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 10 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 12 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.

March 30, 2022 at 12:02am

#1029714

Steely Gaze

While I've done a few short-ass movie reviews in here -- there's another one below -- it's not often that I focus on contemporary TV, apart from nerding out over Star Trek sometimes.

How 'Superman And Lois' Became Our Best Modern Superman

I don't fully agree with everything in this Cracked article, but one thing the author and I can agree on is that this particular show is awesome.

See, I don't usually seek out other peoples' opinions on shows and movies I watch. That may be a bit hypocritical, since I'm displaying my opinions here for other people, but I prefer to draw my own conclusions before finding out what other people think... and then I usually forget to find out what other people think.

But when it's on a website I read anyway, okay, fine. And I'm actually glad someone else likes this show.

It's nuts that it has been 42 years since we've had a Superman movie whose critical reception didn't consist of exasperated sighs at best or angry nerd screeching at worst.

Look, any movie involving a fandom is going to result in angry nerd screeching. It's as inevitable as construction on the DC beltway and almost as annoying. This is true for fantasy, science fiction, and the odd blend of the two that is the superhero movie (or show). Someone's not going to like it. Someone loud. Like a tractor-trailer accident on a two-lane road, there's just no getting around it.

That Christopher Reeve Superman movie got a pass because it came out while the internet was in diapers.

There's just something about Superman that makes him exceptionally hard to adapt on film without crapping the bed in some way or another.

Well, yeah. It's hard to write a relatable story about someone who's indestructible and extraordinarily powerful, especially since, over the 90 some years of the character's existence, they kept coming up with new powers so he could get out of the situations they put him in. It got so bad that, in the comics, they pulled a reset back in the 80s that depowered Superman quite a bit. Because before then, he spent most of the 70s using his godlike abilities to do little more than keep Lois Lane from finding out that he was actually (spoiler alert) Clark Kent.

While we've been sitting here waiting for a Superman movie that's good at something other than inspiring a million internet flame wars, CW's Superman & Lois show has quietly reinvented the Man of Steel by not reinventing him and just sticking to what made him work in the first place. (A revolutionary concept no one else had thought of, apparently.)

There's... more to it than just sticking to what made him work in the first place. I said it was hard to write a relatable story about someone who is, essentially, a god. But it's not impossible.

Also, with TV shows, you get more time to develop characters, backstories, relationships, settings. In the past, lower budgets for TV have interfered with a lot of the spectacle, but even that's becoming less of a problem now.

A major issue in live-action Superman adaptations is that they seem ashamed of Superman.

Which is something even the Aquaman movie wasn't, and that character was the second-biggest joke in comics (after Arm-Fall-Off-Boy from the Legion of Superheroes... don't ask).

Smallville thought showing a blue blur on screen was more acceptable than letting Clark Kent put on his dang costume, while Zack Snyder's Man of Steel and Batman v Superman: Dawn of Justice decided that the only way we'd all believe a man can fly is if he's conflicted about saving people and, oh yeah, kills a bunch of them.

You know, I never actually saw Smallville. It was a noughties thing; streaming wasn't readily available like it is now, and I can't deal with commercials.

As for the Snyder movies, you're not going to get me to say anything bad about Zack Snyder. I've enjoyed every movie of his that I've seen. Yes, even that one. That one, too. And also his Superman movies. My only gripe about his Superman movies is that whatever character he had dressed up in a red and blue costume, it wasn't Superman.

Superman Returns and Justice League came the closest to depicting the idealistic Superman from the comics, which is ... not a good state of affairs.

I also don't care what other people think about Superman Returns; I thought it was a good movie. Kevin Spacey -- before he became a pariah -- made an excellent Lex Luthor, and Brandon Routh was good at both Superman and Clark Kent, even if he was mostly channeling Reeves through the whole thing.

Keep in mind that when we say that there hasn't been a truly good cinematic Superman this century, we're not talking about the quality of the acting. Tom Welling, Brandon Routh, and king of nerds Henry Cavill did a great job with the material they were provided (even if, obviously, they couldn't hold a candle to Nicolas Cage's glorious Superman test footage).

Incidentally, if you haven't seen The Witcher, Henry Cavill is awesome in that.

Superman & Lois's writing is refreshing because it lets Superman be his own uncomplicated self -- the complications come from external factors, like having to raise a hormonal superpowered 15-year-old, and in seeing how such a massively idealistic character deals with them.

How do you introduce conflict and tension in a story that's about Superman? Give him drama that he can't use his powers to solve. Like teenage kids.

Also, while we already praised the other actors, it helps that actor Tyler Hoechlin can play a convincing dork better than anyone since Christopher Reeve.

And my only gripe about S&L is that Tyler Hoechlin is the spitting image of a cousin of mine, and I simply can't get the resemblance out of my head.

Other than that, yes, it's absolutely worth watching.

I'm not generally a fan of horror movies, but I decided to give this one a try on Monday.

One-Sentence Movie Review: X:

While the basic plot of this is straight out of How To Make A Horror Flick, this movie, which, unlike some others of its genre, contains no supernatural elements, elevates the story past simple slasher gore by confronting themes of life, death, movies, fear, love, sex, and why renting an AirBnB is never a good idea.

Rating: 4.5/5

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