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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December 15, 2022 at 12:01am

#1041803

Luck You Too

Today's article is from way back in 2017, but I doubt that matters.

How to Be Lucky
It pays to imagine your life is on a winning streak.

I'm guessing that's easier said than done.

In 1995, a wounded 35-year-old woman named Anat Ben-Tov gave an interview from her hospital room in Tel Aviv. She had just survived her second bus bombing in less than a year. “I have no luck, or I have all the luck,” she told reporters. “I’m not sure which it is.”

Well, I'd say it's luckier to never be bombed at all. I've never been bombed, though, and I don't consider that any luckier than, say, never having been bitten by a rabid coyote.

Then there was Tsutomu Yamaguchi, the guy who happened to be in Hiroshima on... you know, that day. He suffered injuries from... you know, that thing, but he survived. The next day, he was even able to return home.

"Home," for him, was Nagasaki. And, well... you know.

He survived that one, too, but he did have lifelong problems related to radiation, and eventually died. In 2010, at the age of 93. Hell, my father didn't make it to 93, and he wasn't anywhere near Japan when the bombs fell. So... lucky? Unlucky? Well, I guess that depends on your perspective. Which is what this article is all about. Speaking of which, let's get back to it.

The news story caught the eyes of Norwegian psychologist Karl Halvor Teigen, now an emeritus professor at the University of Oslo. He had been combing through newspapers to glean insights into what people consider lucky and unlucky. Over the following years, he and other psychologists, along with economists and statisticians, would come to understand that while people often think of luck as random chance or a supernatural force, it is better described as subjective interpretation.

Getting hit by lightning is a low-probability event (assuming you're not flying a kite during a thunderstorm), and is generally considered unlucky. Winning a lottery jackpot is an even lower-probability event that is generally considered lucky, at least until you realize you're constantly going to be vultured by scammers and "relatives." Neither of those are supernatural forces, despite how lightning was viewed for most of human history. So again, it's a matter of perspective: does the low-probability event help your life, or harm it? If the former, then it's "lucky." If the latter, then "unlucky."

But of course, sometimes things that at first seem unlucky turn out to be pretty lucky. Like if you get hit by lightning, survive, and then meet a woman in the hospital who then becomes the love of your life. But then, naturally, she divorces you and takes the house, the car, and your bourbon stash. So maybe it wasn't so lucky after all.

Psychology studies have found that whether you identify yourself as lucky or unlucky, regardless of your actual lot in life, says a lot about your worldview, well-being, and even brain functions. It turns out that believing you are lucky is a kind of magical thinking—not magical in the sense of Lady Luck or leprechauns. A belief in luck can lead to a virtuous cycle of thought and action.

That's not really what I think of when I hear "magical thinking," but okay, I can run with it.

On the other hand, feeling unlucky could lead to a vicious cycle likely to generate unlucky outcomes. Psychologist John Maltby of the University of Leicester hypothesized that beliefs in being unlucky are associated with lower executive functioning—the ability to plan, organize, and attend to tasks or goals.

This is all getting perilously close to that "positive attitude" bullshit I keep railing against.

He offers a simple example of running out of ink in the middle of a print job. “The lucky person will have got a spare cartridge just in case because they have planned ahead. When the cartridge runs out they’ll say, ‘Oh, aren’t I lucky, I bought one earlier, that’’ fantastic,’ ” Maltby says. “However, the unlucky person won’t have planned ahead, won’t have done the cognitive processes, so when the printer cartridge runs out and they’re left with something to print, they go, ‘Oh, I’m so unlucky.’ ”

Or, if you're me, you congratulate yourself on having planned ahead for such a goddamned predictable occurrence. Like when I hit a deer in South Dakota last year. There were some things I considered lucky that day, not least of which being that it happened within sight of a service station, but even if it had been in the middle of nowhere, I had had the foresight to have insurance and roadside assistance lined up in the event of an emergency. That bit's not luck; it's planning.

While personality and gender seem to play a role, random events could also kick-start a virtuous lucky cycle or a vicious unlucky cycle. Economist Alan Kirman of the École des Hautes Études en Sciences Sociales in Paris realized this could be the case when he worked in an office with relatively few parking places nearby. One guy on his team always seemed to get lucky with parking spots close to the office, while another always had to park far away and walk. To figure out why, the team created a simple game-theory model to simulate the situation. It revealed that if would-be parkers happened to find spots near work early on, they continued to search in a narrow radius in the following days. If they didn’t find spots near work early on, they began to search in a wider radius. Guess who had lucky streaks when it came to finding spots near work? The ones who were actually looking for them.

Again, that sounds to me like simple planning.

Wait, someone found a parking spot in Paris? Sorry, no, now I believe in miracles all of a sudden.

Of course, believing in your own good luck isn’t always a good thing. In gambling, for example, lucky streaks are never what they seem.

In gambling, only the house gets lucky in the long run. But again, that's not luck; that's the ability to calculate probabilities and set payouts accordingly.

Here's what I felt was the most important bit, though:

The key to deciding whether an event is lucky or unlucky is the comparison you make between the actual event and the “counterfactual” alternative you’re imagining, Teigen says. The people asking “Why me?” are making an upward comparison to other people who weren’t assaulted or who avoided an accident. The people who feel lucky to have survived are comparing themselves downwardly to people who had a worse fate. Both are valid interpretations, but the downward comparison helps you to hold on to optimism, summon the feel-good emotion of gratitude, and to weave a larger narrative in which you are the lucky protagonist of your life story.

I don't think this is in accordance with the article, but my default mode of hopeful pessimism serves me really well there. See, if I expect the worst, and the worst doesn't happen (which is most of the time), then I automatically feel lucky. Like, to use another gambling example, if I'm playing blackjack and I'm showing 18, and the dealer's got a 10 up, my assumption is she's going to turn a 10 or a 9. Then when she turns an 8 (for the push), I consider that to be good luck. If it's a 7, then it's excellent luck.

Like any gambler, I still lose in the long run, but for that particular hand it feels damn good if I'm wrong. And if I'm right, I get that pleasure, too.

When times are tough, it might seem frivolous to cultivate a belief in luck. But that belief, psychologists say, can cast a spell that heals our wounds and gives us another shot at success, whether we’ve survived a bombing or just been on a bad date.

And I say luck is no substitute for planning. That's why engineers design redundancies into things. And it's why I try to avoid major cities during wars.

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