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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 20 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 22 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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.

5 Like • Comment • View 3 comments

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