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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:

Kåre Enga in Montana

about 14 hours ago
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

about 18 hours ago
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

a day ago
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

Kåre Enga in Montana

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Montana

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

July 26, 2020 at 12:11am

#989146

One True Pizza

Beer should count as food.

PROMPT July 26th

What food would you like to judge in a Cook-Off?

I'd say "sushi," but that would negate the "cook" part of "Cook-Off."

I think the classic cook-off food is chili. There are so many ways to make it, and so many possible ingredients, that there is no One True Chili. I find the argument about whether beans should go into chili or not hilarious. Because food snobbery should only be applied to pizza.

The chili argument apparently goes back to the Mexican origins of the dish: originally, it was a beef stew with chili peppers and without beans of any sort. But if you're going to make the "origination" argument, you have a lot to answer for; few dishes remain static over generations. Food evolves over time, with different cooking and preservation methods. Hell, originally, they didn't put hops in beer; that was a later innovation that both improved the flavor (up to a point) and acted as a preservative. Nowadays, beer is defined as "water, malt, yeast, hops and maybe adjuncts."

So an argument could be made that adding beans to chili is a major innovation that improved the stuff, and the beanless variety should be consigned to the ash heap of history.

Except, of course, as a hot dog topping.

My personal chili recipe involves ghost pepper sauce, because it can. And beans. And tomatoes, another controversial ingredient. I haven't made it in a while, because when I do make it, it tends to disappear quickly, and then reappear around my waist, which is something I've been trying to avoid.

But let's get back to pizza, the world's most perfect food. I've spent my life on a quest for the perfect pizza, and while I've come close, nothing has ever jumped out at me as "perfect." I do have a couple of hills to die on as regards pizza, though.

1. The One True Pizza will be New York style; that crap they call "pizza" in Chicago can be decent food, but it is not pizza.
2. Avocado has no place on or near pizza.

Given the above, you may be surprised to learn that I stay out of "pineapple as a topping" arguments with pizza. Hawaiian style pizza isn't my favorite, but I'll eat it if I'm hungry enough. Shocking news: people like different things.

But, like with chili, there's a lot of room for diversity in pizza. Still, I never will understand the barbarians who don't eat the outer crust of an otherwise good pie. The kind of dough used on pizza should be a kind that could work on its own as bread. The crust is not, as some would have it, merely a delivery system for the toppings; the crust is the heart and the soul of the pizza, and needs to be respected as such.

So, given all this, I'd totally judge a pizza cook-off, the only problem being that there's just no way I could eat just one bite of the competitors. I mean, maybe I could, but then when I picked a winner I'd insist on eating the rest of the pie and, well, see above about my waist.

But at the same time, I'd be judging the other judges. If I see someone wax poetic about avocado pizza, I'd reject every other opinion they have. Similarly, if someone eats everything but the crust, I'll know it's safe to dismiss their rankings out of hand. And if the competitors include deep-dish Chicago style casserole, I wouldn't even need to taste them. I mean, like I said, it can be tasty, but I reject with prejudice the assertion that it has anything in common with real pizza.

Maybe one day I'll finally find the perfect pizza. Just not outside the northeastern US.

Mini-Contest results!

I appreciated all of the comments, though as I feared, they just made me miss traveling more.

Lilli 🧿 ☕

- Can we take a boat from Key West to New Orleans? I'd love to go the Gulf route.

Starlena

- Yay! You get me! I call that nefarious beast the Accursed Daystar, and while it seems to be necessary for the warmth I require, it's best if I stay out of its direct influence. Also, I'd totally go to Greece.

Satuawany

- Texas is a great place to visit (especially for the chili and beer), and I do like RenFaires. Camping, though? *shudder*

Cappucine

- Je voudrais vraiment visiter la France, et je veux boire du vin et manger des baguettes et du fromage, et je veux voir les cathédrales... but hey, Australia sounds good too.

Sumojo

- (speaking of Oz), now, I've been to Britain, but I'm always up for a return trip, especially if there's ale involved.

Elisa the Bunny Stik

- Yeah, it probably would be cheating, because we already had plans to meet up -- which I still want to do, but not with a raging pandemic going on. Patio beer doesn't count as an outdoor activity if there's shade.

Alexi

- Honestly, I've never even considered the Canaries, but you make it sound inviting. Except for all of the time you want to spend in the gaze of the accursed daystar. Still... it sounds like it might actually be worth lathering up with sunscreen to do all that fun stuff.

Spring in my Sox

- I try to keep poo-flinging in political arguments. Which you don't see monkeys doing. They're definitely more civilized than the children at the zoo.

Prosperous Snow celebrating

- Oddly enough, there are two places I go if I want to have food I've never tasted before, and one of them is your hometown. The other is New York City -- though there, I'm usually too busy trying to find the perfect pizza.

Like I said, lots of good suggestions, and I'd be willing to do all of them (yes, even the outdoors ones, provided I can pack bugspray and sunscreen)... but I gotta give the Merit Badge to Starlena this time, because like you said: outdoors shit is overrated.

But I'll do something like this again soon so everyone will have another chance! Thanks again for commenting, everyone.

11 Like • Comment • View 16 comments

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