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

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 20 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 22 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

2 Like

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.

1 Like

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').

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

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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