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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Write_Mikey_Write!

a day ago
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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.

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Adherennium Plotting something

a day ago
In response to "Global Domination"

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

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Brandiwyn🎶

Nov 12, 2024 at 10:50am
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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"

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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.

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Turkey DrumStik

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

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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).

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Lilli 🧿 ☕

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

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May 7, 2019 at 12:30am

#958436

Subjective Impressions

They say art is subjective. What is art to you?

Subjective.

I had an uncle who was an artist. Created art, taught art at a university, left my aunt for a grad student. Artist.

By which I mean, he mostly made sculptures. Very abstract, clean lines; liked to play with plexiglas, created interesting constructs of geometrical precision.

Now, when I think "art," what I usually think of are sculptures and paintings; you know, the things you find in art museums. But of course, there's much more to art than that: music, drama, architecture, game design, films... writing... and so on. All of that stuff is art, in a sense, even though you won't find it in the Met. Except for architecture. The Metropolitan Museum of Art is, itself, a work of art; so is the Louvre.

You won't find my uncle's stuff in the Met, either. He wasn't all that famous.

I used to have arguments with my aunt, the one who was married to the artist. Well, we used to have lots of arguments, but the relevant one here is that I asserted that art should stand on its own, while she insisted that, in order to fully understand art, one must study it formally, recognize the various "movements" throughout history - surrealism, cubism, pointillism, that sort of thing - and know where some individual piece fits in to the history.

We were both right, I think. And both wrong, mostly for even arguing about it in the first place. Given her background, her position is hardly surprising; the same could be said for me. My parents weren't oriented to artistic endeavors, and all my attempts ended in abject failure. There's not an artistic bone in my body, except for perhaps some small talent in writing. What I admired about my uncle's art had nothing to do with shape, form, composition, and whatever other artistic merit it might have had; no, what I admired about it was its precision. Boxes within boxes, all perfectly cut, stuck together in such a way that there were no gaps or globs of glue.

That focus on precision served me well in my career; I started out as a draftsman. I couldn't paint to save my life, or create a realistic sketch of someone's face, but I could draw the hell out of a road, with all its attendant utilities and other structures. It's precise, it's technical, and it involves little to no knowledge of aesthetic shape, color, composition, and form.

There's a story I heard once about some brave soul who interviewed Jackson Pollack. You know, the guy with all the splatter paintings? The interviewer asked the question above, most likely in a haughty voice: "What... is 'art'?" Reportedly, Pollack answered by dipping his hand in a bucket of paint, throwing the gob against a wall or maybe a canvas, pointed to it with his dry hand and said, "That's art. That's art, because I say it's art."

And, truthfully, he probably could have sold the resulting splat. Not for what it was, but because Jackson Pollack created it. If I did the same thing - and I have, actually, done something very similar - it wouldn't fetch a single cent. Hell, it would probably be worth negative money, since a blank, stretched canvas is somewhat valuable, and anyone but Pollack throwing paint at it would only ruin it.

Pollack was (is? I can't be arsed to look it up) considered an artist, and so was my uncle - and yet, they had very, very different approaches to the practice.

As far as I know, all attempts to define art have failed, or at least fallen way short. This is necessary; it's not like the word itself has any precision. It's like "freedom" or "justice" - an abstract noun. The best I can do is say that, for me, it's art if it elicits some emotional reaction. But even there, the definition fails, because some things (like Pollack's paintings, or the literary equivalent such as James Joyce's oeuvre), certainly elicit an emotional reaction - disgust, mostly, mixed with frustration and a bit of anger - but I'm not going to call them "art."

Others' opinions will vary; hence, the "subjective" bit.

I suppose I could go on writing about this sort of thing, but I'm already deep into tl;dr territory. To summarize: Art is subjective.

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