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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January 18, 2021 at 12:05am

#1002323

Time After Time

PROMPT January 18th

If you could use a time machine, would you travel to the future or to the past? When, and where would you go?

Let's get a couple of things out of the way first:

One, time is not an illusion. While our perception of time depends on several factors, time is as real as anything is -- we can't touch it, but we can sure as hell measure it.

And two, I consider it highly unlikely that time travel to the past (that is, the "past" relative to any given moment of "now) can ever be possible. Time travel to the future will probably be trivial one day, what with general relativity and/or cryogenic technology. But then you're in a new "now" and there's no going back.

Those things said, I'm as much a fan of time-travel stories as anyone, so I'll play the game.

I'm going to assume here that by "time machine" the prompt means, like, the TARDIS or a certain DeLorean. So I'd be able to return. But that really doesn't matter because, either way, it's the future that I want to see, no doubt about it.

This is for one simple reason: the past kinda blows. I mean, maybe the future blows, too, but we know that the past blows. Medical care alone is enough to put me off on the idea for all time (pun intended). Sure, our "system" here in the US sucks ass, but it's fucking sorcery compared with leeches and lack of anesthesia.

And we know a bit about the past. Not everything, obviously, but enough. And if I really did have a time machine, I'd definitely visit, just to answer questions like "What did Helen of Troy actually look like?" "Was Pythagoras as big a dick as I think?" and "Who really invented vodka: Poland or Russia?" But like I said, given the choice of one or the other, I'd pick the future. I want to see how it all turns out, you know.

Well. I know how it all turns out. At some point, everything will just... stop. There will cease to be any thermodynamic processes whatsoever, but long before that happens, life will be unable to exist. Nevertheless, there's a lot of time involved (exactly how much time is a matter for debate, but it's orders of magnitude more than the lifetime of our sun), and so there's plenty of time for further innovation and discovery.

It's that discovery that intrigues me.

So, how far into the future? Where and when? Well, the essence of the future is that while big things like the eventual death of the Sun can be predicted within a billion years or so, there's no way of knowing how far what we call "civilization" can advance or how fast. We can try to extrapolate, but there are externalities that will throw wrenches into the works: war, pandemic, aliens, us choking on our own waste products, whatever. I'd want to go far enough into the future to see cool shit, but not so far that it would be entirely alien to me. So for that, we're not talking trillions or billions or even millions of years.

One thousand years should suffice. Just enough to see if we will really colonize other worlds, and hopefully view them. Alternatively, maybe Earth will be a vast, barren wasteland then. Even that would be good to know.

One final thought for today:

I could, of course, be very wrong about time travel into the past being impossible. We've been known to do things once thought to be impossible before, so it's not completely outside the realm of possibility.

If so, then at some point, we'll be able to visit what will be to us the past. I imagine that there will be people whose job it is to protect the timeline, to make sure that events happen for the best possible outcome.

And if that's true, then maybe we're living in the result.

If that doesn't frighten the living hell out of you, you haven't been paying attention.

Of course, there's absolutely no evidence that this has happened, and plenty of evidence that it hasn't. For example, someone from now going to the past risks contaminating it with Covfefe-19 or AIDS or some such virus that's of relatively recent origin. Right now you're going "well, maybe that's how Trump Mumps actually started, an infection from the future." The problem with that hypothesis is that the virus has been extensively studied, and there's nothing of the future about it. If it had come from the future, it would have shown signs of having evolved longer than anything else on the planet, and there's no indication of that whatsoever.

We're not living in the best of all possible timelines.

We're living in the only possible timeline.

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