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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Elycia Lee ☮

about 5 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

about 18 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

about 21 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

about 23 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

March 25, 2020 at 12:23am

#979097

Gotta Have Space

No escaping human nature. Not yet, anyway.

http://nautil.us/blog/why-we-should-think-twice-about-colonizing-space

Why We Should Think Twice About Colonizing Space

Um, have you read science fiction? We've thought about it more than twice.

And the late astrophysicist Stephen Hawking has conjectured that if humanity fails to colonize space within 100 years, we could face extinction.

The phrase "necessary but not sufficient" comes to mind.

To be sure, humanity will eventually need to escape Earth to survive, since the sun will make the planet uninhabitable in about 1 billion years.

For reference, that's about as long as there have been eukaryotes on land

For example, the astrobiologist Milan Cirkovic calculates that some 10⁴⁶ people per century could come into existence if we were to colonize our Local Supercluster, Virgo. This leads Nick Bostrom to argue that failing to colonize space would be tragic because it would mean that these potential “worthwhile lives” would never exist, and this would be morally bad.

What the hell kind of stupid, irrelevant argument is that? If you're going to assert that any life that could have existed but didn't is morally bad, you have a lot to answer for.

My conclusion is that in a colonized universe the probability of the annihilation of the human race could actually rise rather than fall.

And we could get hit by another dinosaur-killer-sized comet. Which is why people talk about inhabiting other places besides Earth.

Consider what is likely to happen as humanity hops from Earth to Mars, and from Mars to relatively nearby, potentially habitable exoplanets like Epsilon Eridani b, Gliese 674 b, and Gliese 581 d.

Really, I should have stopped reading right there. I'm no expert in exoplanetary science, but really quickly: EpEri b is basically Jupiter; G674b is basically within the photosphere of its star; and that last one is "potentially habitable" only in the most esoterically theoretical sense; it's a super-Earth.

I have heard of no exoplanets that are what we'd consider Earthlike (remember, to an astronomer, a "terrestrial planet" includes such familiar worlds as Mercury, Venus and Mars), and I'm absolutely certain that I would have heard of one if it had been announced. Hell, you would have too. You haven't. You only think you have because science reporters fucking love the merest glimmer of the possibility of a faint hope that maybe something in a star's habitable zone might be life-capable, and play that up.

I will note that, in a rare case of me actually being arsed to look this shit up, I saw that someone had rather humorously vandalized the Epsilon Eridani b Wiki page thus: Epsilon B was famously visited by the mysterious super space pirate who's life was greatly documented by the late Disco Beavers. There it is said he found vast amounts of funky juice and many beautiful space women, as well as meeting his life long friend and later enemy, Diamond Prince. I'd link it but these things tend to disappear fairly quickly.

Okay, my point is that while there might be "potentially habitable exoplanets," especially since we really couldn't find Earth if we were looking at our solar system from, say, Epsilon Eridani b, we haven't found any, and any claims that we have are spurious.

Anyway, back to the philosophizing.

In other words, natural selection and cyborgization as humanity spreads throughout the cosmos will result in species diversification.

I think that's a fair prediction, though speciation takes a very long time. Your guess is as good as mine as to how quickly some of us become the Borg Collective.

At the same time, expanding across space will also result in ideological diversification.

You know, I haven't read everything in science fiction. That would be impossible. But I have yet to read any fiction where a Mars colony doesn't declare its independence, usually resulting in interplanetary war. Maybe this is because it would be boring if they didn't, or maybe it's because SF authors have already figured out this little tidbit about human nature.

The author goes on to explain the difficulties of having a top-down governing system for far-flung human colonies. But as the argument hinges on not having warp drive, I should note that not having warp drive would also make inter-system war very, very difficult. I mean, imagine, say, Planet England and Planet France.

PF: You have infringed upon our trade rights!

(PF assembles a war fleet)

(70 years pass as a war fleet crawls from PF to PE)

PE: What is that war fleet doing here, and why are they signaling for help with their engines and life support systems?

I mean, really, someone should write books about this stuff. Oh, wait, they have.

The lesson of this argument is not to uncritically assume that venturing into the heavens will necessarily make us safer or more existentially secure. This is a point that organizations hoping to colonize Mars, such as SpaceX, NASA, and Mars One should seriously contemplate. How can humanity migrate to another planet without bringing our problems with us?

In the end, though, I can't disagree with the basic premise here. Which doesn't mean we shouldn't do it. It just means we need to account for human nature.

And this last bit, written in 2018, seems prescient enough:

Human beings have made many catastrophically bad decisions in the past. Some of these outcomes could have been avoided if only the decision-makers had deliberated a bit more about what could go wrong—i.e., had done a “premortem” analysis.

And that's enough prognostication for one day.

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