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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Elycia Lee ☮

about 7 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

about 23 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

February 27, 2020 at 12:03am

#976432

The Original Dark Side

First, let me set the mood here.

Got that going? Good.

Now a few words about The Wizard of Oz.

http://www.bbc.com/culture/story/20190808-the-subversive-messages-hidden-in-the-...

The subversive messages hidden in The Wizard of Oz
It’s easy to mistake the 1939 classic as traditional family entertainment – but 80 years on from its release, the musical is more radical and surreal than ever, writes Nicholas Barber.

(Article is dated six months ago, near the actual 80 year anniversary)

In December 1937, Walt Disney Productions released its first feature-length cartoon, Snow White and the Seven Dwarfs. It went on to be cinema’s biggest hit of 1938, a success that not only encouraged Disney to make other fairy-tale cartoons for decades to come, but also encouraged another studio, Metro-Goldwyn-Mayer, to try its own fantasy musical about an orphaned girl and a wicked witch: The Wizard of Oz.

80 years on, Disney has clearly won that particular war. Only now they're making fairy tales about superheroes and Jedi. Whatever works.

The pig-tailed Dorothy Gale (Judy Garland) is so wholesome, the Harburg and Arlen songs are so delightful, and the Technicolor adventures are so exciting that it’s still easy to mistake The Wizard of Oz for traditional family entertainment, 80 years on from its release in August 1939. But it upends the conventions of good-v-evil storytelling in ways that would have had Walt Disney fuming.

Much has been written about the symbolism inherent in Oz. I won't belabor the point here, but there's definitely a darkness to it. The article skips the symbolism, too, opting instead to look at the movie through the lens of world events at the time, as well as the things we've experienced and learned since.

The message is that people will march behind any authority figure who makes a splash, however undeserving they may be. It’s a subversive message in 2019, and it was even more pointed in 1939, when fascist dictators were stomping across Europe.

Or, perhaps, haven't quite learned.

It’s significant, too, that the Emerald City of Oz isn’t the turreted faux-medieval Ruritania where Snow White lives, nor is it the Istanbul-ish collection of domes and spires drawn by WW Denslow in the original book’s illustrations. Instead, it is a modernist mass of neon-striped skyscrapers – and, like almost everything else in the land of Oz, it is blatantly artificial.

Yeah, that's not just an artifact of the limited capabilities of special effects in 1939. A lot of it was deliberate.

A couple of years back, someone decided that it would be a good idea to update Oz (the book and its sequels) into a series called Emerald City. It, too, was a reflection of its time, our time, dealing with modern issues of feminism, gender, and culture clashes, especially the one between science and religion. Critics were less than thrilled. Critics aren't always right, though; in my own opinion, the series started out very strong, but devolved into chaos. I maintain it's worth watching, though it was cancelled after only one season.

Notably, the showrunners knew about and exploited the popular connection between Pink Floyd's Dark Side of the Moon and The Wizard of Oz. If you don't know what I'm talking about, here: https://ultimateclassicrock.com/pink-floyd-dark-side-wizard-of-oz/

It's important, I think, to experience entertainment in the context of its time. A lot of the songs from the late 60s and early 70s, for example, are impenetrable if you don't know about the Vietnam War. But we tend to ignore history, or, worse, forget about it, so the occasional article like this one helps us to keep things in perspective.

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