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 8 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 10 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.

June 28, 2022 at 12:01am

#1034377

Not for the Faint of Art

Time now for some cultural enrichment, courtesy of a certain dick-joke site.

5 Pieces Of Bonkers History Hiding In Classic Artwork

Ready to be enriched, to bask in the glory of classical art? Good. As usual, you'll have to click on the actual link to see the pretty pictures.

5. The Tuscan Dick-Tree Reveals Medieval Witchcraft And Middle Ages-Propaganda

But dongs were less vulgar and more symbolic in the olden days. So the image isn't just funny, it's historical and educational. One theory says the tree simply represents fertility and good fortune. Booooring.

Yeah, thinking they were always considered obscene is a common phallusy.

Another posits that this is possibly the oldest image of European witchcraft, revealing some interesting thaumaturgy. It may depict something later detailed in a Latin "witchfinder's guide" called Malleus Maleficarum, which sounds hella rad and doesn't disappoint in translation: "Hammer of the Witches."

Which I'm sure I've talked about in here before. Hm, now that I think of it, perhaps there's a metaphorical connection between "malleus" and "phallus." I'd look into that, but I'm afraid I might be wrong. Still, it seems plausible at first glans.

An equally slick theory posits that the scene is political propaganda. Around this time, the Guelphs and Ghibellines were political Bloods and Crips allied to the Pope or Holy Roman emperor, respectively, just like modern street gangs.

I suppose those are better names than Sharks and Jets. I get the feeling one or the other would eventually get the shaft.

Okay, no more penis-mightier-than-the-sword puns. For now.

4. Van Gogh's Café Terrace At Night May Be A Last Supper Shout-Out

Vincent van Gogh moved to sunny, wind-whipped Arles in February 1888 to escape the soul-stifling craziness and pretension of big-city Paris.

Some things never change.

Van Gogh only spent about a year in the bright, Mistral-besieged Provencal city but produced many of his most famous paintings here despite, or perhaps due to, his subsistence on coffee and absinthe.

Hopefully not at the same time. That would be nasty. Also, I'm going with "due to."

Among his most gawked at is the Café Terrace at Night, as it's known today. It was Van Gogh's first nighttime thing, painted in September 1888. Actually, since he painted it on the spot and true to life, as researchers dated the configuration of the stars to September 16 or 17th.

One odd effect of having cataracts was that, whenever I'd look at lights at night -- be they streetlights, passing helicopters, the moon, or stars -- I'd see a cluster for every light source. They reminded me of Van Gogh's paintings, leading me to wonder if dude had cataracts, despite his young age.

Some say it's a symbolic Last Supper, featuring 12 apostles breaking bread and French wine, which the maître d' probably spit in. One figure lurks in shadow, the traitorous Judas. The central figure is Jesus ensconced in a robe, with the light above serving as a halo and the window behind forming a cross.

One of the most annoying habits of literary and artistic interpreters is their propensity to find Jesus everywhere.

Also circa late-September '88, Vincent wrote to his brother, Theo, that he felt a "tremendous need for, shall I say the word—for religion." And was apparently already thinking in apostolic terms of 12, while buying 12 chairs for his no-show painting buddies.

Well, okay, in this case there might be a legitimate argument.

The famed eatery still exists and is now Le Café Van Gogh, a tourist trap piss-nest with a 2.8 rating on Google reviews and worse elsewhere.

Yeah, I'm not going to Gogh there.

(I said no more dick puns. I didn't say no more puns. Also, that one only works in American English; the proper pronunciation of his name was something closer to "cock." ... dammit.)

3. Reply of the Zaporozhian Cossacks to Sultan Mehmed IV of the Ottoman Empire Is An Epic Old-Timey Diss

Sounds like something for Epic Rap Battles of History.

The term Zaporozhian Cossacks refers to a bunch of surly, burly, fighty former peasants from Ukraine.

As I've been saying: history echoes.

The supposed setting: after the Zaporozhian Cossacks gave the sultan's forces a bit of an ass-paddling, Mehmed IV sent a rudely worded missive asking for the Z’s to surrender, which isn’t how war works. The response from the mercenaries was one of the greatest fulminating remonstrations in history, real or imagined, utilizing an assortment of archaic insults that probably deserve a comeback.

Despite the promise of this text, which includes a link to the Wikipedia page on the painting, I can't find a list of said insults anywhere (I write these things in less than an hour and I don't have time for a thorough search). It's too bad; I imagine they would have been glorious insults.

2. Bichitr Exemplifies Mughal Art, Throws Shade At Western Kings

Who?

The 17th-century artist Bichitr is so exquisite it's a shame that most of the seven people who have heard of him call him "Bih-s**tter." He was the finest of Mughal court painters, serving as official immortalizer for two prominent emperors, Jahāngīr and his successor-to-the-throne son, Shah Jahān of Taj Mahal fame.

Oh, that Bichitr.

In the painting, Jahāngīr sits on an hourglass and transcends earthly, mortal time. The dudes on the left are in order of propagandic hierarchy. Up top is the Sufi Shaikh, a hot-shot holy man of the Sufi Islamic mystic faith. Lesser than his holiness is the Ottoman Sultan, and lesser still is a hangdog King James I, depicted in a lowly position with a mighty impotent air. His posture is limp, and his expression as resigned and listless as a brachycephalic luxury cat accepting mortality when dinner is a minute late.

I'm just including this text because it's comedy gold.

1. Caravaggio's Seven Works Of Mercy Is A Counter-Reformationist Easter Egg Hunt

Michelangelo Merisi da Caravaggio is unjustly the Michelangelo of lesser renown. He's named after the town where he spent his childhood, like your homeboy Fat Cleveland. Yet, instead of being known for guzzling tallboys (though he was for that too), MMdC earned his appellation through unbridled artistic excellence and mad-hat bat-shittery. Hell, he'd be the Michelangelo had he been born a bit earlier.

One of the fun things about playing video games and, to a lesser extent, watching shows and movies, these days, is finding all the Easter eggs -- references to other works of art. For example, Fallout 4 includes a setting that perfectly mimics the climactic scene of Blade Runner, as the two works of art share a common theme.

Point being, such Easter eggs are nothing new in art.

Anyway, like I said, you'll have to view the article to get the full effect with pictures of the paintings and all. But I couldn't resist adding my own commentary. Art can be nuts.

3 Like • Comment

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