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 5 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 7 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.

April 30, 2024 at 9:51am

#1070196

Mad as Hell

Sometimes I have to be reminded of stuff, because apparently I have a problem with object permanence once an item drops off my Favorites sidebar. In particular, "JAFBG"

[XGC] is still around, under new management, and there have been prompts to tackle since the beginning of the year.

As usual, I picked one at random:

How useful do you think anger can be?

Well, I'm not a psych-talker person, nor are emotions something I know much about. I can barely identify them in myself, let alone in others.

That said, it seems to me that anger is, at base, a manifestation of the well-known "fight or flight" response to a threat. That makes it related to fear, in my book (which I won't write because, like I said, not an expert). The "flight" part probably doesn't involve anger, but the "fight" part does. I hear that, in those cases, anger can be fairly useful to focus one's hostility into action... so long as it doesn't escalate into blind rage, which leads to random flailing around and, eventually, defeat.

I used language related to physical conflict there, but this also applies to the mental realm, like when government passes a law you don't agree with, and anger leads you to mobilize.

"I'm as mad as hell, and I'm not going to take this anymore!"

The quote there is from a 70s movie, Network. I don't recall ever seeing the film, but the line is iconic, with or without the context provided by the story. So of course, I looked it up on Wikipedia,

because it's not like I have to avoid spoilers for a movie that came out before Star Wars.

Which, if you take a look at the link there, provides a nice segué into the other half of the equation. Above, I touched on how useful anger might be to the person feeling it. But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation.

An angry person doesn't think rationally. This is almost by definition: "mad" is a synonym for "angry," but it's also a synonym for "fucking nuts." If you don't want people thinking rationally, if you want them to react with their limbic system instead of their neocortex, as it were... get 'em angry. Convince them, through emotional language, that their social position, their way of life, their very lives are in imminent danger, and you'll sell more torches and pitchforks.

I always thought it would be fun to put a Torch and Pitchfork store into an RPG campaign, to provide helpful tools for the occasional riot. The proprietor would be bored most of the time, so, occasionally, they'd drum up some business by manufacturing some sort of outrage. "The King wants to increase taxes so he can gold-plate his throne! Meanwhile, honest merchants are struggling! Are we going to let that stand?"

Might be a little too close to reality for a fantasy role-playing game, though.

So, to answer the prompt, anger can be very useful... in the people you're trying to influence. Way more useful there, as an agent of chaos, than it can be for an individual reacting to an actual threat.

So, if a headline spurs you to anger... take a step back. While suppressing anger is probably bad (or so I've heard), it may be possible to calm down enough to let some rational thought take over. If I'm right about the fight-or-flight thing, which I can't guarantee, anger is a kind of burst emotion; I imagine a whole cascade of neurotransmitters firing indiscriminately, urging us to action. And that burst can't really be sustained, any more than more pleasant emotions can.

Anger is a useful tool for the people in charge. It's way more useful for the rest of us to recognize when we're being manipulated, and sidestep that.

In my unprofessional opinion.

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