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.

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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.

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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.

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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.

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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.

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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 15, 2021 at 12:04am

#1021645

Dark Side of the Moon

Well, at least I have some experience with this one.

PROMPT November 15^th

You and your two best friends are going on a month-long road trip. Where are you going? What three items must be in your luggage for this trip to be a great experience for you.

While I normally road-trip alone, in August of 2017, two of my closest friends and I traveled halfway across the country to see the solar eclipse. It's something I'd always wanted to see, and Missouri is a bit more accessible than Svalbard, which was where the previous eclipse had been visible from. And we could have gone someplace closer, but I wanted to reduce the risk of cloud cover.

That trip wasn't a month, though; it was more like four days. My limit for being in close quarters with someone, in order for us to remain friends, is something like two weeks. But okay, according to the prompt, I'm going on a month-long road trip. Let's set the trip in 2024, because that's when the next solar eclipse crosses parts of the US. Since an eclipse takes about two minutes, and we need the day before to get there and the day after to get back (solar eclipses tend to not be visible at night), that leaves four weeks to do other stuff.

You know what would be fun? To roughly trace the path of that eclipse as a road trip. Obviously, you wouldn't be able to chase the shadow. Even jets have a hard time chasing the shadow. But just taking the time to do it for the hell of it. Pick the spot where you want to actually see the eclipse, and plan accordingly, but also trace the path as best you can while the sun's actually shining. And visit breweries along the way, of course, because that's what I do.

The path of totality across the US can be seen here.

As you can see, the center of the shadow crosses the border from Mexico just southwest of San Antonio (and no wall is going to keep it out). It kisses Oklahoma (ew), darkens Arkansas, then swings through southeast Misery, Illinois, Indiana, and Ohio, before skipping across the Great Lakes, western New York, a bit of Vermont (saying hi to Canada too) and thence across Maine.

Just for comparison, there's also a map that shows both the 2017 and 2024 eclipses here.

So... "What three items must be in your luggage for this trip to be a great experience for you?"

Well, clearly there will need to be more than three items. Hell, I'll need at least three pairs of underpants alone.

But okay, fine, apart from the obvious.

1. Eclipse Glasses. I still have a crapton of these from the 2017 eclipse. Seriously, folks, don't be an utter idiot and look at the sun before or after an eclipse. Or anytime, really. No, the reason I need eye surgery now isn't because I didn't take that advice.

2. Camera. The last time I saw an eclipse, I wanted to simply experience it, not fuck around with camera settings or have anything between my eyeballs and the conjunction (except for Item #1 above). Next time, I want to try to get pictures. No, the camera on a mobile phone won't cut it.

3. GPS. Unbelievably, there are still people who freak out about GPS. Is it perfect? No. Neither are paper maps. But it's freakin' sorcery, and yes, it will even work during a solar eclipse (if the sun spits out a flare, though, all bets are off, which is why I also always carry a road atlas as backup).

Like I said, though, I'll need way more than just those three things, like earplugs so I don't have to listen to my travel companions snoring (I'll bring spares so they don't have to listen to me snoring). And money. And maybe a change of underwear.

...and everything under the sun is in tune
but the sun is eclipsed by the moon...

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