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

November 30, 2021 at 12:03am

#1022544

Try to November Me

And so it ends.

PROMPT November 30^th

Wow, it's the end of the month! With Thanksgiving behind us (here in the States), and Christmas on the horizon, what, if any, are your Christmas traditions? If you have no holiday traditions, do you have any plans for this month?

As we careen into December, I think it's of the utmost importance to remember the real reason for the season:

Axial tilt.

As the Earth orbits the Sun, the axis stays pointing in the same direction (roughly toward Polaris) (or away from it, depending on your perspective), which changes the apparent path of the Sun across the sky. In summer (assuming a location outside the Tropics), the Sun climbs higher, baking us with more direct rays for a longer stretch of time, while in winter, that bastard can barely get out of bed, and things cool down.

And I'm still not sure which is worse: burning under the pitiless gaze of the accursed daystar, or freezing from its absence.

Okay, I'm going to go with "freezing is worse" because I'd rather be too hot than too cold.

In any case, axial tilt is the reason for the season, and, relative to the Sun, it'll be at its most tilted in about three weeks: The solstice this year (winter for me, summer for the folks hanging upside-down) will occur on December 21 at 10:59 am EST. Why it couldn't wait another minute for a nice, even 11:00 is beyond me. And now I'm tired of coddling the Down Under, so everything else here is going to be presented from an unashamed Northern Hemisphere perspective.

I always celebrate that moment, because it marks the point when days begin to gradually get longer, when the light begins to conquer the dark. Not that there's anything wrong with dark, per se; it's just that, as I noted, less sunlight means lower temperatures, and fuck lower temperatures.

Unfortunately, as weather tends to lag astronomy, the winter solstice isn't usually the coldest day. Obviously that's going to vary from year to year, but on average, around here, that doesn't occur until close to the end of January. Still, I'll take my victories where I can, which in this case means appreciating the solstice.

And then, once I'm done appreciating the solstice, I remember that the next two months are going to be utter hell. The only month I hate worse than January is February. But at least for one moment, I can believe that, as they put it on an episode of Doctor Who, we're halfway out of the dark.

As for other things in December, well, that remains to be seen. Literally. This is the month when I'm supposed to get my eyes worked on. I dread the operation, so I try to focus (pun intended) on the expected result instead. That's hard for me. I've never been one to endure temporary hardship for longer-term benefit. They do one eye, and then a week later, the other eye. Supposedly it has a solid success rate. I'm still absolutely not looking forward (pun also intended) to having them screw around with my eyeballs.

Other things? In the Before Time, I used to go to California every year in December. Travel's a bitch now, though, because a lot of people are infected with weapons-grade ignorance. Perhaps I'll be able to do that again one day, but this year is not that day.

Rather than inflict my presence on those who don't want me around (which as far as I can tell is everybody), a couple of years ago, I began a new tradition for myself on Christmas Day, a day which otherwise holds little significance for me: I walk over to the drafthouse cinema, get drunk, and watch a movie before staggering home. This year, I don't know if I'll be able to do that or not; my second surgery is scheduled for the 23rd. Probably I'll be able to. Unless they botch it, in which case my movie-watching days are over. Hey, it's important to expect the worst, so I can only be pleasantly surprised.

For anyone else who's made it this far through 30DBC, thanks for reading. But I'm on a roll and I don't expect to stop blogging, so tune in next month for a whole lot more of the usual -- and maybe some things unusual.

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