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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Show Recent Comments

July 5, 2022 at 12:02am

#1034711

That's Hollywood

Hopefully we're all aware that movies are fiction. Even the ones based on real events are fiction. Still, they can give us the wrong idea, like how in Being John Malkovich they were able to get from Jersey City to Midtown in 15 minutes. (That was, incidentally, the most unbelievable part of that movie.)

4 Myths About Natural Disasters Movies Want Us To Believe

I'm not sure I'd go quite so far as to say they want us to believe it, Cracked. More like, they want to provide a cool spectacle, realism be damned.

But, just like movies have put it in peoples' heads that the asteroid belt is dense enough to provide an obstacle course for a spaceship (it's not), we've all gotten the wrong idea from watching big-budget movies. Yes, even me.

Anyway, here are some examples of how Hollywood egregiously uses our fear of apocalyptic events to bypass real and accurate science. It’s just that easy, we guess.

Look, I get it. Science is hard, and usually visually boring (except around stars and black holes, which, by the way, Nolan got the time dilation thing wrong there), so you want to go with what grabs a viewer's eyeballs. Even the ones that purport to get the science right (The Martian and Gravity, e.g.) don't. But I'm not linking this article to rag on the movies.

It's even possible to go too far with the "realism" critique. I think it was Neil deGrasse Tyson who pointed out that, in Titanic, the stars were all wrong for that latitude and time of year. I wouldn't know, because I've never seen the movie out of fear of having to listen to Celine Dion. I mean, if the stars were actually essential to the plot, sure. I can get, for instance, noting that the door (or whatever) that the chick was floating on at the end was actually wide enough for both of them. That's plot-related. Again, never seen it, just going by spoilers (the ship sinks, by the way).

Anyway. I'm just pulling those examples off my brain. Presumably, the author here did actual research. But it's Cracked, so you never know.

4. No, Ocean Impacts And Tsunamis Won’t Go Down Like In The Movies

I gotta admit that, after seeing fictionalized tsunamis as a kid, when I saw an actual one (on video of course), it was decidedly underwhelming. I mean, scary, sure, and I don't want to minimize the absolute horror that the victims went through, but it wasn't nearly as apocalyptic-looking as I'd been led to believe.

My sense of betrayal there is largely because I read Lucifer's Hammer, by Niven and Pournelle, as a kid. It was a meteor impact disaster novel before that kind of thing went mainstream. A hipster meteor impact disaster novel. I've forgotten much of the plot other than "meteor slams into ocean, shit floods, people die," but one scene will always be burned into my mind: an entire chapter devoted to a dude catching the giant wave resulting from the impact and riding it on a surfboard.

Great scene. Well-written. Exciting. Disappointing that such great SF writers as Niven and Pournelle could have gotten that science so wrong. In their defense, though, science advances and perhaps that was the best estimate of what would happen, at the time.

Galen Gisler, a senior researcher in physics and geology at Los Alamos National Laboratory... said that “the folklore has been that tsunamis from impactors will be the danger … The splash wave can be very dangerous, out to tens of kilometers, but beyond that, they fall away more sharply.” Gisler calculated that only a tenth of 1% of an impact’s kinetic energy will be spent on forming waves.

Spoilsport.

3. Dormant Volcanoes Don’t Simply Kick Back To Life Like In The Movies

I mean, I'd suggest just reading this one. It's not even so much about dormant volcanoes, but volcano scenes in general. I can excuse the one where Anakin Skywalker gets turned into Darth Vader because that's a fantasy movie set on another world, or the one where Spock nearly gets fried except Kirk tore up the Prime Directive, because that followed the Rule of Cool. But again, let's not pretend that these in any way reflect Earthly reality.

Still, when I was standing on top of Haleakala, I have to admit I was a little bit like, "What if this thing starts up again?"

2. You Can’t See ‘The Eye Of The Storm’ Inside A Tornado

I would have thought this one was obvious. Look at a tornado from the side. What's it shaped like? Usually it's all twisty. Thus, if you're lucky enough to survive the wind wall and you look up, you won't be able to see through to the sky above. I suppose maybe the really strong ones are more upright cylinder-shaped, but good luck making it through one of those.

1. Earthquakes Don’t Open Up The Earth To Swallow People Hole

Typo or pun? You decide.

Anyway, maybe not, but again... it looks good on screen.

So in conclusion, like I said, movies are fiction and we shouldn't be fooled that they reflect reality. And yet, I can enjoy them anyway, even though I know that the square-cube law would prevent anything like Godzilla from actually existing. Hell, part of my enjoyment is noticing where they get it wrong.

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