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.

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February 8, 2021 at 12:02am

#1003941

Let It Roll

Entry #2 of 8 for "Journalistic Intentions"

[18+]

Let It Roll

Back in college, I had occasion to visit the Drama Department every once in a while, mostly in my capacity as newspaper photographer. This department was in what was then a relatively new building, with more contemporary fixtures than in many of the other University buildings. One such fixture was the toilet paper dispensers in the restroom stalls.

Now, there wasn't a lot of graffiti at the school in general. Oh, there was some, to be sure; the scribblings in the Philosophy Department, much older than Drama, were particularly incisive, and I had great laughs at the occasional math pun in the Engineering department (which is where I took most of my shits). But in this particular case, someone had seen that the toilet paper dispensers had a lever with words on it that read: "PRESS DOWN FOR NEW ROLL." Predictably, but still amusingly because this was, in fact, the Drama School, this person had altered the last L to become an E.

Anyone who's been following along should know that I like to learn about language, and play with it. "Roll" and "role," as you might imagine, are... absolutely of the same origin.

Surprised? I was, when I discovered this.

Turns out that, at least according to Dictionary.com, the word "role" split off from "roll" somewhere in Old French, where an actor's part was referred to, in what I suppose is a case of metonymy, from the roll of paper upon which the actor's lines were written.

I absolutely love this sort of thing; it gives me insight into the way peoples' minds work through the lens of language development.

I should also note that it appears that the French word "roue," which translates as "wheel," comes from the same source as well. Not entirely sure of this one, though.

The "role" origin also seems to tie into one of the other definitions of "roll," as in "roll call." Oddly enough, though, the word "scroll" doesn't seem to be related, at least not as far back as they can trace it -- though it should be, describing as it does a roll of paper, parchment, vellum, whatever. No, "scroll" comes from, of all things, "escrow," which itself was an alteration of an older word "scrow," which apparently meant... roll.

Essentially, "roll" is traced from Latin and "scroll" is Germanic in origin -- though both language families, naturally, stemmed from the same source, even further back: Proto-Indo-European, or PIE. This is, of course, what we call it now; no one seems to know what this original language was called or even, with any level of certainty, what its words or structure were.

A while back, I did a blog entry about the invention of the wheel. After much searching, I finally found it. Here: "As the Turn Worlds (or whatever)"

. And in that entry, I refer to another entry from a couple weeks prior, here: "Lox Pie"

. Now, based on what I found out in writing those entries, it seems that people smarter than I am have figured out where the PIE-speakers probably originated from, and that the reason that particular language spread so far and wide was because those bastards invented the wheel, slapped those suckers on a cart, and hooked the resulting contraption to horses. This made the people in that culture incredibly mobile for the time. I mean, anyone with a horse would have been more mobile (Genghis Khan comes to mind), but if you want to take your stuff with you, you need a cart, too. Preferably one behind the horse, rather than in front of it.

So, essentially, the PIEs rolled all over Europe and parts of Asia, bringing their language with them -- a language which then fractured, merged with other languages in different areas, adapted to its speakers' varying needs and environment, and then -- thousands of years later -- maybe started coming together again, in a vastly different form, as English borrows heavily from so many different other languages.

All of which is to say that there's more than half a dozen songs called "Let It Roll," and I haven't even heard some of them, and those that I have, I don't particularly like, so this entry's about roll and not rock.

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