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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Elycia Lee ☮

about 7 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 23 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

February 3, 2020 at 12:07am

#974769

Chewing the Fat

Food.

We all need it, but here in the developed world, at least, it's become a source of anxiety and a victim of misinformation.

Case in point:

https://www.popsci.com/not-in-ketosis/

Sorry, keto fans, you're probably not in ketosis

Then again, you shouldn’t be anyway.

I haven't bitched about nutrition science in here lately, so it's probably about time. To recap, my problem with nutrition science is at least two-pronged: 1) Studies, done with limited participants and often questionable methods, tend to be non-replicable, but only the initial results are usually reported; moreover, reporting tends to be breathless and sensationalized. 2) The funding source for any of these studies can skew the outcome; thus, we end up with numerous articles claiming that chocolate is good for you - many of the studies for which were funded by the likes of Nestle.

Over the course of my lifetime, I've seen numerous fad diets and lousy reporting, as well as things like the see-saw findings about eggs: first they were good for you, then bad, then good, then bad, then only the whites were good, then it turned out that the whole egg is good, but not in excess (to be fair, nothing is good in excess, not even water). Fats were necessary, then to be avoided and replaced with carbs, but then suddenly carbs were poison, and then some fats are okay and some carbs got the green light... about the only constant, so far, has been an acknowledgement that the third macro, protein, is kinda necessary for life. But then people started arguing about sources of protein, leading to eating disorders such as veganism. And then, on the other end of that particular spectrum, you had things like the Atkins diet, which as I understand it (I probably don't, really) is mostly protein.

This is all complicated by the stunning discovery that everyone is different and has unique nutritional needs, thus throwing all of the one-size-fits-all studies into question.

My own weight loss has been made possible by two things: a formal plan that balances the three macronutrients (lipids, carbs, protein), and exercise. I still have a ways to go, but I've made significant progress. It works for me; it may not work for everyone. So, to get back to the point of this particular entry, it's good to find a plan that works for you -- but at the same time, it's important to get all the facts.

And facts can be hard to find amidst all the hype.

Keto is hard. If it's not hard, you're probably not doing it right.

You can say that about a lot of things, not the least of which is the double entendre so obvious that it's hardly (snort) worth mentioning.

The diet gets billed as a miraculously enjoyable diet—eat all the fat you want, just cut out the carbs.

And I'm sure that's some peoples' idea of "enjoyable," but for me, bread is life. I'm fine with mostly eating whole-grain, fiber-rich versions, in moderation, but there's no way in hell I'm ever going to give up bread. I'd stop drinking first. You can pry my baguette from my cold, dead fingers. Hell, someone will probably have to.

Researchers aren’t exactly sure what those differences are, but Fung says it's so hard to get adults into deep ketosis (which is likely deeper than a dieter's target) that often nutritionists don't even attempt it as a therapy.

Nutritionists are commonly a divided lot, so if most of them are in agreement about something, it's probably worth noting.

“Keto is not easy to maintain, it’s not a palatable diet,” says Andrea Giancoli, a dietician and nutrition consultant in California. Getting 80-90 percent of your calories from fat—which is what’s generally required for keto—is actually difficult.

Leaving aside for the moment the question of whether "dietician and nutrition consultant" is an actual thing or if it's something anyone who's read a couple of books can claim, I simply can't imagine eating that much fat. Or, well, sure, I can imagine it, just like I can imagine the carnage afterward.

And when you're (nutritionally) starving, your body will start to break down protein just to get those sweet, sweet carbs. Of course, you have a source of protein in your body already: your own muscles. "When in starvation mode, your body breaks down muscle in your body," says Giancoli. "Ketosis is a way of trying to preserve that protein. It's not ideal, but it's your body's way of saving you."

Biology is insanely, diabolically complicated. I'm always hearing about how people are math-phobic, but math is beautifully simple compared to biology. I can't even pretend to understand more than a tiny sliver of the subject matter. Most people can't. Part of that is because of the way we categorize different branches of science - physics is physics; chemistry is chemistry; quantum theory is quantum theory; fluid mechanics is fluid mechanics; and biology is all of these sciences plus a few more thrown in for good measure, plus all the math. The miracle is that anyone understands anything about it.

Of course, ketosis itself comes with its own risks. Circulating ketone bodies make your blood too acidic, and your body will draw calcium from your bones as a buffer.

See? Chemistry. Both organic and inorganic. Fun!

Without the fiber from whole grains and fruits, you're also likely to get constipated and have other digestive issues. Plus you need fiber to maintain a health[sic] gut microbiome, which tends to come from the kind of whole grains that you can't eat on the diet, and though it is possible to get enough fiber from vegetables on the keto diet you have to carefully monitor your eating to ensure that.

Now look, earlier snark about veganism aside, I'm not here to cast judgment on what anyone eats. For one thing, like I said, we're all different individuals with different needs and desires; for another, I know jack shit. But for your own sake, do the research and try not to fall into the trap of confirmation bias.

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