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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.

December 6, 2021 at 12:05am

#1022813

Astronomically

I just wanted to share this link at some point, and today's the day it came up. I've always loved astronomy, and I found this informative. Also, it's timely right now, because there's a comet

that you might consider viewing, if you don't mind the cold.

How to Find the Best Stuff in the Night Sky From Absolutely Anywhere
A beginner’s guide to admiring stars, planets, and satellites—no mountaintop or fancy gear required.

My utter hatred of being cold is one thing that kept me from pursuing professional astronomy. As you'll see in this article, sky watching is better when you're cold, dry, and high up. I can handle dry and elevated. Cold repels me.

There was a night many years ago when most of the planets were visible in just one quadrant of the sky. Just after sunset, one could view (if I recall correctly) every classical planet except Mars, between the horizon and the zenith. We went up to the Blue Ridge to get a nice view. And we froze our asses off, but got to see Jupiter's moons and Saturn's rings through a telescope.

You’ll also want to figure out what’s going on with the lunar cycle, because the moon can either improve or obscure your view, depending on what you want to see. If you’re looking for the Milky Way, for instance, or faint, distant objects, your best bet is to wait for a moonless night. If you’re planning to look for something on the moon itself, though, you probably want to do so when it’s full.

I'm going to quibble a bit about this one. When the moon is full, some of the central features can get washed out, because they're not casting shadows. It's like high noon here on Earth -- short shadows, which can make it difficult to see contours. There's a lot to see when the moon is in an intermediate phase, especially around the terminator. Without a telescope, the lunar terminator looks like a smooth curve (or line if it happens to be a half-moon). Thorough even binoculars, though, you start to make out details: some crater floors darken before the rims, and mountain tops can be visible even on the dark side.

This page

explains it better, and with helpful images.

I don't really need to quote more of the article; it really is a good beginner's guide if you're even slightly interested in astronomy. Click through to the links for more information; there are apps there that can really help in locating night sky objects.

I once spent a week up in the Rockies (cold, dry, high) with a professional astronomer, and we saw some really cool stuff. That was expensive, but as the article notes, you don't have to travel to get some nice views.

One more personal note: when the eye doctor was talking about new lenses for my eyes (actual eye lenses, not glasses), he said something like, "...and with this kind, you can change focus to near or far, but the downside is that small lights can get halos around them. This isn't usually a problem unless your hobby is astronomy."

Me: "Well, actually..."

Still, the advantages of these lenses outweigh the disadvantages. I'll still probably be able to see stars and planets better than I do right now (I see blurry fireworks instead of lights). But it's likely my stargazing days are behind me. That's okay; I've seen what I can see. The only thing left on my list is viewing the planet Mercury. I have never seen it and known it for sure. It's possible that I mistook it for a star, as it tends to be near the horizon and small. The other visible planets are really obvious, once you know what you're looking for.

So with this, it's my hope that more people will look up at night sometimes. There really is nothing else like it.

5 Like • Comment • View 4 comments

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