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Sunday, Jun. 30, 2024

2:44pm EDT

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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

a day ago
In response to "Je Voyagerai"

I once was allergic to going on a cruise. But... I should... just for the experience. I've gotten out of my comfort zone for experience.

With the crush of tourists these days any travel should include how much you want to put up with "other people's vacation" as opposed to your own.

1 Like

Lazy Writer est 4/24/2008

a day ago
In response to "Je Voyagerai"

Happy traveling..be safe.

Elisa: Middle Aged Stik

a day ago
In response to "It Ain't Wrong"

Ain’t apparently begins as amn’t, a contraction for am not, which you can still hear in Ireland and Scotland today

I mean, an abundant number of Scotch and Irish immigrants ended up in the American South, so this tracks.

While a lot of people consider ain’t improper, it’s a very regular and legitimate part of many forms of English, including in Black English (AAVE). Be mindful that judging someone’s use of ain’t as “wrong” can be a very socially loaded act, to say the least.

For some reason, this reminds of one of the reasons I stopped reading Ask A Manager. A bunch of people were getting real uppity about the use of the phrase "baby daddy". Unsurprisingly, many of these people appeared white. One Black person in the comments finally had enough and threw down with this remark:

"I gotta say, as a black woman it grates to have phrases that come out of AAVE be appropriated, assigned negative connotation (likely because they come from black culture), then discarded with prejudice. If you’re using the phrase to conjure up stereotypical imagery, I can get on board with putting an end to that. But there are plenty of people for whom baby daddy or baby mama are not pejorative at all, just descriptors of the relationship in their dialect, and correcting them to “Father of my child” is just another case of denigrating minority speech patterns and legitimate AAVE grammatical constructions and upholding “white is right” standards."

(For more context, check out this link

1 Like

🌕 HuntersMoon

Jun 28, 2024 at 10:54am
In response to "It Ain't Wrong"

Ain't you on a roll today...

3 Like

Adherennium Dr of Phoolishness

Jun 27, 2024 at 11:53am
In response to "Wave Rider"

Horrifying fact, I once painted a reproduction of the Hokusai wave on our bathroom wall. Eventually of course it was painted over, but I like to fantasise that in some distant future it will be archaeologically excavated and held up as an example of early cultural appropriation, (badly executed too.)

1 Like

Annette

Jun 26, 2024 at 8:16pm
In response to "The Oz Hypothesis"

The nekked wimmin look like something that I might have drawn in first grade.

1 Like

Aiva Raine

Jun 25, 2024 at 10:16pm
In response to "Can't Spell Success without Sucs"

Ok- this made me truly LOL- "sipping a gin and tonic made with low-paid laborer tears"

Thank you. Totally agree with the rest of your article.

Money doesn't solve all problems but it sure as hell eliminates many and makes it easier to go places or buy things that can create some happiness. IE- a property somewhere quiet and peaceful and a trailer with which to chill on said property.

Rich people are typically clueless assholes.

1 Like

Anni Pon

Jun 25, 2024 at 9:30am
In response to "Can't Spell Success without Sucs"

I've never heard anyone say, "That guy is so successful, but he doesn't really make much money." Or anything similar. Sure, you can have a positive impact on the world without making lots of money, but people won't call it success.

2 Like

Annette

Jun 23, 2024 at 1:28pm
In response to "Revisited: "Experience""

This is how I coped:

1 Like

Elisa: Middle Aged Stik

Jun 23, 2024 at 9:51am
In response to "Revisited: "Experience""

Not gonna lie I kinda miss Zoom parties, at least for this community since people on WDC are so spread out. Meeting in person is pretty tricky and/or expensive. It's certainly more socializing than I've been doing regularly since, oh, 2022 or so.

As for server migration, well, I work in the IT world. I've been through that. And honestly, server migration is a bit nerve wracking, as some stuff can go missing. It's still more manageable than a complete redesign. Complete redesigns are awful. You can test in different environments that the public can't see till kingdom come, but things always go wrong when the public version is rolled out. A local grocery store recently redid their website, and the old site held thousands of recipes. When the new site was unveiled, none of those recipes made it over. I ended up reaching out on Messenger and was able to get a copy of a recipe I use somewhat often, but I'm kind of annoyed that the text recipes are currently MIA because somehow that data didn't transfer with the site rebuild. Absolute madness.

1 Like

June 10, 2024 at 9:56am

#1072422

Betelgeuse 2

The person (P.Z. Myers) who writes the blog I'm linking today is a scientist—but he's a biologist, not an astronomer. So his take on astronomy probably isn't any more reliable than mine. This doesn't stop either of us from writing about it.

Betelgeuse, Betelgeuse, Betelgeuse

I'm sharing this because 1) I would love to see a supernova before I kick it (though hopefully not one that's too close) and 2) I'll take any opportunity to call out misinformation.

Something funny is going on 650 light years away…or should I use the past tense? Something funny was going on 650 years ago.

Yes, we know light has a speed, and we don't have good grammatical tenses for it. The way I look at it is: what matters is when we see the event; the distance (or time) when it happened is immaterial.

The star Betelgeuse is/was acting up, dimming and then brightening (well, it’s always been flickering a bit, but this was a greater reduction in brightness than usual.)

This is cool by itself, because it contributes to astronomers' understanding of how stars work.

And now some people are saying it’s about to go supernova!

Well, it is. For cosmological definitions of "about to."

There is a real-time deathwatch on YouTube. “LIVE Betelgeuse Supernova Explosion Is Finally HAPPENING NOW!” it says.

And that's where the misinformation, or at least misunderstanding, comes in.

I'll let PZ explain there at the link, or at least quote actual star-gazer type scientists.

Awww, but it sounds like it will be spectacular when we do get the Giant Space Kablooiee, and not spectacularly dangerous, the best kind of spectacular there is.

The Giant Space Kablooiee is not to be confused with the Horrendous Space Kablooie,

which was, to the best of our knowledge, what kick-started this enormous thing we call the Universe. It's not even in the same ballpark. To quote Jules from Pulp Fiction, it's not even the same sport.

But yeah, the science-talkers who should know stuff about these things don't think it'll do much, if any, harm to Earth when it happens. But "spectacular" is an appropriate adjective.

I do wonder if that guy running the live video feed is prepared to keep it going for 10,000 years.

No, but he's probably prepared to collect ad revenue from YouTube, taking advantage of the misinformed. Which is why I'm not going to click on that video.

Perhaps I am more cynical than PZ.

To be clear: we don't know enough about stellar lifecycles to predict just when Betelgeuse, or any other star of appropriate mass and age, will become a supernova. When it happens, next year or 10 millennia from now or somewhere in between, if there are still scientists around at that point, we'll learn more.

Hell, even the "650 light years" thing might not be correct. For whatever reason (it's discussed at the link I'm about to provide, though with an overwhelming amount of math), it's incredibly hard to estimate that star's distance with high certainty. Wikipedia

puts it at 400-600 ly, which, even on the low end, is still supposed to be comfortably far for a supernova (though close enough to look awesome).

Now, I've written about the probably-not-a-supernova-anytime-soon bit before, notably here: "Stardust"

. Today's update is more about the "live feed" nonsense. And maybe a bit of a distance correction.

If you want a live feed that's actually interesting, I usually remember to check this one

out occasionally around late spring/early summer. Bears!

5 Like • Comment

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