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Tuesday, May. 7, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

🌕 HuntersMoon

a day ago
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

March 25, 2021 at 12:01am

#1007019

Certain Things

Not only is this one ancient now (2014), but it's been hanging out in my queue like panhandlers outside of McDonald's for several months now.

Planning Your Future Is Pointless. The How And Why Of Embracing Uncertainty
Wondering what the future holds is a tough question at any age. Instead of trying to figure it all out, get comfortable with the discomfort of uncertainty.

Look, I'm all for embracing uncertainty, but "planning is pointless?" Clickbait? Or something worse?

Life doesn’t go according to plan, and while a few people might do exactly what they set out to do, you never know if you’re one of those. Other things come along to change you, to change your opportunities, to change the world.

I've found the trick lies in retconning your ambitions. Like, say you're a teenager and want to be a doctor. You end up running a backhoe at the local landfill (nothing wrong with that; it's just about as far from "doctor" as I could think of). So you retcon Teenage You into someone who always wanted to run a backhoe.

Hey, it worked for me; I never wanted to be an astronaut.

So if you can’t figure out the future, what do you do? Don’t focus on the future. Focus on what you can do right now that will be good no matter what the future brings. Make stuff. Build stuff. Learn skills. Go on adventures. Make friends. These things will help in any future.

Nothing wrong with any of those activities, but screw "don't think about the future."

One of the most important skills you can develop is being okay with some discomfort. The best things in life are often hard, and if you shy away from difficulty and discomfort, you’ll miss out.

On the contrary, I find that the best things in life are free, easy, and instantly gratifying. If you work hard, something will come along to change something in yourself or your environment, and all your hard work won't have paid off. Still gotta think about the future, though.

Learning is hard. Building something great is hard. Writing a book is hard. A marriage is hard. Running an ultramarathon is hard. All are amazing.

I've never run an ultramarathon, fortunately, but I've done those other things. I wouldn't call any of them "amazing."

Try writing a blog or meditating every day.

Um, hello.

If you’re good at discomfort and uncertainty, you could do all kinds of things: travel the world and live cheaply while blogging about it, write a book, start a business, live in a foreign country and teach English, learn to program and create your own software, take a job with a startup, create an online magazine with other good young writers, and much more.

You can also do all those things if you're independently wealthy. I don't want to travel cheaply, though. I want to experience the finer things, like great wine and five-star hotels.

All of this is useless if you can’t overcome the universal problems of distraction and procrastination.

Welp, I'm boned.

Learn about how your mind works, and you’ll be much better at all of this. The best ways: meditation and blogging. With meditation... you watch your mind jumping around, running from discomfort, rationalizing. With blogging, you are forced to reflect on what you’ve been doing in life and what you’ve learned from it.

Nah, it's more fun to blog about peoples' deadline-driven articles. As for meditation, I've tried it; I fall asleep every time.

Speaking of which, have you ever had -- I don't know what it's called, but the exact polar opposite of a nightmare? Like, in a nightmare, you wake up sweating, heart racing, maybe even crying out in terror (or in my case, being unable to cry out in terror because no voluntary muscles work). With what I'm talking about, you wake up at utter peace with yourself and the world. And get your mind out of the gutter; there was nothing sexual about the dream, though I don't much remember what it was about. Mostly I just remember the feeling of great well-being, the opposite of the fear that a nightmare engenders.

It's happened to me a few times, most recently just two nights ago. Since there's not a name for it as far as I know, I have no idea if others experience that sort of thing.

I don’t think money is that important, but making money is difficult.

On the contrary, I think money is incredibly important. Enough money and you can basically tell everyone who's trying to tell you what to do to fuck off. You can set your own schedule. You don't have to worry so much about where your next meal is coming from. You don't have to spend time and energy on trying to find the cheapest everything (though it can be fun to do so). Most of all, if you get sick, you can actually get medical treatment (this last bit only applies in backwards countries). That old canard about how money can't buy happiness? Maybe, but it can buy beer, and that's close enough for me. It brings its own set of problems, of course, but they're better problems to have than "oh shit I can't afford this year's rent increase."

Protip: save an emergency fund, then start investing your earnings in an index fund and watch it grow over your lifetime.

Oh, look, one gold nugget amidst all the pyrite.

So, I don't know. The article is obviously aimed at someone much younger than me, and the world has moved on since he and I were starting out. As hard as things were in the 80s, they're exponentially tougher now. I think the basic advice has some merit, but not everyone's cut out for a life of pure uncertainty.

And yet, there's enough uncertainty that we should definitely plan for it.

Good luck with that.

2 Like • Comment • View 4 comments

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