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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

July 19, 2021 at 12:01am

#1013915

The Price You Pay

Another one from "JAFBG"

[XGC]

We all have chores or responsibilities. If you could outsource one task, guilt-free, without it costing you a cent, which chore or responsibility would you pawn off on someone else?

Aging.

Okay, fine, that's not a chore or a responsibility, but the inevitable result of not dying.

Everything has a cost. That's basic economics. The cost isn't always monetary (to address the "without it costing you a cent" part) or emotional (to address the "guilt-free" part) -- often the cost is one of opportunity or time. There are probably other ways to assess cost, but I can't be arsed to provide an exhaustive list right now because that would take too much time. See? Case in point.

While money is our most common way of figuring cost -- it's versatile and convertible; hence "time is money" -- sometimes things that are "free" are simply not worth the non-monetary costs.

For example. "Sign up for our FREE newsletter!" Yes, you don't have to pay money for the newsletter, but we're going to send you emails every fucking day and, moreover, make money ourselves by selling your email address to a bunch of other shady businesses who will then proceed to send you emails every day. You will then have to spend time and effort to deal with emails -- directing them to other folders, unsubscribing (good luck with that), deleting them as they come in, or abandoning your email address and getting a new one.

This was a lesson I learned early on in life, and is one of the few that I didn't need personal experience to absorb. I think I was filling out whatever paper forms there used to be for the Publisher's Clearing House sweepstakes. My dad saw me doing it, and gave me my first (but definitely not my last) lesson in economics.

"Nothing's free," he said, or words to this effect. "Even if you did win that, which is improbable, you'll be hounded for taxes and probably discover cousins you never knew existed, and neither did your mom or I."

Well, he said it in different words and with a bit of a Louisiana accent, but that's the gist of it.

Still, it sure would have been nice to win the million dollars or whatever. That was a lot of money back then. Well, okay, it's still a lot of money, but it's not "tell the world to go to hell" money anymore. Even after taxes, that would have been at least $500K, and then maybe I could have gone to a state university in a Porsche instead of a pickup.

The point is, everything has a cost; the key is to know what the cost is so you can decide whether or not you're willing to pay it. The economic principle is No Free Lunch, and it's absolutely immutable.

In the case of chores or responsibilities, at least part of the cost in outsourcing them is loss of control.

For instance, I've been paying someone else to mow my lawn for years now, because I'm allergic. Not to grass clippings, but to hard work. It comes from growing up on a farm and then discovering that hard work doesn't actually get you anywhere. If hard work was all it took to succeed in life, then sharecroppers would be millionaires. But I digress. Back when I did mow my lawn, I knew every contour, every corner, every bump in my yard, and I could plan the mowing thereof accordingly. I had precise knowledge of where the boundary lines were (I helped survey them when I bought the place, because I could). Now, however, some guy of indeterminate immigration status comes in on one of those lazy-ass standing riding mowers, gives the lawn a buzz cut along with parts of the neighbors' yards (though they haven't complained yet), bills me $50 and buggers off.

It's still worth it to me to avoid having to give the yard a haircut when it's 100F and humid out there, which here in Virginia we call "all summer."

I also outsource a lot of the housecleaning, for much the same reasons except it's usually not 100F inside the house.

One of the other non-monetary costs of outsourcing stuff is harder for me to describe. I don't know what to call it, so I'll give an example. Say your toilet overflows. It happens, and it sucks. If you don't deal with it yourself, you're telling someone else that you think so little of them that you would pay them to clean up your literal shit. "I'm more important than you are, so here's a wad of cash if you just wade through my sewage for a couple of hours."

I guess that's where the guilt comes in. You gotta clean up your own shit and not expect other people to do it for you. Another lesson from Mom and Dad. Mostly Dad, after I stuffed way too much toilet paper down the commode as a stupid kid because all kids are stupid and that's the kind of thing they do. Okay. Lesson learned.

Other things, I simply don't want to pawn off. Like taking care of my cats. Pets are responsibilities (tell that to whatever neighbor let their husky wander onto my deck last night). But would they really be "my" cats if I made someone else do the feeding, hydrating, and litterbox cleaning? No, that's part of living with cats, and while I'll arrange for someone else to do all that if I'm gone for more than a day, it's not something I'd outsource when I'm home, even if I were disgustingly rich.

Driving is another thing. I really want a driver to help me in my goal of visiting All The Breweries, but it's not like I can afford a full-time chauffeur or anything. Besides, even if money or guilt weren't involved, I just like to drive. When I actually have a car, anyway, which is not now.

Cooking? Nah. While I'm entirely too lazy and single to bother doing complicated meals, I find the only way to get food that I really like is to fix it myself. Or go to certain restaurants, but again, that's unsustainable as a habit. I might only be making a ham sandwich, but dammit, I'm making it my way, and no, I don't need someone to cut the damn crusts off; what even is that bullshit anyway?

After all that, I can only think of one chore that I'd outsource under the conditions in the prompt here, and that's landscaping.

I don't mean lawnmowing; like I said, I already pay someone for that, and they do a decent job. I'm talking about the rest of the yard: flowering plants, shrubs, trees, keeping the poison ivy away; that sort of thing. I hate doing it myself because a) it's work; b) it's outdoors; c) I don't have the slightest idea what I'm doing and it always ends up looking like shit again and d) because there are plants involved, everything I want to keep alive dies, while everything I want to kill (like the poison ivy) gets bigger.

You'd think someone who spent his childhood on a farm would know how to keep plants alive, but nope, they take one look at me, determine that they can't escape because they're plants, and simply commit suicide as their only means of getting away from me, like when a cat catches a rabbit. Again, except for the poison ivy and Virginia creeper, both of whom have a nice laugh at my expense. I can actually hear them growing sometimes, laughing the whole time.

So that's a long way of saying that I'm pretty happy with my current chores and the delegation of some of them, but man it would be nice to have an actual landscaper.

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