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 6 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 8 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.

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').

1 Like

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.

January 11, 2022 at 12:06am

#1024481

Moral Hazard

Still going through "JAFBG"

[XGC] at random, mostly to prove to myself that I can write about almost anything.

Write about something people do that isn't necessarily a huge deal, but if everyone did it, it would be total chaos.

Well, the sidewalk snow removal (specifically, lack thereof) thing from a few days ago comes to mind, but I already wrote about that.

Ethics 101: ask yourself, would it still be ethical if everyone did it?

I'm sure there's more nuance to it than that, but I consider it a starting point.

For example. Imagine you're at a university that features an internationally-famous lawn. One of the main entry points to this lawn is along one side. Also, the entrance to the Philosophy department is diagonally across this lawn from that entry point. Sidewalks run along all sides of the lawn, with a few cutting straight across, so you're meant to do a Pac-Man thing and turn right, then left (or go straight then turn right). But you're also taking math classes, so you know that the shortest distance, the one that requires the least steps, is to cut across the lawn past a statue of, say, Homer.

The blind poet, not the Simpson.

So you're running late to your first-year Ethics class one day, and to shave a few seconds off the walk, you cut across the lawn, giving Homer a high-five on the way.

No big deal. People walk and play frisbee on the lawn all the time; one student taking a shortcut isn't going to make a lick of difference.

Problem is, though, that other people see you do that, and they take the high-fiving Homer route. The increased traffic kills the grass and compacts the soil underneath, wearing a strip of dirt into the historic sward. Then it rains, and everyone taking the shortcut walks into Ethics class with muddy Uggs.

Homer would roll his eyes if he had them. And wasn't a statue.

And yes, this actually happened when I was at UVA. Except there weren't Uggs back then. One of my first attempts at satire was a proposal I wrote for the editorial section of an April Fools issue of the student paper, suggesting that, at night, the Grounds maintenance crew should bury claymores along the path. Nothing, I argued, would discourage people taking that shortcut more than watching a few of their classmates' legs get blown off by the shrapnel. As a bonus, the soil would become aerated and could be successfully reseeded.

Sadly, they didn't end up mining the Lawn, but they did install some short steel posts around the corners and stretch a chain through them. Which might have actually been effective if they had taken into account that they're dealing with agile teenagers who can easily hop a steel chain, not aging hippies like me who can't catch air to save their lives.

Total chaos? Eh, no, just an ugly brown streak on the diagonal across the Lawn and, ultimately, a destruction of its pristine (and, as I said, historically recognized) design caused by the installation of the ineffective barrier. But that particular kerfluffle stuck in my head and served me well when I graduated and did public space design for a living. No, I never designed land mines, but I always kept in mind where the probable foot traffic would go and propose sidewalks accordingly.

But I'm sure I could think of other examples that fit the prompt, if I weren't so completely hung over right now (thanks, lunch beers). Here are a few that pop into mind:

Driving a gas-guzzling car

Letting your dog shit in someone's yard and not picking it up

Having a dozen kids

Having no kids

Stealing a Coke from a store

Drinking three high-octane beers at lunch

Not wearing a fucking mask in the grocery store during a pandemic

Speaking of grocery stores, failing to return your cart to the corral afterward.

Littering

In a sense, these are all counterexamples of my Lone Asshole Theory. If you didn't read that previous entry, basically, Lone Asshole Theory states that even if the majority of people act ethically, all it takes is one asshole to ruin your day. I used the example of leaving your car unlocked; all it takes is one thief and your interior is trashed, even if 99% of people walking by would leave it alone. In these cases, however, what you're doing barely registers on anyone's radar, but if everyone did them, there would be, if not total chaos, at least some sort of suboptimal result for everyone.

But they're also kind of moral hazards, aren't they? Like in the example I used above: one person cutting across the Lawn, by herself, makes no discernible difference. But other impressionable students, seeing that, might go, "Hey, if she's doing that, then so can I!" And so you get the bare diagonal trail.

And aren't you glad that not everyone writes a long-winded blog entry every day? You'd have no time to do anything but read blogs or, worse, you'd have no time to read mine.

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