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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October 30, 2023 at 9:32am

#1058293

Martian Law

So you don't want to live on this planet anymore?

Everything About Mars Is The Worst
But this jerk planet is still humanity’s best hope for another home in the cosmos.

The idea of life on Mars (whether alien or transplanted from Earth) is obviously one of the oldest themes in science fiction and fantasy. It's so pervasive that some people take for granted that 1) we will colonize Mars and 2) at some point after we do, the colonists will rebel and declare independence, most likely resulting in a war.

If 1 then 2, because Mars is, after all, a god of war, and because all colonies eventually revolt; but 1 is still questionable.

At first glance, Mars seems pretty nice.

Compared to the other planets we know of that aren't Earth, sure.

No other world in the solar system offers us this chance. Mercury is way too close to the sun. Nearby Venus has far too much atmosphere, whose pressure and noxious gases would crush and choke visitors from Earth.

I vaguely recall a thing I linked here a while back that ran the numbers and concluded that the closest planets to Earth, on average, are 1) Mercury 2) Venus 3) Mars. I think it depends on how you calculate it. If this doesn't make sense, consider all the time that these planets spend on the other side of the solar system from us.

And yet, thanks to quirks of orbital dynamics, it's easier to get to Mars. Which doesn't mean it's easy.

At night, temperatures drop to -100 degrees Fahrenheit. Dust devils and shifting sands cover up solar panels and will test even the most tightly sealed spacesuits and habitats. During dust storm season, Martian winds can stir up haboobs that cover the entire globe in clouds of sun-blotting microscopic particles.

That's with an atmosphere that's less than 1% the density of our own troposphere. Also, "haboobs" still makes me chuckle because I'm actually 12 years old.

Humans have been slinging spacecraft Marsward for 57 years, and we’re still not even batting .500.

It has been pointed out that Mars is the only world believed to be populated entirely by robots.

So far, the U.S. is the only country to land anything on Mars, and we’ve stuck the landing on eight of nine attempts.

If I recall correctly (and I might not), the reason the one mission failed is that someone forgot to convert metric to imperial units or vice-versa, which is actually kinda hilarious.

Anyway, the article continues with details of some of the technical challenges, which I won't go into. Numerous SF authors have, naturally, imagined possible ways around each of them, up to and including terraforming. I've done a bit of fictional speculation along those lines.

But I've read enough science fiction to know that it's not going to end well, in any case.

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