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

August 7, 2022 at 12:01am

#1036196

Milking It

I like food. I like science. I do not like food science.

The Vindication of Cheese, Butter, and Full-Fat Milk
A new study exonerates dairy fats as a cause of early death, even as low-fat products continue to be misperceived as healthier.

Nevertheless, this article seems to have its head on straight. Though that may be confirmation bias on my part.

As a young child I missed a question on a psychological test: “What comes in a bottle?”

The answer was supposed to be milk. I said beer.

Already, I like this guy.

Milk almost always came in cartons and plastic jugs, so I was right. But this isn’t about rehashing old grudges. I barely even think about it anymore!

Which is obviously why you led with it.

The point is that the test was a relic of a time before me, when milk did come in bottles. It arrived on doorsteps each morning, by the hand of some vanishing man.

Dude, that's no way to talk about your biological father.

And just as such a world was alien to me as a kid, the current generation of small children might miss a similar question: “Where does milk come from?”

The grocery store, duh.

Many would likely answer almonds or beans or oats.

But it's so hard to squeeze their little teats.

Indeed, the already booming nut-milk industry is projected to grow another 50 percent by 2020. Much of this is driven by beliefs about health, with ads claiming “dairy free” as a virtue that resonates for nebulous reasons—many stemming from an earlier scare over saturated fat—among consumers lactose intolerant and tolerant alike.

Except that most of those "milks" come from produce grown in the Southwestern US. In case you haven't heard, the Southwest is running out of water. No water, no faux milk.

It all happened quickly. In the 1990s, during the original “Got Milk?” campaign, it was plausible to look at a magazine, see supermodels with dairy-milk mustaches, and think little of it. Now many people would cry foul.

Not that quickly. That was 30 years ago.

Difficult as it may be for Millennials to imagine, the average American in the 1970s drank about 30 gallons of milk a year. That’s now down to 18 gallons, according to the Department of Agriculture.

Stop lumping everyone into one generational bucket.

Full disclosure: I don't drink milk. I'll eat the hell out of milk derivatives, though: yogurt, cheese, butter.

And just as it appears that the long arc of American beverage consumption could bend fully away from the udder, new evidence is making it more apparent that the perceived health risks of dairy fats (which are mostly saturated) are less clear than many previously believed.

By "less clear" you mean "probably nonexistent."

An analysis of 2,907 adults found that people with higher and lower levels of dairy fats in their blood had the same rate of death during a 22-year period.

Decent sample size, fair methodology. I would like to know the ages of the "adults," however. If there were negative consequences to consuming dairy, I don't think they'd show up for adults in the 18-40 (22 years) range.

The implication is that it didn’t matter if people drank whole or skim or 2-percent milk, ate butter versus margarine, etc. The researchers concluded that dairy-fat consumption later in life “does not significantly influence total mortality.”

Pretty sure it's margarine that's horrible for you.

The researchers also found that certain saturated fatty acids may have specific benefits for some people. High levels of heptadecanoic acid, for example, were associated with lower rates of strokes.

Yeah, thanks. I'm not ignorant about chemistry, but I'd never heard of that particular chemical, and the article is silent about where it may be found.

As an aside, yes, you CAN pronounce that if you sound it out, everything consists of chemicals, and we shouldn't be promoting ignorance by smugly claiming we don't eat things we can't pronounce.

I know I've said all that before, but it's important.

De Oliveira Otto believes that this evidence is not itself a reason to eat more or less dairy. But she said it could encourage people to give priority to whole-fat dairy products over those that may be lower in fat but higher in sugar, which may be added to make up for a lack of taste or texture.

Yeah, I think the real problem involves replacing fats with carbs. You're still getting a similar number of calories, but your macros can be way off.

Veganism is not encouraged given a national interest in continuing to consume the dairy the country produces.

Good.

The takeaway is that, from a personal-health perspective, dairy products are at best fine and reasonable things to eat, and avoiding butter and cheese is less important than once believed. While the narrative that cheese and butter are dangerous is changing, it also remains true that dairy isn’t necessary for children or adults.

It may not be necessary, but it's also a decent, lower-cost source of protein. And there's always that middle ground between veganism and not having an eating disorder: vegetarianism, which allows for consumption of dairy products but not meat (which is expensive as shit anyway).

A diet rich in high-fiber plants has more than enough protein and micronutrients to make up for a lack of dairy...

So you're saying that dairy is necessary for an American diet.

...and the vitamin D that’s added to milk can just as well be added to other foods, taken as a supplement, or siphoned from the sun.

Screw supplements, and the alternative is venturing (shudder) outdoors.

At this point, the clearest drawbacks to consuming animal products are not nutritional but environmental, with animal agriculture contributing to antibiotic resistance, deforestation, and climate change.

Well, then, so be it.

Seriously, though, I might take this study more seriously than others, but I can't say the science is settled. All I know is, our predecessors managed just fine drinking whole milk, and we didn't start having neuroticism about food until that stopped.

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