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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

March 7, 2021 at 12:02am

#1005950

Praise Cheeses!

Today, in Adventures in Nutrition Science, another about-face:

Great News, America: Cheese Isn’t Bad for You
Don’t feel sheepish reaching for that manchego. Cheese doesn’t deserve its unhealthy reputation.

Well, at least the headline isn't in the form of a question.

Cheese is among the ultimate guilty pleasures. It’s gooey. It’s fatty. It’s delicious. It just has to be bad for you, right?

I mean, sure, cheese is good, but guilty pleasure? I hate that phrase anyway. Either enjoy something and wallow in it, or don't. No need to be coy about it.

Wrong. A large body of research suggests that cheese’s reputation as a fattening, heart-imperiling food is undeserved.

And this is why people have a distrust of science in general. Personally, I gave up on that field long ago, when they couldn't make up their cursed minds about eggs.

Just to be clear, nutrition science is wishy-washy like this because there are lots of hidden variables, variations among populations, suspect methodology, publication without peer review, incentives to find one result or another (that "chocolate is good for you" study was funded by chocolate companies, e.g.) and, most of all, a lot of really bad, breathless reporting about it all.

In the end, people extend their distrust of nutrition science to science in general, because your average person wants The One True Answer, which science can't give, but in other disciplines it's at least easier to approach. This is how you get shit like vaccine scares or climate change deniers.

“There’s almost no evidence that cheese causes weight gain—and in fact, there’s evidence that it’s neutral at worst,” says Dariush Mozaffarian, the lead author of the 2011 paper and dean of the Tufts Friedman School of Nutrition Science and Policy. “There’s no evidence that cheese is linked to cardiovascular disease, and in some studies, it’s even a little bit associated with lower risk. And then, for diabetes, again, it’s at worst neutral, and maybe protective.”

And so people end up back to believing whatever the hell they want to believe, and cherry-picking whatever study results back up their pre-existing beliefs. This particular quote I just pasted backs up my pre-existing belief, so I choose to believe it. Besides, even if someone posted "cheese will kill you dead," it wouldn't stop me from eating that delicious goodness. Because what's the point of living if you can't do the things that make life worth living, including eating cheese?

So just to recap so far: The evidence tends to show that cheese does not make you gain weight. Why hasn’t this amazing news spread more widely?

Probably because no one has bothered to put it into commercials or fund PR campaigns, the way they do with studies about booze or chocolate.

In the case of cheese, there could be several reasons for the surprising lack of impact on weight, though more research is needed. Cheese is fermented, meaning it has live bacterial cultures. That could have a positive effect on the gut microbiome, which appears to play a role in weight regulation. The fermentation process also creates vitamin K2, or menaquinone, which experimental studies have linked to improved insulin function. Insulin regulates blood sugar levels, hunger, calorie expenditure, and fat storage. (One upshot is that hard, aged cheeses, which are more fermented, probably provide more benefit than soft, less fermented ones.) There’s also some promising research about the benefits of a compound called the milk-fat globule membrane, which is intact in cheese but not in milk or butter.

All of these things are speculations requiring further research.

Other high-fat foods, like avocados, have lately enjoyed a reputational rehabilitation. Cheese, not so much. One reason may be the fact that cheese contains not just a lot of fat but a lot of saturated fat, a major dietary scofflaw linked to higher blood cholesterol and rates of cardiovascular disease. But here, too, the science says cheese doesn’t deserve its stigma. While cheese does have high saturated fat, that doesn’t appear to correlate with higher risk of heart disease.

I should probably make the point here (I didn't see it in the article) that they're talking about cheese. You know... real cheese? Not Cheez-Whiz or Chee-Tos and definitely not the plastic abomination labeled "Pasteurized Process Cheese Food," like a certain well-known brand of "singles."

Anyway, it's worth reading the linked article, if only to see for yourself some of the lines of research that support the conclusion. I still cringe at the writing style, but hey, not everyone can be me.

And while I'm on the subject of cheese, I thought I'd share my favorite dessert. No, it's not cheesecake (which is delicious, but it would take a lot to convince me that it's in any way "good for you"). This particular combination of flavors synergizes well, and, most of all, it's dead easy to prepare.

All it takes is:
Pears
Walnuts
Stilton cheese
Port wine, preferably tawny port.

I indulged in this deliciousness yesterday, in fact. And now with the article I linked above, I can't even claim it's just me enjoying something I shouldn't, as all of those things are now... Good For You.

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