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Saturday, May. 4, 2024

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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 21 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.

August 19, 2021 at 12:02am

#1015838

Curiosity

Scientists have studied cats. This is not news. (Mews?) But these scientists studied why cats like to settle into boxes.

What cats’ love of boxes and squares can tell us about their visual perception
"Vision has evolved to answer questions having to do with boundaries and contours."

Some things about cat behavior might make sense if you think about it. Like how their vision is adapted for lower light levels, and they're often active at night. But the box thing? It's not like cats and boxes evolved in parallel. There were cats. And then there were humans. And then the humans made boxes, and cats lived happily ever after.

It is a truth universally acknowledged—at least by those of the feline persuasion—that an empty box on the floor must be in want of a cat.

Gotta love articles that start out parodying Jane Austen.

This behavior is generally attributed to the fact that cats feel safer when squeezed into small spaces, but it might also be able to tell us something about feline visual perception.

Okay, that's not so obvious.

That's the rationale behind a new study in the journal Applied Animal Behaviour Science with a colorful title: "If I fits I sits: A citizen science investigation into illusory contour susceptibility in domestic cats (Felis silvers catus)."

That title. THAT TITLE.

I am compelled, however to point out that the species nomenclature there is both wrong and outdated. From Wikipedia:

In 2003, the International Commission on Zoological Nomenclature ruled that the domestic cat is a distinct species, namely Felis catus. In 2007, it was considered a subspecies, F. silvestris catus, of the European wildcat (F. silvestris) following results of phylogenetic research. In 2017, the IUCN Cat Classification Taskforce followed the recommendation of the ICZN in regarding the domestic cat as a distinct species, Felis catus.

If you click on the link in the article to the actual paper, you'll see they spelled "silvestris" correctly (it's from "woodland" not "silver"), but the date is this year so they should have left that part out entirely.

This is apropos of nothing, really, but when I see a mistake like that I immediately wonder what other errors they're propagating. But I can't be arsed to confirm and cross-reference everything. I mean, look, it's a study on why cats like boxes; it's not, like, the latest epidemiological research or some shit. Just bask in the knowledge that someone managed to get a scientific paper published about the symbiosis between boxes and cats.

Now, I'm finding it difficult to find other sentences in the article that make sense out of context, and there really aren't any; the text builds upon itself. So I'll just leave it at this, but I definitely encourage you to click on the link. The text is pretty short, but it explains everything well; still, if you can't be arsed to read it, it's worth going there just for all the pictures of good kitties.

Why do we study such things when there might not be any direct rewards to reap from them? Well, first of all, you never know what line of research might yield something useful; for instance, I can see how this might have AI applications. And second, not everything has to be useful; sometimes we're just satisfying our own curiosity.

And in this case, fortunately, curiosity didn't kill the cat.

5 Like • Comment • View 3 comments

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