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

September 18, 2022 at 11:31am

#1037831

Eat, Drink, and Be Wary

I recently wrote about the origins of avocados. Those are in here, but so are some other delicious foods and beverages.

5 Secrets Of Everyday Foods That Go Back Thousands Of Years

Link is from Cracked, so I'd rely more on the comedy than on the facts. But I don't think they got the facts horribly wrong.

It’s fascinating to look back through history to see why we humans ended up eating the food we do. You come upon such explanations as “a company exec thought of it,” “an advertising exec thought of it,” “a marketing exec thought of it,” and—of course—“war.”

There's also "food is scarce so let's figure out how to make this disgusting thing marginally edible." The threat of death is part of the "necessity" that's allegedly the mother of invention (laziness is the milkman).

5. We Drank Milk For Thousands Of Years Before Evolving The Ability To Digest It

And some of us never quite did.

The article talks about a scientific study done to refute the idea that the ability to digest lactase gave some humans an evolutionary advantage. Then:

And then, over the course of the next 3,000 years, that gene variant, which was theoretically so useful ... didn’t spread at all. It remained just as rare. The vast majority of people lacked any ability to digest dairy, but everyone went on eating it just the same. “This cheese is too good to give up,” they said, blasting noisy diarrhea, for millennium after millennium.

The problem with that quote is that while the purpose of cheese is to make milk last longer. It has the side-effect of making it more digestible to lactose-intolerant humans.

We used to tell each other that lactase persistence offered a large evolutionary advantage, but it apparently did not. Maybe this was for the same reason that lactase persistence offers no boost to your life expectancy today.

Fweet. Flag on the play. Life expectancy isn't directly correlated with evolutionary advantage. Evolution only "cares" that you live long enough to reproduce. Anything after that may have some social benefit, but it's not a direct driver of natural selection.

4. We Domesticated Chickens For 7,500 Years Before We Figured Out Eating Them

You know, when you think about it, eating pretty much anything for the first time required some kind of leap of faith. Especially mushrooms, but also eggs. "I'mma eat the next thing that comes out of this chicken's ass. Hold my beer." The birds themselves, though? This chickenless past is genuinely surprising, considering how delicious those descendants of dinosaurs can be, and ancient humans must have witnessed other animals chowing down on poultry.

Take the Bible, which is full of talk of people slaughtering lambs, goats, and fatted calves. Notice what no one eats in the Bible? Chicken. There are special rules if you’re thinking of eating a camel or a badger, or a vulture or an ostrich, but no mention of eating chicken. (Which means we can assume chicken's not unclean, but still, no one in the Bible goes and eats one.)

No, chicken is absolutely kosher. Well, like any meat, it has to be prepared in a kosher manner, but that's a more recent development.

I'd always assumed they weren't in the Bible because they were native to the other side of Asia and hadn't quite spread to the Middle East.

3. The Goats Who Danced And Taught Us To Drink Coffee

That would make an excellent band name.

But we do know it took a lot of trial and error, based on evidence we have of prior attempts to eat that bush. Before humans ever made the beverage we call coffee, they made coffee protein bars by mashing the berries and mixing them with animal fats.

I dislike animal fat (except for delicious bacon), and I also never got a taste for coffee, so all I can say is:

Ew.

2. We Can Track The Rise Of Alcohol To Our Evolution As Primates

This one, of course, is the most relevant to my interests.

Humans have been making alcohol for quite a while, maybe even for 9,000 years. We’ve been drinking alcohol much longer than that, however. The first time anyone drank alcohol was when some ancient ancestor of humans picked up a fruit that had naturally fermented. They ate it, immediately spat it out, then ate some more, because it might have tasted foul but they liked what it was doing to them.

This is, of course, pure speculation, but I like it.

This section gets a little technical for a dick joke site, but it's a fascinating look into the biology involved.

1. Avocados Are So Huge Because Giants Swallowed Them

Fruit really is pretty cool, isn’t it? You may think wild berries don’t taste as good as, say, ham, but we carnivores evolved a taste for ham to recognize the nutrition present in meat, despite pigs not wanting us to eat them. Fruit, on the other hand, evolved to cater to our tastes, becoming more tasty because it wants to be in your mouth. Fruits contain seeds. The tastier the fruit, the more likely animals are to eat the seeds then defecate them somewhere distant, spreading that plant’s genes wide.

This is, obviously, simplistic and overgeneralized. Plenty of fruits don't want to be eaten. Chili peppers, for example. That spicy flavor comes from capiscum, as you probably know, and it evolved so animals would take one bite and go "nope." Humans being humans, though, we're like "I'mma eat that hot thing. Hold my beer. BY QUETZALCOATL'S SWINGING BALLS GIMME BACK MY BEER."

(Capiscum-laden plants originated in Central America and yes, those peppers are technically fruits. Like tomatoes.)

Lots of fruits had bigger seeds before we selectively bred the seeds out and shifted to cloning. Bananas were small and full of lumpy seeds before we fixed them.

I once saw a video made by a young-Earth creationist touting the banana as "proof" that God created the perfect fruit. I laughed. Bananas are evidence that selection pressure can change an organism.

Anyway, I went on about avocados in that entry I linked above, so no need to rehash it here. Re-guac it. Whatever.

Many plants were spread exclusively by megafauna. That means they would have gone extinct the same time megafauna did, had humans not cultivated them. In addition to avocados, these plants include pumpkins and gourds...

When we humans die out, all these crops will vanish. If we care about them surviving after we are gone, we have only one responsible course of action ahead of us: We must bring extinct megafauna back to life, and set them loose as farmers. We must unleash the mammoth, the dire wolf, and—yes—even the Megalania, a 4,000-pound venomous lizard.

I take back what I said up there. The awesome band name would be Megalania.

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