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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Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 9, 2022 at 12:01am

#1038889

Smells Like New Jersey

Entry #5 for the October round of "Journalistic Intentions"

[18+]:

One reason for environmental justice

I have this conceit, had it for a very long time now, that I can write something about anything. This started when I was quite young, when used to find random quotes in printed material (pre-internet, even pre-home computers) to expand upon in paper journals that have, to my vast relief, since been lost. In a way, this blog has turned into an extension of that; I guess I really haven't changed much, except now I make fewer fart jokes.

This prompt, though? This one stretches my abilities. I can't think of one reason for environmental justice.

No, I can think of way more than one.

For starters, there's a reason corresponding to every person living unwillingly in an environmentally-compromised location. Not just the people, either, but the wildlife and even the vegetation.

But I got to thinking: what would environmental justice actually look like? There's eight billion of us on the planet, and we all produce waste. I get the impression that some of the bigger producers of waste are the ones least affected by it, but without doing actual research, I can't be sure about that. Either way, though, it's clear that the bulk of the burden falls on people who are already disadvantaged in other ways.

"Justice" implies that, ideally, we should all feel the same burden. Thing is, then that would suck for everyone equally. And rich people would never stand for it; you have to remember the Golden Rule: "Those who have the gold make the rules."

I had a friend who was like, "Why don't we just shoot the waste into the sun?" Leaving aside for the moment the incredible expense of doing so, in order to do it, you'd have to cancel the load's angular velocity, as it starts out with the same value as that of the planet (give or take a few angular seconds/second, depending on where it's launched from and when). To do that you'd need reaction mass. Where would we get the reaction mass? Why, from the waste itself, of course. Which would mean, basically, burning it all before it even got to the accursed daystar. Along with the space barge carrying it.

But never mind that; it's just that I'm still laughing at the concept. No, in reality, it has to go somewhere on the Earth. And the further you move it from population centers, the more expensive it is to deal with. And then you have to employ people to take care of it—transport it, then bury or burn it—which would mean people traveling to and living near the waste disposal site, which puts the burden on them. We're not advanced enough to automate the whole schlemiel.

Producing less waste would be good, but you're never going to get it to zero. Basic thermodynamics.

So, what can we do? Hell if I know. Yes, being a civil engineer, this sort of thing is in my wheelhouse, but all that means for me is that I'm maybe more aware of the economies involved. And hell, right now I'm more focused on immediate quandaries. Like, it's getting cold at night now, which means I want to fix myself a nice hot mug of tea. The quandary is: why the hell does my tea kettle have the "Max Fill" line on the outside? It's utterly useless to me there, as the kettle is opaque.

I used to have a tea kettle where that line was an actual dimple in the metal, so you could, if you looked at it from the right angle, see it on the inside and then stop the faucet when the water reached that line. That tea kettle burned (NOT MY FAULT) and it became part of America's waste disposal problem a few years ago. But even then, could I ever be sure that, as I held the thing under the faucet, it was level enough so that the water surface at the line was correct? Even a slight tilt could result in more or less water than the desired maximum volume.

Point is, I can't even address this inequity, let alone the larger one concerning waste disposal.

So I'll just leave you with the video that came with this prompt. Maybe you can think of something where I didn't.

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