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 18, 2022 at 12:01am

#1035341

Lift Me Up

Today's article is brought to you by Elisa: Middle Aged Stik

, and it comes from her city.

Twin Cities airport installs mock aircraft cabin to help travelers gain confidence

A newly arrived airliner at Minneapolis-St. Paul International Airport doesn't have wings, or wheels, or engines. It'll never leave the gate. But it will help more people gain the skills and confidence needed to make their travel goals a reality.

Look, I'm not going to rag on this concept. I think it's great. Kind of a dress rehearsal for the real thing.

The former Delta Air Lines cabin simulator, lined with rows of Boeing 737 seats, is now located in Concourse C at MSP, in what may be a first-of-its-kind airport installation.

The article includes some pictures, including the interior of the thing, and while I'm not sure if I've ever been in a 737, it certainly seems to replicate the "packed in like pickles" experience. (Sardines are cliché, and pickles is alliterative, so I'm going with that.)

"The whole goal is giving people — as well as people with service dogs — that preflight experience that helps them overcome some of those challenges without being on a real, live flight," said airport spokesperson Jeff Lea.

Bit awkwardly phrased there, Jeff. Are you saying "people with service dogs" are not the same thing as "people?" I doubt it, but he could have phrased it better.

That Navigating MSP program includes using ground transportation, getting through security, getting in and out of plane seats — everything but getting off the ground, in preparation for an actual trip.

I imagine that one's first time flying can be stressful, or, as the article points out, the first time flying after becoming disabled. It sounds like it's got most of the inconveniences covered.

But I see no mention of it being mechanized. That is, a lot of people have a fear of flying. Not of the inconveniences of getting through security theater with one's dignity intact, or of sitting next to a 400 pound individual; those are just annoyances. No, there's no mention of it being rocked around like crazy, like planes sometimes do, freaking out the passengers who are absolutely certain that the plane's about to crash.

It doesn't look like it can drop 1000 feet in 20 seconds or whatever, putting you in free-fall and validating the seat belt requirement.

It does not replicate the existential terror of knowing that you're only held up by chaos.

I mean that literally. Lots of people think it's the Bernoulli effect, which isn't all that complicated: the faster air moves, the lower the pressure, so if you get air moving faster across the top of the wing, then the pressure on the bottom lifts the plane. While that accounts for some of the lift, in reality, it's far more complicated than that, and no one really understands

all the science.

Think about that. Somewhere in the neighborhood of 100,000 flights

take off every day, worldwide. The airline industry is remarkably safe—you're far more likely to be in a car accident on the way to or from an airport. And yet, the science behind it relies in part on turbulent flow, which is poorly understood. Hell, it might as well be fairies doing the lifting.

And turbulent flow is in the domain of the branch of physics known as chaos theory.

There's no question that it works. But as an engineer, "it just works" is usually an unsatisfactory answer for me.

Another question: does the simulation include screeching children, smelly emotional support animals, and crappy movies that you have to pay extra for and watch on your own damn phone? Because those are the reasons that, given the choice, I'll usually seek other forms of transportation. I have no problem with actual service dogs, but if some asshole with an emotional support badger gets seated next to me, we're going to have a problem.

Again, I think this is a great idea, doing a dry run so people know a bit about what to expect when traveling by air. I just think that when they get to the real thing, they might be in for a rude awakening.

I haven't done one of these in a while, because I've been stuck at home for various reasons. But yesterday, I ventured forth to see this movie. And I hope to get back into my once a week habit soon.

One-Sentence Movie Review: Thor: Love and Thunder

I like comics-based movies, and I like movies that don't take themselves too seriously, and therefore I like comic book movies that don't take themselves too seriously—but this one maybe takes that concept a bit too far, verging on self-parody and making light of some very serious situations; nevertheless, despite some choppy editing in its earlier scenes, it's a fun foray into the MCU, with the best soundtrack since the last Thor movie (though not as good as the ones from the Guardians movies).

Rating: 4/5

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