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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 7 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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').

1 Like

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.

July 7, 2022 at 12:02am

#1034803

Old Rasputin

Everyone knows -- or thinks they know -- the story of Rasputin, the Mad Monk of pre-Revolution Russia. Well, Cracked is here to set us all straight.

How Hemophilia Brought The Mad Monk Rasputin To Power

The article goes into more depth than just the hemophilia angle, but that bit's interesting enough.

Now, Russia's not exactly in everyone's good graces right now, and Putin is proving harder to defeat than... well, you know. But when it comes to the people of Russia, well, I for one have no wish to generalize or stereotype. They're just people: some good, some bad, most just wanting to live their lives. True, there seems to be a high proportion of absolute badasses, but that may be confirmation bias talking.

Rasputin, however, was indisputably a badass.

A picture of Grigori Rasputin without a label or name attached to it still likely carries feelings of discomfort, of deviance, a feeling that this is a man living an alternative existence.

One of my favorite beers is a Russian Imperial Stout called Old Rasputin. It features the famous portrait of the dude, the one with his hand up against his head like he's delivering a blessing or a curse, or maybe just calling you a Loser in Russian.

Incidentally, Russian Imperial Stout is a British style, and its origins come from long before the last Tsar. That's a story for another time (or maybe I've already told it in here), but the point is the beer has fuck-all to do with Russia, so if you're boycotting Russian stuff, well, don't include that in your List.

Rasputin did not need to convince anyone of his qualifications or dark connections with charm or confidence, like a modern cult leader or mystic figure would. If someone who looks like Grigori Rasputin tells you he has powers, you believe him, and you quickly look for someone else to talk to.

Yeah, his eyes were all like, "Look, tovarisch, I've seen shit."

Rasputin was born a peasant in Siberia...

I mean, from there, your life can only improve... or end.

Instead of literacy, Rasputin participated much more heavily in sexual exploration, putting in a large amount of hours in his own debauched self-education. It was from this behavior that he actually earned the name “Rasputin”, meaning “Debauched One,” which is metal as hell.

I know very little Russian. I have no idea, for instance, which part of the name means which. This matters because a certain leader of a certain country has a truncated version of that name, and I'm guessing it either means "debauched" or "one," either of which makes him scarier than Rasputin.

Here is as good a place as any to mention another misconception, one that even the introduction of this article may have furthered. Though the manners of his belief and practice were far from orthodox, Rasputin was not a follower of Satan, pagan gods, or any such thing. He was a devoted believer and follower of the Christian god, and it was through him he claimed any powers he may have. In fact, Rasputin was likely closer to a modern day cult leader than the lightning-shooting, incantation-repeating wizard figure he’s sometimes portrayed as.

Anyone still thinking he wasn't Christian with that behavior needs to look up the "No True Scotsman" fallacy.

Nicholas and Alexandra Romanov were the current Tsar and Tsarina of Russia. However, their line of succession was terrifyingly frail, as they only had one son, Alexei, and he suffered from hemophilia.

Oh, it was even more frail than they thought.

The article then goes in to the whole hemophilia cure thing. To summarize, whatever he did seemed to work, and history was made.

His reign and influence on the Romanovs is often portrayed as dark and mysterious, a shadowy figure who invited in evil elements. There is evidence however, that he was far from evil, and still held a great deal of concern for the peasants and lower classes of the empire that he’d been born into.

One of the worst lies foisted upon us by religion is the idea that promiscuity is, by itself, evil. Sure, if the other participants aren't willing, it can be. But not sex in itself.

There are records of notes written begging the Tsar and Tsarina to share food surpluses in times of hardship among the lower class.

I don't suppose "let them eat cake" would have gone over very well.

The notes betray Rasputin’s continued illiteracy, e.g. “kind dear apologies forgive me much meat is needed, let Piter eat, listen help rosputin,” but they warn of a coming revolution if the starvation was ignored, an event that did come to pass.

I gotta think that, had he survived, post-Revolution Russia would have taken a drastically different turn.

The article then delves into the whole assassination thing, most of which is probably mythological, like Washington chopping down the cherry tree.

Rasputin’s own daughter alleges her father was killed by a simple bullet to the head. Which is, to be certain, contains a whole lot less grandeur than a legend of useless poison, tanked bullets, and of course, a massive penis preserved in a jar. (Records of Rasputin’s autopsy say his bits were all present and accounted for, and the general thought is it’s a cow’s penis.)

Never mind the tortured grammar there; I can forgive some lousy editing on a comedy site. But "cow's penis?" Come on.

Anyway, maybe you knew all this stuff, but as usual (apart from some editing problems), Cracked puts a great spin on it.

So I'll just leave you with this horribly historically inaccurate, but catchy, song (the video is also at least a little bit not safe for prudes, just to warn you):

3 Like • Comment • View 1 comment

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