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Thursday, Oct. 17, 2024

11:27pm EDT

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Complex Numbers

by Spirit HalloWaltz

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:

Wordsmitty ✍️

a day ago
In response to "Can We?"

"corrugated ridges add strength " - and this is so you can't use your heel to crush it like you do a beer can.

1 Like

The Ghost of Jayne's Lost Sock

a day ago
In response to "Can We?"

The top and bottom seams fail first. *As reported by a friend who used to play "can bowling" with their coworkers (overnight shelf stockers at a discount grocery chain)

2 Like

Annette

a day ago
In response to "Can We?"

By the way:

"A significant new market for the can had become apparent after 1935 when, following the development of suitable internal coatings, the first beer cans were test marketed."

Evo lution of the tin can

1 Like

Annette

a day ago
In response to "Can We?"

As much as all the modern people decry canned foods, there seems to be a fierce competition on who gets to take credit for inventing tin cans.

In a small museum in the country side in France, I learned that French scientists finally achieved to retro-engineer the exact alloy for tin cans sometime in the 18th century. They claimed to have cracked a German secret.

This article that you quote credits an English person.

Funniest part is that I was convinced it was an AmeriCAN invention.

1 Like

Mummified_Mike

Oct 15, 2024 at 10:45am
In response to "Pushing the Button"

I wonder how many of the NASA guys involved in making the Apollo 11 decision had already read Crichton's The Andromeda Strain - published about 2 months before the Moon mission - and allowed it to influence their decision-making process?

1 Like

🌕 HuntersMoon

Oct 14, 2024 at 10:13am
In response to "DeTECHtion"

I had a B.S. detector but, with the election season here, it was more like a smoke detector with a weak battery - It just went on and on and on...

2 Like

Kåre Enga in Montana

Oct 13, 2024 at 6:18pm
In response to "Spit in the Ocean"

Worthy of a long comment. Yes, I could; but, I won't.

As for your list... I'm a bag of mixed nuts.

1 Like

WakeUpAndLive️🦇Elf'sSpell

Oct 13, 2024 at 11:48am
In response to "Revisited: "I Wood Knot""

1 Like

Elisa: Stik or Treat

Oct 12, 2024 at 10:46am
In response to "A Few Observations"

Oddly enough, I've encountered similar unisex style restrooms in the US. If you wander back up to Minnesota and go to the old Schell's brewery in New Ulm, they have them. Mi amor and I joke that they're likely the most progressive bathrooms in the entire (rural) county.

1 Like

Spirit HalloWaltz

Oct 12, 2024 at 10:13am
In response to "A Few Observations"

Elisa: Stik or Treat

, oh, yeah, I'd intended to mention those things but forgot.

On accessibility: Most of the places I visited were old; in some cases, older than the US. Spiral stairs, tiny passageways, multiple levels, and those blasted cobblestones. Difficult, if not impossible, to retrofit enlightened disability access features. The few places that were more modern did it better. That's one downside to the older stuff, I think.

Restrooms: tiny, few stalls, mostly unisex. The unisex thing doesn't bother me, and I expect it better accommodates trans people... but not disabilities. I saw pay-to-poo restrooms, but I managed to avoid having to use them.

Now, some people unfamiliar with Europe might wonder how the unisex bathrooms work. There's usually one or two urinals tucked behind a corner or something, but I imagine it might be shocking for some folks from the US to peer around a corner and see some dude standing at one. But the view is from the back, and it's not like you see anything important and/or disgusting. Also, the stalls don't have those flimsy walls with big gaps and cracks; each toilet stall is usually (for the ones I used) completely enclosed, with a locking door, and private. The hand-washing facilities are shared, and sometimes not even within the restroom itself. It is entirely possible that I missed some etiquette thing, but in general, one merely needs to acknowledge that, shockingly, we're all humans and we deal with waste in much the same ways. Basically, it's just different if you're used to the US (and mostly UK) segregated style, but it's not privacy-killing.

What was more shocking to me was the near-complete absence of bidets, even in hotel rooms.

October 15, 2024 at 7:42am

#1078334

Pushing the Button

This BBC article (whose headline resembles a Cracked headline) is a few years old now, but all that means is they might be missing a few recent ones.

The moments that could have accidentally ended humanity

In recent history, a few individuals have made decisions that could, in theory, have unleashed killer aliens or set Earth's atmosphere on fire. What can they tell us about attitudes to the existential risks we face today?

Already, I have a bad feeling about this. Sounds like an Agenda, which I don't expect from the BBC.

In the late 1960s, Nasa faced a decision that could have shaped the fate of our species. Following the Apollo 11 Moon landings, the three astronauts were waiting to be picked up inside their capsule floating in the Pacific Ocean – and they were hot and uncomfortable. Nasa officials decided to make things more pleasant for their three national heroes. The downside? There was a small possibility of unleashing deadly alien microbes on Earth.

No. No, there wasn't.

They thought there might be such a possibility, so I understand the caution, but in hindsight, there was never any danger from that.

A couple of decades beforehand, a group of scientists and military officials stood at a similar turning point. As they waited to watch the first atomic weapon test, they were aware of a potentially catastrophic outcome. There was a chance that their experiments might accidentally ignite the atmosphere and destroy all life on the planet.

Again: no, there wasn't. They thought there might be such a possibility, and they went on with the tests anyway, which says more about humans than I care to admit.

This sort of writing demonstrates some sort of fallacy that, well, I don't know what it's called, and it's hard to back-search these things.

The people in those cases were acting on imperfect knowledge. I mean, we all are, all the time, but not usually with imaginary stakes so high.

Let me maybe give you an example. You're walking through the woods one day, and you come across a pedestal with a single, big, red button on it. There's no writing, no pictograms, nothing to indicate what (if anything) pressing the button might do. From cartoons, you know that pressing the Big Red Button is generally a Bad Idea, but those are cartoons, not reality.

You're human, and you're faced with a mysterious button in a strange location to find a button. The urge to push it is real, palpable. How can you, curious ape, just... walk past it? You can't. Billions of years of evolution can't be overcome by the moral lessons in cartoons, not easily. Maybe it opens a secret cache with 10 million dollars in nonsequential, unmarked bills. Maybe a genie would come out to grant wishes, and you could wish for more than $10M. Okay, that's unlikely. Maybe nothing will happen. That seems most likely. Maybe a piano will drop onto your head when you press it, killing you. Or maybe it connects to the gigantic planet-exploding device hidden deep in the core. Wow, that's really unlikely.

What you're doing, whether you realize it or not, is risk assessment. Each possible event has a probability. Each possible event also has a desirability. Assigning probabilities is tough, and it's even tougher without Bayes' Theorem to help you assess the probability through the lens of prior knowledge. It's also very difficult if you have no way of knowing what every possible outcome is. But the point is, if you can do that, even roughly, and multiply the probability by the desirability, you can more easily rank your choices. Finding a stash of money? Low probability but desirable. Blowing up the planet? Really low probability, but undesirable.

In the end, billions of years of evolution win out, and you press the button.

Only through hindsight can you really know if you made the right choice or not.

Getting back to the examples from the article, yes, of course, I'd heard about them before. Take the astronauts thing. High probability: no deadly space microbes. Result: no disaster. Low probability: deadly microbes. Result: Astronauts die, Earth saved. Or, Result: Everyone dies. That last one would be Bad, so I don't blame them for hedging with the "deadly microbes" scenario.

It's kind of like the Trolley Problem, only your action shifts the trolley onto a random track where a random number of people are tied.

I don't know if I'm explaining all this very well, but my main point is, the article does a really bad job distinguishing "what we thought the probability was at the time" and "what the probability actually was, in hindsight."

The remainder of the article does a much better job explaining risk and our reaction to it, but the way it's described at the beginning stuck in my craw, so I could no more resist posting about it than you could resist pushing that Big Red Button.

2 Like • Comment • View 1 comment

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