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 22, 2022 at 12:01am

#1040914

Northern Exposure

Nope.

I Spent the Winter Solstice in One of the Darkest Places on Earth
During the phenomenon of polar night, parts of the Arctic don’t see the sun for weeks or months at a time. The darkness drives some people insane, but for others, it opens a gateway into wonder and peace.

Well. A qualified "nope," anyway.

About eight years ago, I stepped through the unlocked door of a 1915 cabin-turned-chapel in Wiseman, Alaska, an Arctic settlement of about a dozen people roughly seven hours north of Fairbanks.

I looked up Wiseman when I found this article. The "dozen people" thing appears to be true, though some sources say less. Oddly, there is a bed and breakfast there, called Arctic Getaway.

It is quite literally in the middle of nowhere.

The pastor, who had lived in Wiseman for decades, described the inexorable march of darkness as a force both terrifying and beautiful. She spoke of chopping wood, preserving berries, and squeezing the joy out of every moment of daylight before a winter in which, for more than a month, the sun never rises above the horizon.

That's the actual definition of "existing north of the Arctic Circle." You also get a month or so of permanent daylight in the summer. Given my complicated relationship with the accursed daystar, I'm not sure which is worse.

The notion of such sustained darkness in a remote corner of the planet unnerved me. Residents of the Arctic tell stories of people losing their minds in the black of polar night. But I also felt strangely curious—and drawn to return one day.

I, too, admit to some curiosity. But not enough for me to actually go haring off to the Arctic. It's cold and there's probably no internet. On the plus side, there's the aurora borealis. I wouldn't mind seeing that once.

It’s not exactly easy to get to at any time of year and services like hotels and transport are few.

Well, there is that B&B. And being Alaska, don't they all get around by small aircraft? Also, Maps shows an actual state road going through it (apparently built to support the Alaska Pipeline), but I can't be arsed to see if it's passable in the winter.

I did, however, note that there's a place just south of Wiseman called Coldfoot. I immediately assumed that this referred to frostbite, but Wikipedia has other ideas:

Coldfoot is a census-designated place in Yukon-Koyukuk Census Area in the U.S. state of Alaska. The population was 34 at the 2020 census. It is said that the name was derived from travelers getting "cold feet" about making the 240-some-mile journey north to Deadhorse.

So apparently there is also a town (or whatever you want to call it) named Deadhorse. Okay, Alaska.

But last summer, a friend forwarded me an email about a tiny off-grid six-person retreat center that had just opened outside of Wiseman. The owners were hosting a week-long trip that included yoga and exploring the Arctic wild with skis, snowshoes, and dogsleds, and the dates fell right on the winter solstice.

Nope, nope, nope, and nope. Also nope. But at least it's (probably) not hot yoga, though I'm not sure if freezing-your-ass-off yoga would be any better.

I’m not exactly a cold-resistant creature: I’ve suffered from hypothermia multiple times and frostbite that turned my feet white and wooden. I’m generally dressed in a sweater and jeans when my friends are wearing shorts and flip flops. Even at much more temperate latitudes, seasonal affective disorder runs in my family.

I consider anything below 70F to be "cold." Anything below about 60F is "too damn cold." Like, tonight, it was around 40F and in order to take my recycling to the curb, I had to put on my battery-powered vest, scarf, heavy coat, and ushanka hat. I experience seasonal depression, too, but it's not the darkness that would stop me; it's the temperature, and, again, I can't emphasize this enough, no fucking internet.

I also contemplated the wisdom of traveling during a pandemic, and the carbon emissions of flying long distances.

Look how virtuous I wanted to be, but I did it anyway, tee hee.

Soon after arriving, I tugged my snowpants over my jeans, donned both my down jacket and an insulated parka, and pulled on my warmest hat for a short walk.

Whyyyyyy?

The cold blew through it all in seconds. My eyelashes froze and my nose hairs crinkled. The liquid on my eyeballs felt like it was turning to slush. Even the slightest breeze lacerated my cheeks, and my mind felt tight with a barely concealed panic.

Look, I'm not going to berate anyone for stepping outside their comfort zone. I need to do it myself, from time to time. But there's leaving your comfort zone, and then there's going to goddamn Alaska in cocksucking December.

Between November 30 and January 9, the residents of Wiseman, Alaska, do not see the sun. They lose about 12 to 15 minutes of light each day until the solstice and then gain it back just as quickly. The future always looks scarier from the confines of imagination, and polar night was not so unnerving once I was in it. It was actually brighter than I anticipated—locals like to say that on the winter solstice, there are still five hours when it’s light enough that you can’t see the stars.

That's the thing a lot of people don't get about the Arctic Circle (or its southern counterpart). Sure, there are stretches of time when the sun is below the horizon, but depending on how far toward the pole you are, this can be basically a really long twilight.

Not that I've ever ventured that close, myself. The furthest north I've ever been was southern Scotland, unless you count the path my plane took to get there (which happened much closer to the summer solstice, so I could see the midnight sun shining through the plane's windows).

Anyway. The rest of the article attempts to wax poetic about the author's experience and, while I can appreciate the language, every other sentence just made me yelp "nope" again.

One night, I wandered out of my cabin, wrapped in a sleeping bag I had brought just in case, and watched slack-jawed as the northern lights whirled across the dome overhead like a luminous river. After many days, the formidable peaks of the Brooks Range finally disrobed from their mantle of clouds and shone resplendent in the moonlight.

That bit, though... that almost makes me want to visit.

Almost.

At least there's pictures and descriptions that I can see and read because I have a freakin' internet connection in a heated house.

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