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

April 5, 2023 at 8:56am

#1047527

Portrait of a Rant

The final prompt from this round of "JAFBG"

[XGC]...

What current topic/craze are you sick of hearing about?

Well, I'm old enough to say "All of them."

But I suppose that at the top of the list sits TikTok.

"Oh, is it because the government is trying to ban it?"

No. I don't care.

"Because China uses it to spy on us?"

Nope.

"Because it's what kids these days like?"

Whatever. Kids have always liked, will always like stupid things. I did, when I was a kid.

No, the reason I wish TikTok would go away (though I don't agree with a government ban) can be summed up in one alliterative, two-word phrase:

Vertical video.

Vertical video is, in 99.9 percent of cases, an abomination against everything that is good and right.

Let me provide a bit of background.

One of the earliest computers I played on was back in the 70s. It barely even qualified as a computer, being dedicated to word processing—a typewriter with a floppy disk drive. Not one of those solid-case tiny plastic disks, either, oh no. Nor one of the larger, but still below-average, 5-1/4". No, this word processor took massive, throbbing 8" disks.

But the other "feature" of this glorified typewriter was its screen. We're used to looking at paper—typewriter paper, e.g.— in vertical orientation. So the makers of this early word processor (I think it was Wang) oriented their green CRT monitor in a vertical orientation to emulate a typewritten page. And no, it didn't have a preview function; all your codes (think early versions of {i} and {b} and {indent} here on WDC) were visible on the screen but not in the printout, so you had to print a test copy.

Anyway, the point is, this seemed weird to me even then. Paper is paper, but screens, monitors, and TVs were, by that point, standardized in a horizontal format. Mostly something like 3V:4H.

Through time, I've had several different computers, and their screen evolution went something like this: CRTs with 4:3 ratios, monitors with even more horizontal exaggeration, actual widescreen monitors. Movies released in widescreen format often had to be edited or letterboxed to fit a 4:3 monitor. Anyway, point is, I think we finally settled on an ideal ratio, whatever it is, but it's horizontal. This conforms with human vision, which has a much greater range side-to-side than it does up-to-down.

Which is one reason that the inability of some mobile phone video makers to turn the fucking phone sideways enrages me to the point of apoplexy.

One of the earliest examples I saw was some dude trying to capture the aftermath of, I don't know, a car and truck accident or something. Holding his goddamn phone vertically, he had to frantically pan side-to-side in order to catch all the carnage (not literal carnage; I don't recall there being bodies or anything). If he'd only turned the assmunching phone sideways, he could have captured the entire scene.

Since then, I have refused to watch any video in vertical format, on my phone or on the laptop. Well, with two exceptions:

1. Explosions. I don't really care about format if I'm watching shit blow up.

2. Rocket launches (with or without explosions). These are the only events that demand vertical format, because, well, everything important happens vertically.

As an aside, I don't have the same issues with still photographs or paintings. When I did photography, I'd orient the camera as appropriate: vertical for portraits (hence "portrait" mode) and horizontal for groups or landscapes (hence "landscape" mode). But videos are different.

As I blame TikTok for the appalling rise (pun intended) in vertical video, I choose to rant about TikTok in general.

So that's it. Tired of hearing about it, so writing about it. Paradox!

6 Like • Comment • View 3 comments

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