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').

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.

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

1 Like

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

December 4, 2020 at 12:01am

#999616

Barbaric

Well, this is an interesting combination. Let's see how it turns out, shall we?

"30-Day Blogging Challenge ON HIATUS" [13+]:
4. Waiting for the Barbarians Day!
Ok, tell us what you're up to and how you are preparing.

"JAFBG" [XGC]:
What was the worst gift you ever received and what was that person even thinking when gifting it to you?

A while back, when I was still married, there was some fad going on around the holidays involving group gift swaps. For all I know it's still going on, but I wouldn't know because even before the pandemic I rarely hung out with people around the holidays, except for my friends in California, and they had the great good sense not to participate in this activity.

I don't remember what it's called. Gift stealing? Something like that. Some of you probably know and can remind me in the comments. But as I recall, it goes something like this: Each person brings a themed gift, usually with a price cap. Like, maybe the theme is "red" or "fish" or "barbarians."

This particular gift swap was hosted by my then-in-laws. (Hey, can I claim credit for "then-in-laws?" I don't think I've seen it anywhere.) Since we all lived in Virginia, the theme for this one was "Made in Virginia," and the price cap was like $25 or something like that. Like I said, I don't remember too many details, including what I or my wife brought. No, ham wasn't involved. There might have been Virginia wine; I honestly can't recall.

So, like, all our names go into a Santa hat on little slips of paper, and one name comes out. That person picks a gift and unwraps it. Okay, great so far. Next random person can either pick a new, wrapped gift from the pile, or steal the first person's gift, which means the first person has to pick another wrapped gift. Third person can then pick a new gift or steal one of the first two. And so on. I think that when all the gifts are gone, the first person gets to swap with any of the now-unwrapped presents, because they didn't get a chance to steal before this.

Anyway, what I recall about this is that I started out with something really nice that I don't remember, and ended up with a giant can of Virginia peanuts.

I fucking hate peanuts. I mean, I hate them with an all-consuming fiery passion (for some reason, though, I like peanut butter; go figure.) If I were an asshole -- okay, if I were a bigger asshole, like the kind of barbarian who claims their annoying little yippy mutt is a service animal so they can bring them into restaurants, or the like -- I'd claim I was allergic to peanuts just so no one would break them out when I'm around. But I'm not allergic; it's just that my level of abhorrence goes off the scale. I avoid trail mixes with peanuts. I can't stand Snickers bars. Peanuts suck. Not only do I hate peanuts, but I loathe peanut breath more than is probably reasonable. If someone around me is eating peanuts, I'll go open a can of surströmming, pop a sewer manhole, or clean a cat's litter box just so I can smell something more pleasant than their exhalations.

(I do make one exception: Thai or Vietnamese food with peanut sauce. I don't know why, but they work for me in those cuisines. I justify this by classing it in my head as "peanut butter.")

Now, as far as I can recall, I never told my then-in-laws that. My wife knew, of course, and maybe she told them; I don't know. And like I said, I don't even remember the nice gift I started out with. But what has stuck with me all these years is the smug smirk on my shithead stepfather-in-law's face as he handed me this giant can of nasty legumes.

And even after all of that, I'm at a loss to find an artistic segué into the barbarian prompt. I mean, I could say something like "anyone who forcibly swaps me a can of hideous peanuts for pretty much anything can only be described as a barbarian," but that really does barbarians a disservice. At least most cultures commonly referred to as "barbarians" had their own code of honor, and this guy had no honor.

See, history was written not only by the victors but by people with the ability to write, and write in such a way that their writings stuck around a while. If your writing system consists of charcoal marks on tree barks, it's not going to last. If it's chiseled in stone or it's done with ink on parchment, well, then you get to define history, and so the people you don't like can get slapped with the "barbarian" label.

But people are people, and most of them have cultures that work for them, even if those cultures are not understood by the ones who call themselves "civilized." The most obvious examples are the Nordic people, who had a strong culture, but were feared by the more city-dwelling cultures as barbarians. To be fair, they did invade a lot. But just because you don't like someone doesn't mean they follow a failed culture; just a different one. Like... Europeans who settled the Americas considered the natives to be savages or barbarians, but from the native point of view, the invaders would have been the barbarians.

Except that some of these same American natives were the ones cultivating peanuts, so they obviously weren't completely blameless.

Point is, everyone's a barbarian to someone. Keep that in mind next time you're tempted to think some other culture is inferior in some way.

Oh, and Waiting For The Barbarians Day

commemorates a novel written in 1980, which sounds awfully pretentiously literary but I can't be arsed to delve into it further; also, according to my surely-infallible source, it's observed on November, not December, 4. December 4 is, among other things, National Cookie Day.

(I'm sure MirandaCookies IS IN COLLEGE

will be happy to know this.) Just don't put any gods-be-damned peanuts in mine, thanks.

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