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Wednesday, Dec. 18, 2024

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Complex Numbers

by Waltz Invictus

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:

Aiva Raine

about 8 hours ago
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

about 10 hours ago
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

about 18 hours ago
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

Ichabod Crane-- Mince Pie

about 19 hours ago
In response to "Need for Speed"

One of the last jobs I had was with Delta. I did ask the maintanace guys about that tape once and they pointed out high speed will rip duct tape or any other tape right off. They used it personally for their boats sometimes because of its strength. But looking out a window at 400 miles an hour and 30 thousand feet up and see duct tape on the wing. Certainly made me nervous first time I saw it.

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Fork You Too"

Now that I know forks are the center of dining controversy, I'll have to apply this at all family gatherings where demon dining utensils are in use.

1 Like

tj-Merry Mischief Maker

Dec 15, 2024 at 1:06pm
In response to "Revisited: "Those Naughty Brits""

I don't think there's such a thing as kinky sex anymore. Why? Based on the general definition, kinky sex refers to sexual activities and preferences that fall outside standard sexual practices. Nowadays it seems anything goes and there are no norms or standards.

Oh, another thing; do you remember if the link had pictures? Asking for a friend.

1 Like

Wordsmitty ✍️

Dec 15, 2024 at 12:45pm
In response to "Revisited: "Those Naughty Brits""

Waltz Invictus - As they say, you're a more proper man than I. I may even check out the other links with those key words.

2 Like

Waltz Invictus

Dec 15, 2024 at 12:32pm
In response to "Revisited: "Those Naughty Brits""

Thanks for that, Wordsmitty ✍️

. Sometimes I make the effort to find an article from a dead link. This time, I did not.

1 Like

Wordsmitty ✍️

Dec 15, 2024 at 12:29pm
In response to "Revisited: "Those Naughty Brits""

I think we're all glad that society has advanced so far in a decade and a half. Which way is up to you.

For those interested, there's a "Search" option in the Times main page. Page 6 of the results comes up with this link: https://www.thetimes.com/article/4d5f3e63-22c9-45c0-837b-c3ad9d236ae8

2 Like

Elisa: Snowman Stik

Dec 15, 2024 at 9:35am
In response to "Revisited: "Those Naughty Brits""

Damn, another entry that has me wanting to jump in and offer tons of commentary! Let's have at it.

2008 Me: Journalism may not be dead yet, but it's starting to wander and stink.

Dead now. Mostly.

Clearly I need to highlight more non-profit outlets in my link rundowns. I'm finding lots of good reporting on that front. Perhaps I also need to add more examples of this to "Watch This Space!"

[18+]. Even 60 Minutes is still able to provide some good in the weeds reporting. Point is, journalism is struggling, but good reporting is still out there. I'm doing my damnedest to shine a light on the solid examples. I could use a bit more help in the process.

2008 Me: What happened to investigative journalism? Hell, what happened to comprehensive news stories?

Gods, 2008 Me was so young and naïve.

They still exist. Strangely, I have found that at least in my area the local stations actually put some of their most comprehensive stories online. Also, at least the NBC affiliate has a post-local news show that does more comprehensive reporting. Some pieces are fluffy, but others are not. They are more locally focused, but that's probably a good thing these days. Local news has struggled the most, and we're paying the price for that.

2008 Me: Why is this in the "women" section? Men don't want to read about kinky sex? Please.

I'm guessing men are less likely to consider it kinky, outrageous, or naughty. But what the fuck do I know (pun intended)?

Strangely, this seems almost prudish by British standards, especially in light of the fact that Time magazine reported on BDSM in one of its print issues four years prior. And as we know, Time magazine isn't exactly known for sectioning off its content by gender. The Time coverage did include a photo taken in a BDSM dungeon that wasn't too explicit but really captured the D/s component really well. Honestly, I'd expect a UK publication to do that before a US one, not the other way around.

That said, I suspect the articles may have been written for slightly different audiences. If the audience is primarily women, there is likely to be more implicit shame in the picture, as women have been historically more likely to be primed to be ashamed about their sexuality in any capacity. Don't get me wrong. I know men are also subjected to this shaming. However, it tends to be less persistent and less pervasive in terms of different facets being shamed. In a general audience, the shame element can be downplayed. In a women-focused audience, you have to address the shame angle in some capacity, reassuring them that this isn't such a bad thing. This isn't something I'm directly involved in, but given the lack of women on WDC willing to write about this sort of thing these days, I might have to take one exploring this concept in depth. Ladies, if you're thinking about writing something (journal entry, static item, whatever) about this topic, this is your sign to go ahead and do it!

These days, I have preconceived ideas about headline questions: First, if it's a yes/no question, the answer is probably "no."

Snopes is ready to challenge that assumption.

2 Like

December 6, 2024 at 9:37am

#1080874

No Space for Time

The Big Think article that came up for me today involves math. Fair warning so you don't end up defenestrating your device.

Time: Yes, it’s a dimension, but no, it’s not like space
The fabric of spacetime is four-dimensional, with three for space and only one for time. But wow, time sure is different from space!

Time is different from space? I never would have guessed, what with them having different names and all.

When did you first realize that the shortest distance connecting any two points in space is going to be a straight line?

I'm not sure that's a fair question. It's something I'd consider intuitive. What's hard to grasp, sometimes, are the cases where the shortest distance between two points isn't a straight line, because that runs counter to our everyday experience.

In fact, that realization, as far as human knowledge is concerned, comes from a place we might not realize: the Pythagorean theorem.

"In fact," I think they've got this backwards. The Pythagorean Theorem may quantify the "shortest distance" intuition, but I'm pretty sure humans knew about the straight-line thing before they had numbers or geometry. (Also, the idea predated Pythagoras by hundreds or thousands of years; the Greeks didn't invent everything.)

Taking all three of these dimensions into account — so long as we assume that space is still flat and universal — how would we then figure out what the distance is between any two points in space? Perhaps surprisingly, we’d use the exact same method as we used in two dimensions, except with one extra dimension thrown in.

I feel like the only "surprising" thing here is that the math is basically the same.

Thinking about distances like we just did provides a great description of what we’ll wind up with if we consider flat, uncurved space on its own. What will happen when we fold time, as a dimension, into the equation as well? You might think, “Well, if time is just a dimension, too, then the distance between any two points in spacetime will work the same way.”

At which point, unsurprisingly, the math isn't basically the same.

There are two fundamental ways that time, as a dimension, is different from your typical spatial dimension. The first way is a small but straightforward difference: you can’t put space (which is a measurement of distance, with units like feet or meters) and time (which is a measurement of, well, time, with units like seconds or years) on the same footing as each other right from the start.

Feet? Footing? Get it? Haha.

Fortunately, one of the great revelations of Einstein’s theory of relativity was that there is an important, fundamental connection between distance and time: the speed of light.

Yes, and the invariance of that speed is still very, very hard to wrap your head around, because it, unlike the "straight line" thing, runs counter to everyday experience.

However, there’s also a second way that time is fundamentally different from space, as a dimension, and this second difference requires an enormous leap to understand. In fact, it’s a difference that eluded many of the greatest minds of the late 19th and early 20th centuries. The key idea is that all observers, objects, quanta, and other entities that populate our Universe all actually move through the fabric of the Universe — through both space and time — simultaneously.

Turns out that everything in the universe is moving at a constant speed... through spacetime. Once that was pointed out to me, a whole lot of other stuff started to make more sense.

It turns out that the faster (and the greater the amount) you move through space, the slower (and the lesser the amount) you move through time.

Like that, for instance.

There’s an even deeper insight to be gleaned from these thoughts, which initially eluded even Einstein himself. If you treat time as a dimension, multiply it by the speed of light, and — here’s the big leap — treat it as though it were an imaginary mathematical quantity, rather than a real one, then we can indeed define a “spacetime interval” in much the same fashion that we defined a distance interval earlier...

Great. Wonderful. Now we'll get "time is imaginary" on top of "time is an illusion" nonsense. I'd forestall this by pointing out that imaginary numbers aren't actually imaginary (or at least they're no more abstract than the "real" numbers), but that's not going to stop the airy pseudophilosophy.

There is, of course, quite a bit more at the link. I'm not sure if it's really that useful; I feel like there's either too much or not enough math to keep people interested enough to follow the arguments. But if all we can get out of it is "spacetime is weird, and time is different from space," maybe that's good enough.

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