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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

about 12 hours 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

a day ago
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

April 14, 2021 at 12:02am

#1008482

Wagging Tales

Just an interesting tale of tales today.

Fairy Tales Could Be Older Than You Ever Imagined
Jack may have been climbing that beanstalk for more than 5,000 years

Though I'm not sure how solid the foundation for this finding is, it's about storytelling, so it's actually relevant for once.

In a new study published in the journal Royal Society Open Science, a folklorist and anthropologist say that stories like Rumpelstiltskin and Jack and the Beanstalk are much older than originally thought. Instead of dating from the 1500s, the researchers say that some of these classic stories are 4,000 and 5,000 years old, respectively. This contradicts previous speculation that story collectors like the Brothers Grimm were relaying tales that were only a few hundred years old.

Before writing, we're pretty sure people passed down their stories mouth to ear. There's no reason to believe that wouldn't have continued; not everyone in every society was - or is - literate.

It turns out that it’s pretty hard to figure out how old fairy tales are using simple historical data. Since the tales were passed down orally, they can be almost impossible to unwind using a historian or anthropologist’s traditional toolbox. So the team borrowed from biology, instead, using a technique called phylogenetic analysis. Usually, phylogenetic analysis is used to show how organisms evolved. In this case, researchers used strategies created by evolutionary biologists to trace the roots of 275 fairy tales through complex trees of language, population and culture.

While that's interesting and all that, I'd be inclined to want to hear how they justify using phylogenetic analysis for something for which it wasn't intended.

Using the Aarne-Thompson-Uther Classification of Folk Tales, a kind of über index...

More like Uther index, amirite?

...that breaks fairy tales down into groups like “the obstinate wife learns to obey” and “partnership between man and ogre,”...

Both groups being what today we'd call "fantasy."

...the team tracked the presence of the tales in 50 Indo-European language-speaking populations.

It might also be illuminating to use this technique (if it's not found to be fatally flawed) on non-PIE-derived languages. There's a rich culture of fairy tales, or their equivalent, in the East.

But not everyone is certain that the study proves fairy tales are that old. As Chris Samoray writes for Science News, other folklorists are finding fault with the study’s insistence that The Smith and the Devil dates back to the Bronze Age—a time before a word for “metalsmith” is thought to have existed.

Clearly, skepticism is warranted (and encouraged), but in that particular example, so what? The thing about oral traditions is that each generation adds its own spin on the tradition. If environmental, cultural, or (in this case) technological changes occur, they'd probably morph the story to make it more accessible to a younger generation, one that lacks historical context. If you don't believe me, go watch one of the approximately two thousand remakes Hollywood puts out in a year.

Point being, maybe that particular story started out as "The Flint-Knapper and the Evil Spirit." No need for "metalsmith" to have been invented; that would come in after metalworking got going.

So, yeah, I wouldn't take anything here as hard evidence of the age of stories... but we're pretty sure storytelling itself is as old as humanity. Maybe it's when our ancestors started telling stories that we could finally call them "human."

But of course a writer would think that.

5 Like • Comment • View 2 comments

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