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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 1, 2024 at 8:39am

#1065309

They Ain't Here

The truth is out there... but you don't believe it. From New York, not to be confused with The New Yorker:

No, Aliens Haven’t Visited the Earth
Why are so many smart people insisting otherwise?

Just to get this out of the way: Smart people can still draw bogus conclusions. They can be fooled. They can be exceptionally good at fooling. I wouldn't give a smart person's opinion more weight, unless it's in a field they're credentialed in, any more than I'd trust a rich person's opinion over that of a poor person.

Last month, Sean Kirkpatrick, the head of the Pentagon office responsible for investigating unexplained aerial events, stepped down.

The article's from the end of January, so I guess that happened in December?

He said he was tired of being harassed and accused of hiding evidence, and he lamented an erosion in “our capacity for rational, evidence-based critical thinking.”

I feel that.

He may have been pushed over the edge by a pair of events from the past summer. In June of last year, Avi Loeb, an astronomer at Harvard, announced that he had found some tiny blobs of metal by dragging a magnetic sled over the bottom of the Pacific near Papua New Guinea. He claimed that these blobs were metallic droplets that had melted off an interstellar object that might have been “a technological gadget with artificial intelligence” — the product of beings from another star system.

That's the same numbskull who claimed that the interstellar wanderer called 'Oumuamua

was a product of technology.

Look, most of the time, I see Avi Loeb's name, Harvard gets mentioned. I think that's an attempt to give him some credentials. But it has the opposite effect on me: instead of being more inclined to believe Loeb's crackpot "it-must-be-smart-aliens" conclusions, I become more inclined to dismiss anything that comes out of Harvard.

In other words, that guy's such a disgrace to that prestigious university that his association with it (along with a few other Harvard-associated people recently) decreases its prestige.

Oh, and sure, random blobs of metal could have come from an interstellar visitor. I don't consider the odds of that to be very high. Claim like that, though, you're going to have to rule out terrestrial origins, first.

Thoughtful, sensible-seeming, non-crankish people at Harvard, at The New Yorker, at the New York Times, and at the Pentagon seemed to be drifting ever closer to the conclusion that alien spaceships had visited Earth.

One of those things is not like the others...

To be clear, we should investigate these claims, within reasonable budgetary limits. Though any debunking that gets done sure won't convince the UFO nuts. On the other hand, show me definitive evidence, and I'll be convinced.

Yet even after more than 70 years of claimed sightings, there was simply no good evidence. In an age of ubiquitous cameras and fancy scopes, there was no footage that wasn’t blurry and jumpy and taken from far away. There was just this guy Grusch telling the world that the government had a “crash-retrieval and reverse-engineering program” for flying saucers that was totally supersecret and that only people in the program knew about the program. Grusch said he had learned about it while serving on a UAP task force at the Pentagon. He interviewed more than 40 people, and they told him wild things. He said he couldn’t reveal the names of the people he interviewed. He shared no firsthand information and showed no photos.

And that, folks, isn't evidence. It's crackpottery, albeit perpetrated by someone who would be in a position to know. It makes me wonder what he was paid to distract us from. Yes, I have my own conspiracy leanings, but they involve human activity, not purported alien.

Now, the article goes on for a while, and it's absolutely worth a look, at least a skim. I won't comment on it further, though, except to say that our collective fascination with the subject says nothing about aliens and everything about humans.

2 Like • Comment

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