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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

February 15, 2023 at 12:06am

#1044859

Brutal

I don't have much of an opinion about Brutalism.

This entry is for "Journalistic Intentions"

[18+]; specifically, it's about one of the picture prompts: Brutalist Bench.

Some people love it; others hate it. There doesn't seem to be any middle ground... except for me, of course.

On a recent visit with my cousin, an architect, we had a discussion about Brutalism whilst driving around the Jersey Shore, which features several buildings done in something approaching the Brutalist style. The name isn't based on the English word "brute," or the deodorant brand (which I was surprised to find is still around), but on the French word "brut," meaning, essentially, "raw."

In this context, from what I can gather, it basically means the whatever-it-is—usually a building, but sometimes something else, like a bench—isn't painted or otherwise covered, relying instead on the colors and textures of the raw materials of its construction (hence the "raw," I suppose). It's generally the triumph of function over form, which appeals to me as an engineer. My cousin wouldn't come out and say it, but I don't think he's a fan. Considering some of his work, this doesn't surprise me. The stuff I've seen come out of his office ranges from traditional (including some buildings right here in Charlottesville) to what I can only describe as avant-garde. Yeah, I know, sorry, more French. Point is, I don't think he's done many Brutalist buildings, if any.

Brutalism is most commonly associated with bare concrete, but really, any (mostly) unadorned material would do. Including brick. I do have an opinion about brick, which is: painted brick sucks.

But no, the reason I don't have an opinion on Brutalism is that, to my untrained, function-over-form eye, there's a great deal of variation among examples of the style.

This is actually one of the cool things about concrete (which I devoted an entire entry to a couple of days ago): as it is a liquid (actually a slurry, but whatever), it can be molded into many different shapes.

If you look at the Wikipedia page on Brutalist architecture,

you'll see a lot of different examples of the style. They're all different. Some suck. Others don't. Which is which is a subjective decision.

I've said this before, but architecture is art with function. That is, it's meant to do more than be looked at; people live, work, or pass through architecture, while pure art just kind of sits there daring us to criticize it. But make no mistake, it's also there to be looked at. I mean, usually, you can't miss it, so you might as well make it pretty.

Same for a bench, though on a much smaller scale. The primary purpose of a bench, in my view, is to be a place to sit. Lots of forms can achieve this purpose. The one from this prompt, though? Well, to me, it just kind of looks uncomfortable. Like the designer was more concerned with form than function, its flowing shape at odds with the heaviness of the material.

On the other hand, it's indeed made of concrete, so at least we know it'll handle some weight. That's important, these days.

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