I know you have to come up with challenges for the character, and there's only so much you can do with Kryptonite before it gets boring. The trick, which is used in S&L, is to focus on the few things the character isn't super-powered at; for instance, dealing with the aforementioned teen drama. (For those out of the loop, S&L features a Clark who is married to Lois and they have two teen sons.)

4. His Powers Aren't Random: He's A Walking Catalogue Of Active Superpowers

The Complaint:

"Why does Superman have to have so many powers, anyway?! Look at Batman, he's got none and he's cool!"

Doesn't he, though? Like I said, I haven't seen the Snyder Cut, but in the Whedon version (or maybe it was an earlier Affleck Batman), I remember a line of dialogue like, "What's your superpower?" "I'm very rich."

Batman also, traditionally, has the superpower of knowing exactly what tool he needs for any given situation. And he also gets advantage on Stealth rolls.

These aren't sold as "superpowers" because Batman is more of a "I could be him if I were rich and worked out a lot" kind of character, but no amount of money or weight-lifting can help you match wits with Joker.

First of all, the ability to be hugely rich and not be a massive piece of trash is a more fantastic superpower than any of Superman's, so there's that.

Also this.

But also, Superman's powers didn't come about because someone said "let's just make this guy impossible to beat," like video game developers creating an annoying final boss. They grew organically across several decades in a way that's intrinsically linked to the evolution of the superhero genre itself.

That's a good point, and the article goes into the character's changes over the years.

3. He Continually Ups The Stakes ... For ALL Superheroes (Even In Other Companies)

The Complaint:

"Dude's got it too easy! He can stop any crime in like two seconds! And all his villains suck!"

I'mma stop this complaint right here. Lex Luthor is (usually) an excellent villain (especially the John Cryer version, though he's only on Supergirl for now). One exception is the Jokerish version of Lex Luthor foisted upon us by the Justice League movies, thus demonstrating that you really have to match the villain's strengths to the hero's weaknesses.

Speaking of Lex Luthor and rich people, has anyone else noticed Jeff Bezos' uncanny resemblance to Luthor?

Most Superman baddies are merely excuses to force Superman to face important issues he can't easily solve -- human greed, the dangers of media manipulation, what if a being from the fifth dimension turns your friends into cows, etc. But we'll admit that not all writers realize this, which is why they keep trying to make it look like a guy who makes explosive teddy bears could be a real threat to a near-omnipotent being.

I mean, seriously, this article goes into even weirder stuff than this, and it's worth reading if only from a writer's perspective.

2. He Isn't Just Wish-Fulfillment; He's Aspirational

The Complaint:

"He's just a power fantasy! Grow up and read about more complex characters, like the suave super-spy who has sex with lots of attractive women!"

I mean, really, I've been saying this for years now. Not the "complaint" part but the bolded part. In the best Superman stories, the powers are incidental -- they're more interesting for the ethical aspects.

Superman, on the other hand, embodies traits that are actually desirable in a functioning society, like restraint and responsibility. Couldn't the writers do that without giving him so many powers, or at least giving him crap ones? Sure, look at Spider-Man, whose most commonly used power is "using his wrists as spider butts." However, the importance of Superman's sense of discipline is way more effective coming from someone who could accidentally kill you while sneezing...

In our world, most of us don't have superhuman powers, but the things we do even as worker drones or ordinary peasants can and do have an effect on others. If it takes writing fantasy about someone who could fly through an airplane in mid-air, killing hundreds of people, but doesn't, to work as a metaphor for those limited powers that we do have -- so be it.

I mean, lots of us have the means to end at least one person's life in an instant (knives, guns, whatever), but a glance at the news will tell you that some people don't have the responsibility needed to not use them.

1. Some Of His Best Stories Don't Even Involve Powers (That Much)

The Complaint:

"Nice try, Cracked, but I still think he's too powerful! Also I don't know why I'm shouting every sentence, please help me!"

This sort of thing is why I keep reading Cracked.

The rest of this section is a summary of some great Superman stories where he didn't have powers and pushed through anyway.

If it seems like you have to jump through a lot of hoops to make a good Superman story, that's because you do. Superman's invincibility offers a more structured form of storytelling -- he's the superhero genre's equivalent of trying to write a haiku that's also an acrostic that also doesn't contradict 83 years of continuity (or some nerds will get very angry at you). So when those writers and studio execs say it's "impossible" to tell interesting Superman stories, what they mean is that it's "harder." But when they put in the effort, the results can be transcendental.

And when they don't, well ... watching him punch robots is also fun, to be honest.

That last line really sums it up for me. While seeing nothing but that would get boring fast, there is always the spectacle. And isn't that what visual entertainment is for?

Anyway, the article is a pretty good essay on how to write overpowered characters and keep them interesting. I remember a time, maybe in the 70s or 80s, when Superman stories were heavily about the hoops he jumped through to keep Lois from discovering his secret identity. Yeah, the character has had his ups and downs. But mostly ups. Ups. And aways.

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