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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:

GeminiGem of House Lannister

a day ago
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 Udon Thani

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 Udon Thani

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.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

November 1, 2018 at 1:29am

#944617

Labyrinth

Ever wonder if knowledge inhibits creativity?

I have.

I remember once when I was a young child, there was an illustration in a children's book that featured a night sky through a window. I don't remember the context or anything else about the book, but I do remember a crescent moon and stars against a dark blue background seen through the classic rectangular grid of a double-hung window.

Or I might be remembering some details wrong. Memory is like that. But what I am certain of is that there was a stylized star between the horns of the crescent moon.

"That can't happen," I told the teacher, or babysitter, or caretaker, or whatever. "The moon is still there; it's just dark. It would block the star." Okay, I probably didn't use a semicolon, but you can't know that for sure, because I spoke it, and it's not like I would have vocalized the punctuation.

"It's just a drawing," the adult explained. Or something to that effect.

That's the first time I remember using knowledge I'd gained to revoke someone's artistic license. I've done it countless times since then, most often when some science fiction show depicts the asteroid belt as a dense field of tumbling rocks. In reality, the view from any given place in the asteroid belt might feature one or two extra points of light, but is otherwise identical to the view from, say, the other side of the moon. Asteroids are widely separated. If they're not, they turn into smaller asteroids which then become widely separated. The result is barely even a hazard to astrogation, and you're not going to be able to use it to lose the bad guys who are chasing you.

And yet, such a depiction immediately conveys "asteroid belt" to the viewer, just as stars - misplaced or not - and a crescent moon on a dark blue or black background says "night."

Knowledge is like a labyrinth, in a way. A really big and convoluted one. Fixed, but with dark, twisty corridors and branches that lead nowhere, but must be traversed anyway. It has a floor, and walls, and a ceiling, and even adjoining spaces might be miles apart as one traces one's path. It occurred to me recently that the idea of such a maze might have even been inspired by the naked human brain, with its lobes and crevices - the brain, of course, being the container of knowledge.

The classical labyrinth housed the Minotaur, a bull-man chimera with a penchant for munching on humans. The Minotaur never made sense to me, either. Even if you allow for divine intervention or genetic engineering, the circulatory, muscular, nervous, lymphatic, and other bodily systems of human and bovine are simply incompatible. Not to mention that, in its classic depiction as a human figure with a bull head, its teeth would be that of a ruminant: dull, flat, and entirely unsuitable for chewing on any kind of meat - long pig or otherwise.

If the labyrinth represents knowledge, the Minotaur would have to represent ignorance. Fake news. Unfounded fears; the monster in the shadows. Illusions.

Theseus had to a) defeat the Minotaur and b) escape the labyrinth. For the first task, he had a sword. Swords are extremely useful for such things, but only if you know how to use them. For the second task, he had a ball of string helpfully provided by some chick who probably realized that, while Theseus could have simply marked his path with the sword so that he could follow the marks back out, doing so would reduce the sword's effectiveness at its primary task; to wit, turning the Minotaur into prime rib.

I think that part of the story was put in there to illustrate that Cretan women were smarter than Athenian men. But I digress.

It occurs to me that Theseus could have accomplished both tasks if, instead of a sword, he'd carried a sledgehammer. Go in, find the Minotaur, bash it between the horns, and then knock down some walls. But I suppose that would be cheating.

Hang on, though - isn't creativity all about cheating? Did Alexander the Great cheat when he used his own sword to cut through the tangle of the Gordian Knot? Did Captain America cheat when he knocked down the flagpole to earn a ride in the Jeep?

As I mentioned above, two adjacent spaces in a labyrinth could be very far apart if one traverses the maze like a good little lab rat. But let's give the lab rat some supplies - say: a) a chocolate-chip cookie; b) a paper clip; and c) a toy truck. Rats are both smart and creative, so can our Theseus-rat MacGyver his way out of the labyrinth?

This is where creativity comes in. It turns out that each chocolate chip is actually a concentrated block of C-4. Go to a wall, press one of the chips to it, stick one end of the paper clip in the chip, attach the truck's ignition to the other end, and use the starter motor on the truck to ignite the explosive. Boom. No more wall. Step through, see another wall. Pull out another chocolate chip. Repeat as necessary, and you're out. As a bonus, you can even eat the cookie when you're done. And maybe even the truck.

There, now - just because something is absurd doesn't make it impossible, you know. Especially if you're a rat - or a writer.

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