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!

about 9 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 10 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').

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

February 9, 2022 at 12:03am

#1026325

Into an Abyss

Time for another one from "Journalistic Intentions"

[18+].

"Trauma does not make you stronger. Trauma makes you traumatized. The end."

Thinking is fun.

That's the basis for philosophy. Literally, "philosophy" means "love of wisdom" where wisdom is assumed to be based on thinking about shit.

Historically, every branch of thinking was deemed to be philosophy. Physics, for example, used to be called "natural philosophy," as if other forms of thinking were unnatural or something.

Anyone who's been following along in here knows that I'm not above making fun of philosophy that I think is ultimately bullshit. And I have, in the past, railed in particular against a certain quote from Nietzsche, commonly translated as "What doesn't kill you, makes you stronger."

Now, I'm not ragging on Nietzsche here. Dude was epic, and I'm not just talking about his mustache. But of all his commonly misrepresented sayings, that one ranks right up there with "God is dead."

The sense of the original quote is that it is possible, in certain circumstances, to grow from misfortune. It's probably a better attitude to have than crumpling up into a little ball of tears whenever something doesn't go your way -- but that's not the only alternative. The problem comes in when you start thinking that only the people who grow from suffering are worthwhile, and everyone else is automatically a pussy.

I've mentioned my uncle in here before. An American Jew of partially Polish descent, he entered the Army in WWII, and was in one of the Allied crews that discovered Dachau.

Before that, per my limited understanding, the full extent of the horrors visited upon the perceived enemies of the Nazis, including but by no means limited to Jews, wasn't really known in the US. There were rumors, of course, but it would have been hard to accept that the arguably most advanced civilization in the world at the time would engage in such atrocities.

He found out otherwise, and once the full knowledge of what the concentration camp overseers had been doing hit him... well, he was never the same again. Today, we'd call it PTSD. Back then, it was "shell-shock," and carried its own stigma. And there was no real treatment for it. His health gradually declined, and he eventually died of pneumonia in the early 90s.

The irony that the Germans of the time might have been influenced by Nietzsche is not lost on me.

The point is, sometimes the shit that doesn't kill you, destroys you in other ways. Trauma isn't something that we just "get over." Some people do, I suppose, but their experience isn't universal, and thinking that it is can only lead to marginalizing those who have real issues. "He just wasn't strong enough." Bullshit. If you're going to sit there and tell me that someone whose life you can barely even comprehend just should have shown more intestinal fortitude, you can fuck right off with that shit. And sometimes, becoming "stronger" means losing empathy, compassion, or some other attribute that is essential for being fully human.

I'm curious about one thing, though: in English, "strength" may refer to purely physical strength (which I don't think is the intent of the Nietzsche quote), or mental resilience, or perhaps competence, or several other attributes. Writing in German, Nietzsche proclaimed, without much elaboration, "Was mich nicht umbringt, macht mich stärker." I'm just wondering what sense "stärker" has in that language, or maybe had in the late 19th century when he was active.

I always assumed the quote referred to mental resilience, but you know what assumptions do.

Also ironically, the best counterpoint I can think of to that aphorism comes from the same source, and it's one that, by contrast, I have always considered to be essential life advice -- and I think it's especially relevant to my uncle's experience:

Whoever fights monsters should see to it that in the process he does not become a monster. And if you gaze long enough into an abyss, the abyss will gaze back into you.

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