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Monday, May. 6, 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:

🌕 HuntersMoon

about 17 hours ago
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

a day ago
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
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

May 3, 2024 at 11:58am
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

April 2, 2021 at 12:02am

#1007556

Scout Chip

I know I've talked about this before, but here's Cracked's take on it.

5 Facts About That Enigmatic Object That Invaded Our Solar System

A couple years ago, a big hunk of something flew into our solar system. It took a bit longer for the world to really stand up and pay attention to it, and we're still keeping an eye on it, even though it's now billions of miles away now.

You said "now" twice. So it's twice as now?

Hey, look, I'm not claiming I never make mistakes. I also don't have an editor.

5. First, The Most Important Part: What's With That Name?

Just kidding, the name isn't the most important part, but let us set this thing up for you.

In October 2017, astronomers were looking up at the sky, which is their favorite thing to do. They thought they spotted a comet, so they named it, using the standard comet designation protocol. Then they looked closer, saw no hint of a tail, and figured it had to be an asteroid, which meant they instead needed the asteroid designation protocol. But as they got out their notepads and tried to trace its path, they realized it had to have come from outside the solar system. It was the first ever identified interstellar object, which was very exciting, because it meant they had to create a NEW designation protocol.

You know, the way this is generally reported, it makes it sound like nothing ever passes through this solar system from outside it, that this is a once-in-a-billion year happening, that it's unique and precious.

Of course, we don't know how often it happens because we've only recently had the tools to detect that it's a) an object that b) comes from outside the solar system. This was the first detected, not the first ever. Probably. Or the last. Probably.

Cracked, unlike any other article I've seen on the subject, goes into detail about the origin, prononciation, and meaning of its name: 'Oumuamua. So, way to go, dick joke site?

4. We Don't Actually Know What This Thing Looks Like

At the top of the page, and copied again right below these words for your convenience, is an artist's impression of 'Oumuamua:

For which I'm afraid you're just going to have to click on the link. We only get a limited number of image files here, and I'd rather not use up all of mine on rocks.

Scientists, and the media, unanimously refer to this as "cigar-shaped," which is the term they use when they want you to think about dongs but are too afraid to just come out and say "dong."

Way to go, dick joke site.

All this is to say -- and this will become important in a minute -- there's a whole lot of very basic info about 'Oumuamua we don't know. But we have been able to monitor its movement, and that's what's stumped so many scientists.

Surprised they didn't joke about it moving in and out and in and out and in and out...

3. 'Oumuamua Is Moving Very Weirdly

As those astronomers were tracking 'Oumuamua (during the few breaks allowed to them by their droves of oversexed groupies), they realized their interstellar visitor could not have come from the fairly close star system Vega just 600,000 years away, nor could it have come from any of the first several other close alternatives they checked out. The paths just didn't line up. Instead, 'Oumuamua might have spent billions upon billions of years traversing the cosmos before it landed in our solar system.

Which is really kinda cool. By the way, they're not saying Vega is 600,000 light years away; I believe they're implying that at the speed this thing was traveling, it would have taken it that long to get here from the vicinity of Vega. If you want to know for sure, ask a Vegan. (You have NO IDEA how long I've been wanting to make that joke. NO IDEA.)

But that's not the weird part about its motion. The weird part: 'Oumuamua was accelerating. Not just accelerating due to the Sun's gravity, as expected, but also accelerating due to ... something else.

And this, as I mentioned before, inspired a lot of people, including a supposedly respectable dude from Harvard, to start shrieking "ALIENS! ALIENS! AAAAALLLLIIIIIEEENNNNS!" But, as Cracked loquaciously puts it:

2. So, Is This An Alien Spaceship?

No.

2. C'mon. Couldn't It Be Aliens?

Okay, we can't rule out that aliens made it. But if you've heard the theory that 'Oumuamua is an alien spaceship -- and if you heard of 'Oumuamua before today, you probably heard this theory -- you should know that despite all the coverage it got, this idea comes from just one dude. Well, two dudes, counting his paper's co-author, who doesn't seem to chase the spotlight as much. Well, three dudes, if you count the ancient aliens guy who has endorsed the theory.

And since I harped on this before, I'm just leaving this bit here.

1. Let's Go And Find Out For Sure

There's a reason we're still talking about 'Oumuamua today, and that reason is Project Lyra. This group of scientists is considering sending an interstellar ship to catch up with 'Oumuamua, even though it's already billions of miles away and going 500 million miles farther every year.

Are they nuts? No, no they are not. We absolutely do want to catch up with interstellar objects eventually, and 'Oumuamua's a good candidate because at least we know where to find it. The other plan for reaching an object like that has been approved by the European Space Agency and involves sending a vessel into space by the end of the decade and then just chilling for a while till we figure out where to direct it. Compared to that, Project Lyra seems downright reasonable.

The idea that we can chase this sucker with a probe is *almost* as farfetched as the "alien ship" hypothesis.

Before they launch, Project Lyra also needs funds, and personnel, and manufacturing facilities, and lots more. The group behind it (the "Initiative for Interstellar Studies"), they're just some scientists writing a proposal, not a space agency. But we're absolutely rooting for them to come up with a plan that makes sense and to put it in the hands of those who can make it happen. And if they do discover that 'Oumuamua is a functioning spacecraft, we will admit we were wrong, and then we'll commandeer the vessel and fly it back to where it came from.

But we're clever (except for the guys shrieking about aliens); we can find a way.

Because that's what we do. When we're not making dick jokes, anyway.

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