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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 an hour 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.

Simply Me

about 6 hours 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)

🌕 HuntersMoon

a day ago
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

1 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.

September 10, 2019 at 12:04am

#965912

Facts

PROMPT September 10th

Today’s prompt was submitted by LdyPhoenix

!

What are some of the weirdest facts you've ever learned? Wow us with your bizarre knowledge!

Today's vocabulary word, boys and girls, is epistemology.

Essentially, that's the study of how we know what we know.

I bring this up because possibly the weirdest fact I ever learned is that we don't know how we know what we know. If that makes any sense. If it does, then congratulations; you're eligible to become a philosopher.

"Fact" is a slippery concept, and I don't just mean for politicians. Most concepts are slippery, I think, if you examine them too closely. It's like... you can see a cat and identify it as a cat, but when you zoom in really close, all you see is the fur. And fur could belong to any number of animals. And what's an animal, anyway? At the smallest scales, the lines get blurred.

This is especially true if you keep zooming in. Eventually, you get to molecules and atoms, and then protons, neutrons, electrons - what we commonly call the building blocks of matter. Zoom in further on a proton or a neutron, and you get to quarks, which are more like electrons. What I mean by that is, not only do we not know where a particular electron or quark is at a given moment, we can't know where it is at that moment. It's all smeared out in a probability distribution.

At that scale, even the distinction between matter and energy is fuzzy. And also the distinction between "is" and "isn't."

How, then, can we know anything?

As I've mentioned before, I only claim to know anything to some degree of probability. I live in Virginia, so I'm close to 100% certain that Virginia exists. But then, what is this concept of "Virginia?" An area enclosed by a surveyed boundary, probably, or perhaps the political system, or its checkered history.

I've been to New York City, so I'm pretty sure it exists, too, though less certain than I am of Virginia. And I've never been to Afghanistan, so I have to take other peoples' word that it exists. I don't have a problem doing this. But if you came up to me and told me you've been to the lost continent of Atlantis, I'm going to need more in the way of confirmation.

So it is with anything we take as "fact."

At some point, to avoid falling down the rabbit-hole, you have to put things in different boxes: "reality," "not-reality," "save for further investigation." But we all tend to put different things into those boxes. Flat-earthers come to mind; the idea of a somewhat spherical Earth goes into the "not-reality" box for them (unless they're just fucking with us, which is a possibility we have to consider). Meanwhile, sane people put the round Earth into the "reality" box. Other people might accept as absolute truth the literal existence (as opposed to the literary existence, which is largely undisputed) of fairies.

Science helps with that sort of thing, but one has to be open to its findings, as well - though science, and science reporting, are not without their problems.

So I don't know if that qualifies as "bizarre." I think it does, if you give it real thought - the idea that there might always be limits to our knowledge, and that facts are actually probabilities.

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