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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 8 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

about 23 hours 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 26, 2022 at 12:01am

#1031393

Engage Tractor Beam

Every so often, I'll see an article relating scientific advancement to ideas from science fiction in general and Star Trek in particular. This is one of them.

Star Trek has tractor beams. So do we.
But so far, they can't grab anything bigger than a dime.

And, like all the rest of them, it's disappointing and ends up increasing, rather than decreasing, misconceptions about physics.

Like, a while back, there was a slew of articles about teleportation. Of a single particle, but with all the hype attached to it, I'm sure people thought transporters were just a few years away.

Spoiler: they're not.

This is similar.

The “tractor beam” has been a reliable narrative device in science fiction for nearly 100 years, deployed whenever the plot requires seizing a runaway spaceship or manipulating objects at a distance.

That's right; Trek didn't invent the concept. As an aside, as a kid, the phrase "tractor beam" in Trek confused me. Being on a farm, I knew damn well what a tractor was. It was only later, after four years of Latin in high school, that I finally figured out that it comes from the same root as "attractor."

Okay, maybe it didn't take me that long, but remember, I started watching Trek at a very young age.

Anyway, the root, trahere, means "to pull," and we can already do that mechanically, or with magnets.

But most sci-fi fans probably know it from Star Wars and Star Trek — the sinister Death Star and the mighty USS Enterprise each boasted a frequently convenient tractor beam system.

Ugh.

Star Wars is great, too. I don't take sides. What I do insist upon, though, is that Star Wars isn't science fiction; it's fantasy with SF props.

The first thing to know about real-life tractor beams is that they work more like another sci-fi concept: force fields.

No. No, they really do not. As the article explains, it's sorcery with light and sound waves. The completely fictional idea of force fields is about energy without mass. One could argue, of course, that creating a barrier of focused sound waves would do a lot of what a force field is supposed to do; that is, keep people and things from crossing it.

Anyway, look, I'm not knocking the science, and I'm only knocking the article for being a little misleading up front; the author does go on to explain the technology in some detail (without getting too technical). As the author points out,

We won’t be using tractor beams to yank around Corellian-class smuggling ships, but we could certainly use them to accomplish touchless manipulation of tiny objects in places where precision is at a premium.

The abdominal cavity, for instance. Subramanian says that in the future, acoustic tractor beams could be used to move tiny nanobots inside the body for targeted drug delivery, when a particular medicine needs to be delivered to a particular area of tissue.

That seems to be a very niche application, but I think it's an important one, and pretty cool.

Drinkwater concedes that it will be “a fair few years” before any of this happens — and many more before we can start scaling up to spaceship-sized tractor beams.

Yeah, more like centuries for the latter, absent some sort of massive breakthrough.

Just don't expect to see these at your neighborhood Tractor Supply Store.

3 Like • Comment • View 3 comments

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