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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Write_Mikey_Write!

about 6 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 8 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.

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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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

July 12, 2021 at 12:02am

#1013491

Politics

Might as well accept that I'm going to be doing more prompts from "JAFBG"

[XGC].

What pisses you off most about politics?

Sometimes, the other side has good ideas.

I want to say that nothing is black and white, but that's an absolutist stance, so it's wrong. Some things are actually black/white, like "genocide is a bad thing." While there are people who will disagree with that statement, they're, by definition, evil. Almost everything else has gray areas.

Here in the US at least (I'm not embedded in other countries' systems), we tend to treat politics like a team sport. There are exceptions, of course, but a large number of Republicans are Republicans because that's their team, and the same with Democrats, regardless of actual policies and effects. It's like if you're a Yankees fan, you're going to support the Yankees whether the sportsball team is doing well or not.

It gets so that anything a Republican president does is routinely opposed by the Democrats, and vice-versa. That's even if it was their idea in the first place, like with universal health care. They stopped listening to each other and started gathering into tribes -- with, of course, a few exceptions, as always.

Now, I'm not saying "both sides are bad" here. I definitely have a "side," a bias in my political leanings. I don't talk about it much in here because I don't want to start pointless arguments, but I'm sure you've figured it out.

But I don't generally see the other side as evil. Some individuals, sure, but not the entire bloc. And that's what pisses me off the most about politics: that so many people see it as us vs. them, my people vs. their people, when the reality is the solutions are almost always somewhere else.

You all know what happened in my town four years ago: a rally to protest the impending removal of a Confederate general's statue from a prominent position downtown, a gathering that resulted in at least three deaths. Well, you may or may not have heard that yesterday, the people in charge of such things finally removed the statue.

Was it the "right" thing to do? I couldn't tell you. I can see arguments on both sides, but in the end, I don't think we need participation trophies, and I think it was a reminder of a past best left in history books, not endlessly glorified in bronze. But the political process here is supposed to be based on words and ideas, not guns or homicide by motor vehicle. Virginia may not have invented that ideal, but we sure as hell popularized it 250 years ago. (I will also remind everyone that most of the bad actors in that situation were bused in from out of state.) The tribalism, though -- that circumvents the exchange of ideas and replaces it with, in extreme circumstance, an exchange of punches, or gunshots.

Tribalism begets violence. Not always, and not immediately, but when you get people identifying with a certain subgroup and they find themselves at odds -- ideologically, or over resources -- with another group, eventually it usually comes to blows. Don't get me wrong, though; sometimes violence is necessary, or at least preferable to rolling over and accepting whatever other people decide is your fate. I remember a discussion from a social studies class in high school. Someone said, "violence never solves anything," and the teacher rattled off a list of the things that violence definitively solved, like in World War 2.

And yet, the root cause of that war was... tribalism. If certain people hadn't decided that their culture was somehow inherently superior to all others, perhaps it would not have happened. It's fine to identify as whatever, and even be proud of it (depending on circumstances), but it might help to remember that we're all one people, interdependent with the other species on Earth.

A friend, yesterday, challenged me with the question: "Have humans ever done anything good, ultimately, long-term?" The context was not just the statue controversies but the existential threat posed by human-caused climate change. I was hard-pressed to come up with an answer, and I consider myself a humanist. I mean, there's space travel, right? So we can exploit other worlds. There's art and architecture, but do they really improve upon nature? (That one's obviously very subjective.) For every asshole who kicks puppies or abandons kittens, there are ten or more people working to rehabilitate animals; but they wouldn't be in that condition if not for us. Many people work to raise other people from poverty, but, as she pointed out, it was humans who put them into the societal structures that made them poor in the first place.

In the end all I could come up with was "beer," and that was invented something like 10,000 years ago.

I genuinely believe that most humans are good, or at least try to be. But peoples' definitions of "good" vary, too, and what one group considers a benefit, another will consider to be a downside. The example I usually hold up is Mother Teresa: I think she genuinely thought she was doing good, but she was, in the end, as evil as they come. I'd cast her as a villain, not a saint -- but the reality, again, is somewhere in the middle.

The point being, we can have the best intentions and still do wrong. That's why we have to listen to the other side, sometimes, and not just demonize them and reject everything they say or do simply because they're on another team.

I've rambled on long enough, but hopefully I've conveyed exactly what pisses me off about politics. It's still mostly a Forbidden Topic here, but I reserve the right to delve into it occasionally as it is, in fact, my platform to raise statues on.

Or remove them.

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