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 14 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 16 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.

April 16, 2022 at 12:01am

#1030814

Nothing to Celebrate

I was involved in designing several planned communities in my career, so this sort of thing catches my eye.

How the Disney dream died in Celebration
Disney invented Celebration as an ideal American small town. But recession, a brutal murder and a suicide have killed the magic

First of all, when I see "ideal American small town," fear ignites in the pit of my stomach. And that's before the murder part.

This is because some Amercans' idea of the "ideal" is racist, sexist, and theocratic. But I repeat myself.

Guardian link, so British spelling.

There once was a place where neighbours greeted neighbours in the quiet of summer twilight. Where children chased fireflies. And porch swings provided easy refuge from the cares of the day. The movie house showed cartoons on Saturday. The grocery store delivered. And there was one teacher who always knew you had that special something. Remember that place?

Yeah. I also remember the screams in the middle of the night, the black eyes, the furtive glances.

The first paragraph of this article is lifted verbatim from the town's original sales brochure from 1996, written by Disney's famous "Imagineering" team.

Verbatim? I doubt it. I can't imagine Disney's team would have spelled it "neighbours."

The words were matched in visual form by the new town's seal, which was stamped on everything from storm drains and mugs to golf towels. It showed a little girl with a ponytail riding a bicycle past a wicket fence and an American oak with her dog running dutifully behind her.

Why does the American oak have a dog?

The news that Celebration – population 11,000 –had suffered its first murder with the bludgeoning to death of a 58-year-old retired teacher? Followed less than two days later by the sight of a Swat team in an armoured car staking out a house, which ended with the homeowner's suicide? How do you fit these events into the Disney dream?

You'd think they'd have learned by now: one person's heaven is another's hell.

There are shops with names such as Day Dreams selling Barbie dolls, the streets are lit by olde worlde lanterns and you are followed everywhere by muzak from the 40s and 50s piped out of speakers hidden beneath palm trees. Jingle Bell Rock, Santa Claus Is Coming to Town, Oh Come All Ye Faithful – the theme is unrelentingly Christmas.

Case in point. No, it wouldn't drive me to murder or suicide, but it would drive me right out of town.

But of course that's what Disney's all about: the stitching together of reality and fantasy so finely that you can't tell where the one ends and the other begins.

Which of course is perfectly fine in their movies. Desirable, even. But at the end of the movie, only memories remain.

Why he was so drawn to live here? "I think it was because of the way Disney does things. You knew the parks and the streets would be clean, the grass cut, all the houses would look the same, it would be family-oriented and everything would work just right. A lot of people wouldn't like to live in that sort of place, but for me . . ."

Tonight, I shall have nightmares.

On 29 November Matteo Giovanditto was found bludgeoned and strangled inside his apartment where he'd lived with his chihuahua.

Ah, the motive becomes clear. No one likes to live near a chihuahua.

Walt Disney always had a dream to build a model town. It was part of his master plan to create a new magic kingdom in Florida.

I think the guy who designed The Villages

beat him to it. That place also freaks me the fuck out.

"Walt Disney was an authoritarian. Yes he was an artist, but he was also a control freak. People tend to see the Disney creativity, they often miss the centralised control that lies behind it."

This is a big part of my hatred for the concept. It's like the worst diabolical HOA you can think of, raised to an exponent of fakery.

Then they boiled all the ideas down into a pattern book that dictated every detail right down to the plants that could be grown in the yards. There were only six house styles permitted, and only a limited range of colours – white, blue, yellow, pink and buff, all in pastel shades.

All your creativity are belong to us.

The extent to which the Disney corporation went to control the warp and weft of Celebration speaks to one of the central paradoxes of modern American life. For a country that prides itself so fiercely on its untrammelled individualism and freedom, it has a strong streak of conformism.

Such an outsider perspective is often needed. Other people have said this before, better than I can, but every time you get a bunch of individualists together, they always end up converging on some style or another. Counterculture becomes conformity.

Yet Celebration has not been immune from the devastating economic collapse in America. As a symbol of its economic ailments, a couple of days after Thanks- giving, its cinema, a striking mock art deco building by the lake, shut down.

I should have noted this before, but this article is from 2010, when the worst problem we had seemed to be a recession that everyone knew wouldn't last forever. Now? I look back on that simple time with the same nostalgia that Celebration was meant to inspire.

If I cared, I'd try to find out how that planned community is doing now, after a pandemic, rampant inflation, another housing crisis, and World War III.

But I don't care. I'm staying well away from that little slice of Hades.

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