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.

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').

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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

January 5, 2021 at 12:00am

#1001443

Behind the Eight Ball

PROMPT January 5th

If you could learn the answer to one question about your future, what would the question be?

Hey, why limit it to one question? You can ask all the questions you want and get a true, accurate prediction right here:

Ask The Magic 8-Ball!

Just think of a question that can be answered "Yes" or "No", concentrate very, very hard, and click on the "Ask" button.

Hey, Magic 8-Ball, will I ever get married again?

Cannot predict now

Whoa, that's a tough one, huh? At least I can choose to avoid doing so. How about something easier?

Magic 8-Ball, will I become owned by another cat this year?

My sources say no

Well, that's kind of a relief. Between my two and my housemate's other two cats, we have reached the limit of catitude just below that of Crazy Cat People (the formula is C=N+2, where N is the number of humans in the house and C is the maximum number of cats before you are a CCP).

Anyway, now we're getting somewhere.

So, Magic 8-Ball, will the guy who's supposed to rebuild my deck this fall do a good job?

As I see it yes

Definitely a relief, and this also tells me that he'll actually do it and not make up some excuse to put it off until 2022!

Now, Magic 8-Ball, will I manage to end this year sipping cocktails on a tropical island, being fanned by beautiful bikini-clad women while reading good books?

Cannot predict now

Dammit! We're back to that again. I'll just have to find a way to make that happen on my own.

...I actually had one of those things when I was a kid. Yes, they've been around that long. I guess some relative or other presented it to me as a birthday or Hanukkah gift. I messed with it a little bit, but it didn't take me long to figure out how it worked: 20-sided die (okay, this was before D&D but I already knew about platonic solids) suspended in a semi-opaque liquid inside the black ball. Pretty neat, incredibly simple toy and very clever.

It does not, however, predict the future. Of course.

Sometimes, amusingly, I could get the icosahedron to balance on an edge at the window, rather than a face, thus representing the true uncertainty of any predictions about the future.

There are some things about the future that can be predicted to incredible accuracy, like the next few thousand years' worth of solar eclipses

as seen from Earth. Orbital dynamics is well-understood and predictable to a great degree of precision -- and yet, all these predictions assume that nothing happens like a black hole swooping through the neighborhood, or space aliens coming by to mess with the planetary orbits. (Why would they do that? Who knows? I can't tell you how humans think, let alone space aliens.)

On the other hand, we can't predict the weather to any degree of accuracy out past a week or so, and the further you go the less accurate the weather report is. The systems involved are simply too complex.

Hell, last time they predicted "snow tomorrow," we got a few millimeters of ice and that's about it.

Kid Me with his Magic 8-Ball wanted to believe. He really wanted to know what the future actually held, but something in him even then realized that a dollar's worth of plastic and glass wasn't going to cut it.

Now, though? My actual answer to the prompt above is "No, I don't think so. Nothing." It's like, when I had a lot of future I wanted to know what would happen, but now that there's not a lot of future left for me, I think I'd rather be surprised.

Especially since the alternative would be, as Kid Me eventually discovered, profound disappointment.

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