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Wednesday, Dec. 11, 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:

Wordsmitty ✍️

about 6 hours ago
In response to "On the Label"

"June 25, 2024 5:40 AM" = Need time zone.

"Note to self:" = self-driving taxis

Some people definitely can be labeled, but we don't say those in polite company.

2 Like

Annette

about 6 hours ago
In response to "On the Label"

Only one label matters. Cat person or dog person. If the answer is anything except "yes" the person's life choices defy all logic analysis.

I wonder what is the difference between psychotherapist, psychologist, and psychoanalysis. Or are they like po-ta-to, po-tuh-two?

3 Like

tj-Merry Mischief Maker

Dec 9, 2024 at 2:51pm
In response to "Analog Log"

As time progresses, it amazes me how we have moved from tide clocks/analog computers to people eating tide-pods.

1 Like

Aiva Raine

Dec 9, 2024 at 2:37pm
In response to "Story of the Century"

Fantastic entry. Have I mentioned how I love your snark? Thanks once again for the grins and your awesome sarcastic responses to the articles.

1 Like

Anni Pon

Dec 8, 2024 at 10:01am
In response to "Revisited: "Millennials Killed Millennials""

It's kind of funny to me that people still use "millennials" to refer to "those darn kids" when most of them are in their 30's now with kids of their own. As an elder millennial (a Xennial, if you will), I haven't really found the generational definitions too useful for myself. I'm right on the border of generations, so I have a lot in common with both Millennials and Gen X. Growing up in silicon valley with tech savvy parents, I had earlier access to computers and technology than a lot of other people my age. But in terms of cultural milestones, I lived through and am old enough to remember things like the 90's, Y2K and 9/11. The borders are sort of arbitrary, but on a population level I can see how sometimes it can be useful to group people into generations.

2 Like

Elisa: Snowman Stik

Dec 8, 2024 at 9:40am
In response to "Revisited: "Millennials Killed Millennials""

*rubs hands together* Time for me to dive right in!

2. In this system, I'm early Generation X, which hardly anyone ever talks about, opting instead to rag on "boomers" or "millennials" or "Gen-Z." Or praise them, depending on the author's beliefs and age.

Looks like I need to change my name to "hardly anyone". Also, there has been more and more chatter about Gen X in some circles, but curiously this seems to be happening in institutions that have dropped labels (which I'll touch on in a second here).

On top of which, you'd have to convince me that our 1983-born X-er has more in common with someone born in 1966 than with someone born in 1986.

Very, very, VERY few organizations that have ever in their history used generational labels have set the Gen-X border any later than 1980. While I'm still very salty that Pew stopped this practice (one of the few things they've done wrong, IMO), I still use their Gen-X demarcation of 1965-1980. That demarcation nicely strikes the median of most demarcations. SO in this case it's not a matter of you needing to be convinced. It's a matter of the author leaning on a REALLY bad source.

Much of what sets "generations" apart, in modern terms, has to do with technological advancements and societal changes...then other people had to think that the original wearer was cool enough to start a fashion trend.

And how that message has been conveyed has drastically changed among generations, to boot.

So you know what I want to see Millennials finally kill?

Generations.

As a Millennial, I can tell you that this is probably the last thing our generation will try to kill. It's something that's persisting even as we reach middle age. And that's another thing. We're aging very strangely, namely that physically we're aging more slowly than people in Gen Z. We could have killed the concept of aging a certain way, but Gen Z's circumstances have effectively torpedoed that revolution.

Also, if anyone wants to read the original article, you can certainly try throwing the URL in the Wayback Machine. It doesn't always work, but it's worth a try. There are some other methods as well, but I haven't tried them myself yet. (Speaking of which, has anyone here tried Zette? I've been meaning to sign up for that but haven't had a chance. It's one of my goals for 2025.)

2 Like

Wordsmitty ✍️

Dec 7, 2024 at 12:41pm
In response to "Story of the Century"

Maybe it helps if you can dance - Dick Van Dyke and Coldplay - although he's just 99. (I assume everyone has read about that)

4 Like

🌖 HuntersMoon

Dec 6, 2024 at 10:07am
In response to "No Space for Time"

I think you just added clarity to the Twilight Zone...

3 Like

tj-Merry Mischief Maker

Dec 5, 2024 at 4:05pm
In response to "How To Give A Fock"

How many photons does it take to make a torpedo? Who gives a Fock! Actually, wasn't Fock Spock's half brother?

3 Like

Jayne

Dec 5, 2024 at 10:23am
In response to "How To Give A Fock"

Versatile Fock States is the name of my next thrash album.

3 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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