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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 20 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

a day 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

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

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.

March 13, 2021 at 12:01am

#1006274

Crow Magnon

We've all known for a while that corvids are smart (and smart-asses), but it's always good to have some science backing things up.

Brainiacs, not birdbrains: Crows possess higher intelligence long thought a primarily human attribute

As usual, I have a few quibbles with the article, but I think it's generally effective in communicating what's going on.

Whether crows, ravens, and other “corvids” are making multipart tools like hooked sticks to reach grubs, solving geometry puzzles made famous by Aesop, or nudging a clueless hedgehog across a highway before it becomes roadkill, they have long impressed scientists with their intelligence and creativity.

Don't know why they had to put the quotes there. And it's not only scientists who are aware of these bird's big brains -- anyone who's watched them long enough can tell you they're scary smart.

Research unveiled on Thursday in Science finds that crows know what they know and can ponder the content of their own minds, a manifestation of higher intelligence and analytical thought long believed the sole province of humans and a few other higher mammals.

1) This article is dated last September. "Thursday" doesn't help much now.

2) I've known some humans who can't "ponder the content of their own minds."

3) The author uses the adjective "higher" here and in a couple other places. I don't like it. "Higher" implies a hierarchy that can be misleading, or it could mean "more intelligent," in which case it's redundant in an article talking about intelligence. No species is really more advanced than any other, in general; they've merely evolved different survival strategies, only one of which is what we'd call "intelligence."

Knowing what you know is also a form of consciousness, and the discovery that more and more nonhumans seem to have it raises tricky questions about how we treat them.

Unless they're also delicious, like pigs.

The article goes on to describe, in detail, an experiment that demonstrated the cognitive abilities of crows, and another that studied the neuroanatomy of sky rats. Er, I mean, pigeons. I have to admit I kinda zoned out on this part, more interested in the conclusions than the methodology.

“Besides crows, this kind of neurobiological evidence for sensory consciousness only exists in humans and macaque monkeys.”

Snort. He said "macaque monkeys." Snicker.

“In theory, any brain that has a large number of neurons connected into associative circuitry … could be expected to add flexibility and complexity to behavior,” said Herculano-Houzel. “That is my favorite operational definition of intelligence: behavioral flexibility.”

I do wonder about this, but the woman quoted there is a scientist and I'm just an onlooker. Either way, though, I find this sort of thing fascinating. It's like how a crow can't resist a shiny thing, or how a raven can't resist annoying an angsty poet.

5 Like • Comment • View 5 comments

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