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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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day 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

a day 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.

July 31, 2019 at 12:32am

#963475

Scrambled Eggs

Today, kids, it's time for another episode of Adventures in Epistemology.

https://www.theatlantic.com/health/archive/2017/02/eating-toward-immortality/515...

Eating Toward Immortality

Diet culture is just another way of dealing with the fear of death.

Well, you can't spell "diet" without "die."

Knowing a thing means you don’t need to believe in it. Whatever can be known, or proven by logic or evidence, doesn’t need to be taken on faith.

And here's where the epistemology

comes in - right in the lede. Normally, assertions like this set off my bullshit detector and make me Stop Reading There. To be specific, there's a philosophical argument that we can't really "know" anything. I don't really agree with that argument, but as I've pointed out before, there are degrees of knowing. Drop an apple and it will fall - well over 99% certainty. Anthropogenic climate change - maybe 98% certainty. My assertion that the relevance of a logical conclusion is dependent on the validity of the initial premise(s)? Well, I'm an engineer, not a philosopher, so mmmm... 95%.

Point is, few things are ever "proven" in science. And that goes orders of magnitude more for nutritional science.

Which brings me back to the article. Now, it's a long one, and I'm only quoting a few excerpts here; I recommend reading the link before proceeding. Or don't, and skip this entry. Or do whatever you want. Whatever.

Eating is the first magic ritual, an act that transmits life energy from one object to another, according to cultural anthropologist Ernest Becker...

Ah, yes, cultural anthropology. A discipline known for its diligence and scientific rigor. {/sarcasm}

That's not cultural anthropology; that's metaphysics. Philosophy. And that sentence is opinion masquerading as fact. A few opinions, actually, standing on each other's shoulders wearing a lab coat to try to look like legitimate science.

...the desire for more life—not just delaying death today, but clearing the bar of mortality entirely—grew into an obsession with transforming the self into a perfected object that might achieve a sort of immorality.

You know what I love more than just about anything that wasn't fermented? It's when someone misspells something and doesn't bother to edit, only spell-check, and the result is hilariously appropriate. "Immorality," indeed.

If it weren’t for the small chance of death lurking behind every food choice and every dietary ideology, choosing what to eat from a crowded marketplace wouldn’t be considered a dilemma.

Yeah, yeah, we're all gonna die. If not from salmonella-infected romaine, then by getting hit by a meteor or run over by a bus or, at the end of a lifetime of self-denial, slipping in the fucking bathtub. You know, one of these days I'm going to go to Japan. When I do, I want to try fugu. Now there's an actual risky food.

Almost all children go through a phase of pickiness with eating. It seems to be an evolved survival mechanism that prevents us—once we are mobile enough to put things in our mouths, but not experienced enough to know the difference between safe and dangerous foods—from eating something toxic

Now, I'm no expert on child-rearing (actually I'm about the furthest thing from that you can get), but it seems to me that "almost all children" also go through a phase of shoving every goddamned thing they can find that will fit in their pieholes - worms, hamsters, dog shit, Dad's cufflinks, whatever - once they're "mobile enough to put things in our mouths." How is that an evolved survival mechanism, Ms. Evolutionary Psychologist Wannabe? No, Mom has to go "What's in your mouth? Spit it out SPIT IT OUT!!!" That's how we've survived as a species - diligent parents, not by being picky eaters.

Only those with status and resources to spare can afford the most impressive gestures of renunciation. Look at all they have! The steel-and-granite kitchen! The Le Creuset collection! The Vitamix! The otherworldly glow! They could afford to eat cake, should the bread run out, but they quit sugar. They’re only eating twigs and moss now. What more glamorous way to triumph over dirt and animality and death? And you can, too. That is, if you have the time and money to spend juicing all that moss and boiling the twigs until they’re soft enough to eat.

Okay, that bit is legitimately hilarious.

People willingly, happily, hand over their freedom in exchange for the bondage of a diet that forbids their most cherished foods, that forces them to rely on the unfamiliar, unpalatable, or inaccessible, all for the promise of relief from choice and the attendant responsibility.

Okay. Okay. You got me. Yeah, I can admit it. Though one quibble: I didn't do it willingly or happily. I keep asking myself why I'm putting myself through a weight-loss regimen, and one answer I don't come up with, ever, is "fear of death." Though I can't honestly say why I'm doing it - except maybe because I want to see if I can do it.

Look, I'll give you a little analogy. Think for a moment, I mean, really consider (though you don't have to tell anyone): what is the one thing that brings you the most joy in life? Maybe it's your kids. Your dog. Your life partner. The satisfaction of successful completion of a task. Something else, something unique to you. Maybe it's more than one thing; if so, just pick one of them. Now, someone comes up to you with scientific and anecdotal evidence that is extremely compelling, evidence that this thing that gives you the greatest joy will shorten your life, and you'll have a life expectancy of, I dunno, say another five years, if you give it up. No guarantees - there's always the chance of slipping in the bathtub - but the possibility. Really think about this. Never seeing your kids again. Always failing at everything. Never being able to have a dog. Whatever it is.

Could you do it?

That's me with food.

What's the point of maybe living longer if you have to give up the thing that makes life worth living?

Humans are the only animals aware of our mortality...

*fhweeet* Assertion without evidence! Flag on the play!

...and we all want to be the person whose death comes as a surprise rather than a pathetic inevitability. We want to be the one of whom people say, “But she did everything right.”

NO. I want to be the one of whom people say, "He lived life on his own terms." Maybe even "He died doing what he loved," though I don't have enough hands to simultaneously eat, drink, smoke, and play blackjack whilst receiving oral sex. Okay, maybe that's TMI. Actually, I couldn't care less about the oral sex; I just threw that in there to make a funny.

But diet culture is constantly shifting. Today’s token foods of health may seem tainted or passé tomorrow, and within diet culture, there are contradictory ideologies: what is safe and clean to one is filth and decadence to another.

Well, DUH. Possibly the oldest known dietary restrictions we've placed on ourselves can be found in the Old Testament. There might be some in older Babylonian texts, but I'm not a historian, either. Point is, though, even those were designed to set one people apart from another. This has gone on ever since. "Our rivals eat pork for their rituals to the Wrong God, so we must not eat pork."

Nutrition science itself is a self-correcting series of refutations.

Fixed that for you. Very little in nutrition science is set in concrete, except maybe that we shouldn't eat concrete. "But it's rich in calcium!" "Shut up."

To eat without restriction, on the other hand, is to risk being unclean, and to beat your own uncertain path.

I always had restrictions. Notably, I won't eat anything smarter than I am. That leaves out octopus, cat, calamari (most of which is actually made from cuttlefish, not squid, which would be only marginally better), border collie, elephant, dolphin, etc. Also corvids, and yet I find myself eating crow on a regular basis. Well, metaphorically, anyway.

Cows? Chicken? Fair game. Jury's still out on pigs.

Look, I'll accept that this author knows more about nutrition than I do. That's why I bothered to read the article. And she raises some interesting points. But I'm still not convinced that nutrition "science" has many legitimate or useful answers. Why, in my lifetime alone, eggs have swung from good to bad to good to bad to good to bad so many times that they're basically all scrambled at this point.

Still, it's good to read this sort of thing from time to time. And comment on it. Helps me focus my own thoughts, you know, what with fewer calories available to power my immense brain.

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