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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 4 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').

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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.

August 11, 2021 at 12:06am

#1015475

Whipshrek

This one's about a movie, but not one I've seen recently. No, earlier this year I was treated to the Guardian's brutal takedown of a 20-year-old movie. And it's just now come up for me to review the review.

Shrek at 20: an unfunny and overrated low for blockbuster animation
The fairytale comedy was a hit with critics and audiences but its toilet humour, glibness and shoddy animation mark it out as a misfire

So... audiences liked it, critics liked it... but some elitist asshole is here to tell us all that we're wrong and we should be ashamed of ourselves by calling it a "misfire."

Hopefully y'all know by now that I like different kinds of movies. Highbrow, lowbrow, unibrow, whatever... I won't be browbeaten into joining the consensus or rebelling against it. And I'm certainly not above laughing at juvenile humor, though I can also appreciate fancy French art flicks.

Hating on Shrek is just plain snobbery.

I'm not saying it's a great movie, mind you. But it certainly doesn't suck.

Shrek has an outhouse with a working toilet.

Oh yeah, let's open up by pointing out the bathroom humor. It's a kids' movie for shit's sake; there's going to be bathroom humor.

Twenty years later, that flushing sound seems to signify the moment when blockbuster animation circled the drain.

Right, because no animated movie has succeeded since then.

Shrek is a terrible movie.

No it's not.

It’s not funny.

Yes it is.

It looks awful.

No it doesn't. Okay, yeah, animation has come a long way in 20 years, along with the computers that generate it, but it's perfectly fine for the time.

And yet Shrek was a sensation with critics and audiences in 2001.

So, this guy is positioning himself as the ultimate arbiter of how good a movie is, and not other critics or, you know, the people who actually pay money to see them. Look, I get it; I have opinions, too, and not all of them align with the consensus. I disliked The Matrix. I liked Cats. I never saw Titanic and don't plan to. Whatever. We all like what we like and hate what we hate. I'm waiting for an objective argument to support the author's hate.

Even the stuffed shirts at the Cannes film festival, who usually separated Hollywood summer fare from its official selections, brought it into the competition slate, where it premiered alongside new work from world masters like David Lynch, Jean-Luc Godard, Hou Hsiao-hsien and Jacques Rivette.

You do not get to look down your nose at a lowbrow children's comedy (that also has adult appeal) and then call the Cannes judges "stuffed shirts." Pot, meet kettle.

Worse yet, it encouraged a destructive, know-it-all attitude toward the classics that made any earnest engagement with them seem like a waste of time. Those once-upon-a-times were now rendered stodgy and lame, literally toilet paper.

Yes. That's how storytelling evolves. Shakespeare was sop for the working classes of London; his works only gathered a patina of respectability with time, and people continue to riff off of the plays. As I noted in an entry a while back, it's likely that the works we know as "fairy tales" changed multiple times, with new audiences and new context, before they were written down in their "final" form. Playing with those ideas, updating them for the times, is what storytellers do.

But the balance in Shrek is off on both ends: there’s an excess of anachronisms and buddy-movie riffs from Myers and Murphy that have little relation to the backdrop and a woe-is-me soppiness to the love story between two lonely, misunderstood freaks. (Nothing screams “unearned gravitas” like slipping in a cover of Leonard Cohen’s Hallelujah.)

YOU SHUT YOUR PUDDINGHOLE RIGHT NOW.

But once Shrek and Donkey cross the kingdom on a quest to bring Fiona to Farquaad, the storybook references are all but abandoned. Even when Robin Hood and his Merry Men appear in the woods, the film blows past that boring old mythos in order to pay homage to The Matrix and Riverdance.

Yes. That's a feature, not a bug.

Last year, the National Film Registry added Shrek to the Library of Congress, which seals its canonization, but it’s remarkable how much of an early aughts relic it’s become, an amber-preserved monument to phenomena (Mike Myers, Smash Mouth, Michael Flatley) that hasn’t stood the test of time.

"Noughties," not "aughts." Come on, even the BBC has adopted my word for that decade. Probably not on purpose, though.

Even the film’s referential style looks resolutely slow and unhip next to the whirring pop Cuisinarts of Lord and Miller productions like The Lego Movie and Spider-Man: Into the Spider-Verse or even IP-heavy Disney fare like Wreck-it Ralph and its sequel.

Yes, because those movies came out later and had the benefit of better technology. This is like comparing the original Godzilla to this year's Godzilla vs. Kong.

Don't get me wrong; I like the Guardian in general (their harsh review of Hudson Yards sticks out in my mind as the greatest piece of architectural criticism in the history of ever, though to be fair, it's an opinion I already shared), but this would be something like the British equivalent of a small-town newspaper from Alabama reviewing Broadway plays. They're entitled to their opinion, but they're not the target audience. And the UK is hardly innocent in producing lowbrow humor. (I should note that for all I know, the reviewer is from the US, but he's still writing for a British journal.)

So, okay, I get it; he didn't like the movie. Fine. A lot of the reasons he hated it are the same reasons I liked it. I hope I never see his opinion of Despicable Me. Or, actually, I hope I do so I can rag on that review, too.

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