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

September 1, 2021 at 12:02am

#1016461

Never Say Die

Welcome to September.

I'm not participating in "30-Day Blogging Challenge ON HIATUS"

[13+] this time around, because I volunteered to be a judge for a week later this month. That's right; I'm judging YOU. (If you're participating.)

There's a lot going on in September. Not just the WDC 21st birthday celebration, but it also contains my own account anniversary, and soon, people can start signing up for "October Novel Prep Challenge"

[13+]. That's right, November is coming up faster than you could possibly want, so it's not too early to start thinking about NaNoWriMo if you're into that sort of thing.

It is, however, too early to start thinking about the December holiday season.

The best thing about September, though -- actually, one of the only good things, because I'm one of those weirdos who prefers the heat of summer to the dreariness of autumn -- is Oktoberfest.

Why, you ask, am I associating Oktoberfest with September? Good question. The official Oktoberfest season is mid-September to early (really early) October. I have no desire to go to the original (and couldn't this year anyway because it's cancelled), but that doesn't mean I can't take the opportunity to enjoy a fine Märzen lager. So that's one thing I always look forward to in September. Another is, believe it or not, pumpkin beer. A lot of them suck, yes, but occasionally you find a good one.

But with the coming of cooler weather here in the Best Hemisphere, one thing I can appreciate more than I do in the summer is darker beers. Specifically, stouts. While I'm not above enjoying any beer in any season, some of them just work better in different weather conditions. And my favorite style of stout is Russian Imperial Stout, which isn't actually a Russian style but a British one, and that's a story for another time (if I haven't already told it somewhere in here).

And, to segué into today's article, my favorite Russian Imperial Stout is made by North Coast in California, and it is called Old Rasputin.

It even has a portrait of the Mad Monk on the label. Its motto is "Never Say Die," because Rasputin was famously hard to kill.

...or was he?

How Exactly Did Rasputin Die? The Real Answer (And The Weird Theories)

In the winter of 1911, a group of high-ranking Russian priests gathered in secret to lay a well-planned trap. Their target was none other than the notorious Siberian mystic Grigori Rasputin, who seemed to have established a mysterious hold over the Tsar. Rasputin had been lured to the meeting by his former friend, the Archimandrite Iliodor. But as soon as he breezed into the room, he was rushed by Iliodor and the “holy fool” Mitka Kolyaba, a one-armed epileptic who had been a previous favorite of the royal family. They grabbed hold of Rasputin’s penis and squeezed it, demanding that he confess his sins, while a hysterical bishop began beating him around the head with a huge crucifix, screaming “Devil!” with each blow. After all, it’s like the old saying goes: “Problems with a mad monk? Try crushing his junk!”

In case it's not blindingly obvious from that intro, or from previewing the URL at the link, this is from Cracked.

From the safety of the palace, he had his enemies exiled. But perhaps he should have listened to them, or at least learned some lessons about attending mysterious meetings alone. Because five years to the day after the priests’ attack, Rasputin agreed to pay a late-night visit to Prince Felix Yusupov, the richest man in Russia. His body was pulled from a frozen river the next day.

Some people never learn, I guess. But I suppose if you're buddies with the Tsar and banging the Tsarina (well, okay, allegedly), you might start to believe you're bulletproof.

As we all (hopefully) know, no one in that court turned out to be bulletproof.

You’ve probably heard the conventional version of the story, where he survived being poisoned, shot, beaten, stabbed and castrated, before finally drowning clawing at the ice of the frozen river Neva. That story actually has more problems than that math textbook Jay-Z wrote, but don’t worry! Everyone at Cracked is dressed in the full Lara Croft outfit, archaeology shorts included, and we’re ready to solve this historical mystery.

There follows the usual Cracked numbered list, but I'm not going to paste it all here; this has already gone on long enough and, frankly, the remnants of a hurricane are blowing through here right now, along with rain, floods, wind, thunder, lightning, and cats and dogs living together, and my internet could go out at any moment. Especially since I just (11:25 pm) got a tornado warning on my phone.

Still, some highlights:

And no matter what a certain animated movie may have claimed, he definitely did not rise from the dead to lead the Russian Revolution.

Look, anyone who thinks that ANY historical Disney movie has anything more than a wispy connection to the facts is beyond such warnings.

In 1914, he was stabbed by a noseless follower of the mad priest-monk Iliodor.

Why Disney has to make shit up when there's awesomeness RIGHT THERE is beyond me.

Seriously, though, the rest of the article is fascinating, poking holes in the "official" history while suggesting alternatives.

But one thing remains: whatever the actual facts are -- and we will probably never know for sure -- the mythology and speculation around those last years of the Tsars is compelling enough without needing to invent Zombie Rasputin.

Great beer, though.

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