Complex Numbers (Book)

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Complex Numbers

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/

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10 Most Recent Blog Comments:

Writer_Mike

about 6 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 22 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

April 22, 2023 at 9:55am

#1048605

But the Sun Is Eclipsed By the Moon

I was hoping something would come up from the depths of my blog fodder folder today that was somehow related to Earth Day, so that I could once again remind everyone that we're doomed and there's nothing we can do about it, so we might as well just enjoy the slide into oblivion.

All You Touch and All You See: ‘Dark Side of the Moon’ at 50

Close enough.

Storm Thorgerson and George Hardie’s image of a light shining through a prism and emerging as a spectrum of color—pitched dead center, slightly raised, cast against a black backdrop—is probably the most famous in rock history.

One might even say... iconic? Younger Me spent many an hour listening to the album while contemplating the relationship of the cover art to the music. And I never did come to a satisfactory conclusion. Like most album art at the time, I figured it was either an in-joke or something way beyond my teenage depth, like a New Yorker article.

Only thing I could come up with was the traditional rainbow has seven colors and a musical scale has seven notes. But it couldn't have been as simple as that, not from Pink Floyd... could it?

I hadn't tried drugs yet.

There's a lot more at the link, and even if you're not a big fan of Pink Floyd, it's an interesting (if somewhat circuitous) read.

...the band spent four years indulgently experimenting, with mixed results, before it pared back, tightened up, and achieved global fame with the breakthrough of The Dark Side of the Moon in 1973.

Some of their output before Dark Side is... how shall I put this charitably... unlistenable. But it did give the world such glorious song titles as "Several Species of Small Furry Animals Gathered Together in a Cave and Grooving with a Pict," so it wasn't a total bust.

The week of Dark Side’s release, the no. 1 album in both the U.S. and U.K. was Elton John’s Don’t Shoot Me I’m Only the Piano Player, whose singles, “Crocodile Rock” and “Daniel,” demonstrate the two dominant trends of that moment. The first song is a kitschy, glammy celebration of rock ’n’ roll; the second is a rippling, downbeat soft-rock song about a wounded Vietnam War vet returning home.

As different as they are, I want to emphasize that Don't Shoot Me is also on my list of greatest albums of all time.

So Dark Side is likely something that someone would be exposed to for the first time as a teenager, which also means it’s exactly what someone would play when they’re getting into weed.

As I mentioned above, I became a Pink Floyd fan before I smoked a single joint. I appreciated the music for what it was, absent mental amplification.

That said, yeah, okay, weed helps. But if I had to choose between Pink Floyd or weed, I'd pick the former every time.

The album begins at birth and then presents the three major preoccupations of modern life—the scarcity of time, the dominance of capital, and the conflicts between people and nations—before ending by basically asking listeners, “You will go mad and die, so how will you resolve these challenges?”

I noticed that thematic progression on, I don't know, my thirty-third or thirty-fourth listening. Figured maybe it was just me projecting my own biases onto the music, like it was a musical Rorschach test or Pollock painting.

For the album’s 40th anniversary in 2013, the acclaimed playwright Tom Stoppard wrote Darkside, a comedic radio play about various moral philosophers (Nietzsche, Kant, Hobbes) that’s scored by the entirety of The Dark Side of the Moon.

You know that now that I know about that, I have to find it.

Few, if any, teenage totems have come close to Dark Side’s longevity and reach. Its claimed sales are 45 million copies, making it the fourth-best-selling album of all time.

I was curious, so I looked it up: the three that beat it are Jackson's Thriller (fair), AC/DC's Back in Black (also fair), and the soundtrack to The Bodyguard (wtf?)

I would be remiss if I didn't point out #5 and #6 which are, respectively, The Eagles Greatest Hits, and Meat Loaf's Bat out of Hell, both of which are on my own list of greatests. (Source

)

Anyway, the article delves into the (sometimes unfortunate) history of the band and its members, and I won't quote much more from it. I just want to point out this bit:

Perhaps no example of the mythos around Dark Side is as well known, or as ridiculed, as The Dark Side of the Rainbow, the act of syncing up Dark Side to The Wizard of Oz. In 1995, an article in The Fort Wayne Journal Gazette described how specific songs complement scenes from the first 45 minutes of the 1939 movie.

Obviously, I knew about this, but never could be arsed to take the time to sync the album to the movie, myself. Fortunately, someone did that and posted it on YouTube; the video is embedded in the article I'm referencing here. But a few years ago, NBC released a show called Emerald City, which was—how to describe this—a dark and gritty postmodernist reimagining of Wizard of Oz. It was roundly panned and canceled after one season, but I liked it—at least at first; it later devolved into a Game of Thrones parody. In it, Dorothy (with Toto, who's either a German shepherd or Belgian Malinois, I forget, in this one) meets the Wizard, who is also from Earth, maybe halfway through the season. And during this peak, important, emotional scene, when she encounters the Wizard (played by Vincent D'Onofrio, who is excellent in everything), he's got Dark Side of the Moon playing.

They knew. Damn right they knew.

There's a lot more in the article, but I'll just leave you with this tribute video below. I've posted it before, but it's very appropriate here: an AI-generated music video for the entire album, in honor of 50 years of Dark Side.

"There is no dark side of the moon really. Matter of fact it's all dark."

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