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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 1, 2021 at 12:16am

#1014825

If You Choose Not To Decide, You Still Have Made A Choice

Great... it's Hangover Night and the random numbers give me this one. Oh well, I have no choice but to post it. Just don't expect much in the way of commentary.

The clockwork universe: is free will an illusion?
A growing chorus of scientists and philosophers argue that free will does not exist. Could they be right?

Oh hey look, a rare example of the answer to a headline question not being "No."

But because it's philosophy, it's not necessarily a "yes" either.

It isn’t unheard of for philosophers to receive death threats.

Just in case you were thinking, "Oh, I think I'll become a philosopher. Sounds like a nice safe profession."

They argue that our choices are determined by forces beyond our ultimate control – perhaps even predetermined all the way back to the big bang – and that therefore nobody is ever wholly responsible for their actions.

That last bit doesn't necessarily follow from the first bit. Coyne (he gets quoted later in the article) distinguishes between moral responsibility and practical responsibility. But he's a biologist, not a philosopher.

“This sort of free will is ruled out, simply and decisively, by the laws of physics,” says one of the most strident of the free will sceptics, the evolutionary biologist Jerry Coyne. Leading psychologists such as Steven Pinker and Paul Bloom agree, as apparently did the late Stephen Hawking, along with numerous prominent neuroscientists, including VS Ramachandran, who called free will “an inherently flawed and incoherent concept” in his endorsement of Sam Harris’s bestselling 2012 book Free Will, which also makes that argument.

This is a rare case of someone whose writings I follow showing up in a major news publication (yes, I consider The Guardian a major news publication). I've linked to Coyne's blog here before, probably mostly for his (to me) decisive takedowns of panpsychism. He's had quite a bit to say on the subject of free will in there, too.

Arguments against free will go back millennia, but the latest resurgence of scepticism has been driven by advances in neuroscience during the past few decades. Now that it’s possible to observe – thanks to neuroimaging – the physical brain activity associated with our decisions, it’s easier to think of those decisions as just another part of the mechanics of the material universe, in which “free will” plays no role.

And this is why it's more than a philosophical question: there's science backing it up. It's also why scientists get to have an informed opinion on the topic. I should note, however, that it's not necessarily settled science.

Arguably, we would be forced to conclude that it was unreasonable ever to praise or blame anyone for their actions, since they weren’t truly responsible for deciding to do them; or to feel guilt for one’s misdeeds, pride in one’s accomplishments, or gratitude for others’ kindness.

"Arguably" is right, because if there is no free will, we also can't help feeling gratitude, guilt, or pride.

The conviction that nobody ever truly chooses freely to do anything – that we’re the puppets of forces beyond our control – often seems to strike its adherents early in their intellectual careers, in a sudden flash of insight. “I was sitting in a carrel in Wolfson College [in Oxford] in 1975, and I had no idea what I was going to write my DPhil thesis about,” Strawson recalled. “I was reading something about Kant’s views on free will, and I was just electrified. That was it.” The logic, once glimpsed, seems coldly inexorable. Start with what seems like an obvious truth: anything that happens in the world, ever, must have been completely caused by things that happened before it. And those things must have been caused by things that happened before them – and so on, backwards to the dawn of time: cause after cause after cause, all of them following the predictable laws of nature, even if we haven’t figured all of those laws out yet. It’s easy enough to grasp this in the context of the straightforwardly physical world of rocks and rivers and internal combustion engines. But surely “one thing leads to another” in the world of decisions and intentions, too. Our decisions and intentions involve neural activity – and why would a neuron be exempt from the laws of physics any more than a rock?

That's about all I'm going to quote from the article because, like I said, hangover. But that last paragraph pretty much sums up the argument; the rest is commentary. And if you're sitting there going "Well, that's utter bullshit," well, I suggest you go to the link and read the whole thing. I mean, I don't blame you -- you can't help it -- but part of who we are is our experiences and memories thereof, all of which contribute to our decisions, including on what to believe and not. And articles are part of those experiences and memories.

I will say this, though: the opposite of "free will" isn't necessarily "determinism;" the article goes into a brief discussion of quantum effects, which are by nature unpredictable. And then there's chaos theory, which has a specific meaning in physics, having to do with the unpredictability of certain systems, such as the weather, past a certain point. And I'm betting the brain is at least as complex as a weather system.

But the most important corollary, in my estimation, has to do with moral responsibility, and the article delves into that at length.

So there it is. Hell of a way to start a new month, but what can I say? I'm a slave to the random number generator.

Because I choose to be.

6 Like • Comment • View 1 comment

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