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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August 28, 2022 at 12:03am

#1037010

Past Tense

Everything I need to know about medieval peasants, I learned from Monty Python and the Holy Grail.

And maybe today's article.

What Did Medieval Peasants Know?
The internet has become strangely nostalgic for life in the Middle Ages.

The period, which spans roughly 500 to 1500, presents some problems for people trying to craft uncomplicated stories. “No age is tidy or made of whole cloth, and none is a more checkered fabric than the Middle Ages,” Tuchman wrote. Historians, she noted, have disagreed mightily on basic facts of the era...

We know enough for comedy movies. You know Holy Grail is almost 50 years old now? Still one of the greatest and most quotable movies of all time.

...the Middle Ages have been a common hobbyhorse for people of all political persuasions who suspect modernity might be leading us down the primrose path, especially as the internet has become a more central and inescapable element of daily life. Our ancestors of the distant past can be invoked in conversations about nearly anything: They supposedly worked less, relaxed more, slept better, had better sex, and enjoyed better diets, among other things.

Sure, you can prop up straw men and just make shit up about the past. But look, Hobbes (the philosopher not the tiger) described our lives as "nasty, brutish and short" for good reason. Namely, the past sucked. It sucked hard. I can kind of understand the Renaissance Faire people, but even they know they're just role-playing, and if one of them got a foot infection, they'd go to a modern hospital and not have a barber treat them with leeches. Or whatever.

The present is no walk in the park, but the problem isn't the internet; it's endgame capitalism. And at least we have global trade and laser eye surgery.

The problem is that these assertions about our glorious history usually don’t quite check out—they tend to be based on misunderstandings, disputed or outdated scholarship, or outright fabrications long ago passed off as historical record. But that doesn’t stop people from regularly revisiting the idea, counterintuitive though it may be, that some parts of life were meaningfully better for people who didn’t have antibiotics or refrigeration or little iPhone games to play to stave off boredom. What, exactly, is so irresistible about a return to the Middle Ages?

Boredom? I'd bet the peasantry had lots of problems, but boredom wasn't one of them.

But as the article points out, I don't know shit.

He now thinks that English peasants in the late Middle Ages may have worked closer to 300 days a year. He reached that conclusion by inspecting the chemical composition of fossilized human remains, as well as through evidence of the kinds of goods that urban peasants in particular had access to.

This is in reference to the claim that European peasants worked like 150 days out of the year. Which, even on the face of it, has got to be suspect. After all, what was the ruling body of Europe, the one thing that defined and united the continent during what we call the Middle Ages? The Church, obviously. And what's the Official Doctrine of the Church? "Six days shalt thou labor..." It's right there in the beginning. Now, sure, there's question about how much the lower classes could actually read, but they'd have gotten the message. So 365 minus 52 Sundays, and take off maybe a few other days for Suck the Duke's Dick Day or whatever, and 300 sounds reasonable.

Compared to that, our standard two-day weekend is luxury itself.

Clark and his colleagues have revised their estimates upward, but the school of thought his previous numbers belonged to still has many academic supporters, who generally base their estimates of how much peasants worked on records of per-day pay rates and annual incomes. “This other view is that they were quite poor, but they were poor kind of voluntarily, because they didn’t like work and didn’t want to do a lot of work,” he told me.

There were probably signs up at medieval convenience stores: No Onne Wonts Too Werk Anymoor.

“It allows people to make their own medieval mythology and cling to that,” Janega told me. “They’re just kind of navigating on vibes.”

Like I just did. Only I know I did it.

The clear delineations that people assume between work and personal life just aren’t particularly tidy for peasants doing agrarian labor. “They’re thinking of these people as having, like, a 9-to-5 job, like you’re a contracted employee with a salary and you get vacation days,” she told me. “The thing about having a day off is like, well, the cows ain't gonna milk themselves.” So while people are correct that European peasants celebrated many more communal holidays than modern Americans, in many cases, that just meant they weren’t expected to do a particular set of tasks for their lord. Minding the animals, crops, and themselves never really stopped. Their vacations weren’t exactly a long weekend in Miami—after all, they didn’t really have weekends.

I'm imagining a feudal serf clocking in and out of their field. Er, their lord's field. Whatever.

There were, of course, some other obvious downsides to medieval life. A huge chunk of the population died before the age of 5, and for people who made it out of childhood, the odds of seeing your 60th birthday weren’t great. There wasn’t any running water or electricity, and there was a very real possibility that mercenaries might one day show up and kill you because your feudal lord was beefing with some other lord.

At least there was beer. On the other hand, there was a nonzero chance that you'd be burned as a witch for brewing it.

If you’re looking for a vision of history where people were generally peaceful and contented, though, you might want to check in with societies outside of the Middle Ages. Perhaps look for a group of people not perpetually engaged in siege warfare. “Medieval peasants are a weird one to go to, because, you know, they were rebelling constantly,” Janega noted. “Why are they storming London and burning down the Savoy Palace, if this is a group of happy-go-lucky, simple folk who really love the way things are?”

And maybe don't be so hyperfocused on Europe? I get it; the vast majority of Americans have (relatively) recent ancestry there. But other areas have much to teach us as well, as indicated by, for instance, the Chinese invention of the compass, the Indian invention of the concept of zero, and the Arabic invention of higher mathematics—all of which took place during the European Middle Ages.

The article actually has quite a bit more relevant information than what I've copied here, but the takeaway, at least for me, is: don't look at the past with rose-colored glasses (spectacles were a medieval European invention).

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