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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 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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

March 14, 2024 at 9:54am

#1066257

Current Events

Remember Rebecca Black's squirmy earworm song, "Friday?" No, I will not link it. Well, this morning, I was tempted to do a parody in honor of today called "Pi Day," but my cats talked me out of it, so we all get to keep our sanity.

Unlike this guy, apparently. From Big Think:

Tesla’s pigeon: How the great inventor fell for a bird
"She understood me and I understood her. I loved that pigeon.”

The main trouble with Nikola Tesla's legacy is that his name was appropriated by a mediocre band, and then run into the ground by one of the most successful con artists in history. The only way to reclaim what shouldn't be a laughingstock, I think, is to remember the inventor's life, accomplishments, and, yes, even his apparent madness.

On a February morning in 1935, a disoriented homing pigeon flew into the open window of an unoccupied room at the Hotel New Yorker.

That hotel, though it's been through a few changes over the decades, still exists. I've stayed there. I don't think the windows open anymore; with air conditioning, it's not necessary, and maybe it prevents some jumpers.

While management debated what to do, a maid rushed to the 33rd floor and knocked at the door of the hotel’s most infamous denizen: Nikola Tesla.

When I was there, I even made a pilgrimage to the 33rd floor.

“Dr. Tesla … dropped work on a new electrical project, lest his charge require some little attention,” reported The New York Times.

"Charge?" They just couldn't resist, could they? Revolting how some people just plug in the most obvious puns.

Nikola Tesla—the Serbian-American scientist famous for designing the alternating current motor and the Tesla coil—had, for years, regularly been spotted skulking through the nighttime streets of midtown Manhattan, feeding the birds at all hours.

He invented way more than that. Some say, rather poetically, that he invented the 20th century. While a bit hyperbolic, it's not that far off the mark.

In the dark, he’d sound a low whistle, and from the gloom, hordes of pigeons would flock to the old man, perching on his outstretched arms.

Look, all I'm saying is, that's a remarkable image and if someone hasn't painted that, someone definitely should.

Tesla said that he and his bird could speak to one another mind to mind, and that sometimes, as they silently conversed, beams of light would shoot from her eyes.

This is the sort of thing I meant by "madness" above. But is it really? Or was he operating on a different level of reality? With genius like Tesla's, there's always that seed of doubt: maybe he was right, and it's the rest of us who are blind.

Tesla’s love of pigeons was an obsession with a capital O. Likely followed by a capital C and a capital D. He seems to have suffered from obsessive compulsive disorder, and his case was severe.

Anyone with only a passing knowledge of Tesla's life, and of pop psychology, would immediately jump to the same conclusion. While it can be dangerous and rude to diagnose someone from some distance in space and time, all the signs were absolutely there. I'd even throw in the possibility of autism.

I won't quote more from the article, which is fairly long, but, and I can't emphasize this enough, beautifully written. It weaves quite the tapestry of history and science, and, of course, there's a bit of Mark Twain in there.

This is where I'd usually relate the subject of the article to Pi Day, but all I could think of was the relationship between pi and the sine waves of electrical current, but maybe that's a bit too obvious while at the same time too esoteric. So I'll just leave it at that and go find me some pie.

6 Like • Comment

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