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:

Write_Mikey_Write!

a day 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

a day 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.

October 3, 2019 at 12:28am

#967155

Metropolitan Mistake

https://www.theguardian.com/artanddesign/2019/apr/09/hudson-yards-new-york-25bn-...

I don't comment on these things often because it's not really my métier. (See? I learned me some French.)

But I spent my career in what you might call an architecture-adjacent field, and my cousin in an architect in NYC, so this caught my eye.

Horror on the Hudson: New York's $25bn architectural fiasco

It is a billionaires’ playground where haircuts cost $800 and high-rise duplexes go for $32m. So why do the angular towers of Hudson Yards look so cheap?

For $800 I'd better get the world's best blowjob with that haircut. Without the snippers being in the room, just in case.

That aside, like I said, I've been working with architects for most of my adult life, and this article is the best takedown of an architectural abomination I've ever seen.

He can now look down on his co-creation every day from his new office in one of the development’s towers and see hundreds of people climbing up and down Thomas Heatherwick’s Vessel sculpture, like tiny maggots crawling all over a rotting doner kebab.

And it only gets more creative from there.

I mentioned my cousin. His offices used to be on - I think - 33rd Street, overlooking this pile of manure. A couple of years ago, they moved to a Park Avenue address (they are very good architects). I asked him why the move - they'd had a sweet three-floor space overlooking the river; prime NYC real estate.

"Because we didn't want to look at Hudson Yards," he said.

I also asked him if he'd seen this article. He knew exactly which article I was talking about, and he just smiled, laughed, and nodded.

The surprising thing isn’t that such a development has happened. The real shock is that it’s quite so bad.

Yeah, everyone knew the land would be developed at some point. It's NYC. Apart from Central Park, I think this was the most obvious place to turn undeveloped property into skyscrapers, and Central Park is sacred space.

There were some interesting engineering challenges, such as building the fucker over an existing, active, and very, very busy train yard, and those naturally interested me more than the visual design aspects, so I kind of ignored the latter for as long as I could. It was hard to ignore the former; every time I took a train into Penn Station, I got to see some of the construction up close. But I won't bore you with the details.

Yet it all feels so cheap. From the architectural zoo of convulsing angles to the apparent lack of care spent on the details, this is bargain-basement building-by-the-yard stuff that would feel more at home in the second-tier city of a developing economy.

I really hope the architects who designed this towering mess have a good supply of aloe on hand, because the burns are just beginning.

It climbs up into the sky in ungainly lumps, with a triangular observation deck wedged into its side near the top, forming a pointy beak that gives it the look of an angry chicken.

That sounds like it cost a few bawks.

It is a tableau that almost elicits pity, like chubby fowl engaged in their first awkward mating ritual.

I'll stop with the fiery quotes, now; you can go to the article for more. Their description of the monstrosity known as Vessel is particularly satisfying.

I've seen scathing movie reviews, brilliant takedowns of crappy scientific studies, and deconstructionist critiques of everything from fashion to art - but this piece will stick out in my memory as the most eloquent trashing of a giant pile of ugly that I've ever had the pleasure of experiencing.

There have been a few architectural disasters in my lifetime - the Hyatt Regency skywalk, for instance, and that bridge in Florida - but those were matters of structural engineering, not their aesthetics. And when something collapses like that, it's perfectly obvious to everybody, whether they're an engineer or not, whether they have any design sense or not, that Something was Wrong.

Well, I have no aesthetic design sense, except that when I saw this thing, I thought "ugly." This article eloquently describes exactly why it is ugly. It almost makes me wish the whole thing will collapse like those other architectural failures I mentioned, except that doing so would be tragic - not to the architecture of NYC; it would be a boon to that - but to people who had nothing to do with its design.

The best that we can hope for is that it'll be a financial failure, and it'll get demolished in a controlled way, to be replaced by something that New Yorkers can be proud of. Sadly, the Hudson will probably rise to claim it long before that happens.

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