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 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.

March 22, 2024 at 10:33am

#1066720

Trope-Osphere

Good writers shun clichés, and avoid them like the plague. Then there's Cracked:

5 Storytelling Clichés That Started Out Very Different

Every cliché starts from somewhere.

As I've said so many times it's become cliché: Every cliché was once profound wisdom.

Often, these tropes don’t simply stay exactly the same until we’re all sick of them. Hunt down their debut, and you’ll see something different from what you’re now picturing.

There is an important distinction between a cliché and a trope, but I can't be arsed to get into that right now. But it's like... in action movies, explosions are tropes. Hard to have an action movie without at least one explosion. It's part of their charm, along with car chases and fight scenes. But the action hero calmly walking toward the camera while shit explodes in the background? That's cliché. But the first time we saw it (I don't remember what movie), it was fresh, cool, and different.

5. The First Damsel Tied to Railroad Tracks Was a Man

If you’re a mustachioed villain, and you have a helpless maiden in your clutches, you know what you have to do.

The villainous mustache has transcended trope and cliché and achieved icon status.

The article delves into the history, here, and it's worth looking at if only for the photo of greatest villainous mustache of all time. But, as the header suggests, the first fictional track-victim was male. I'm sure someone could get their college thesis done on just this topic alone. In any case, this seems to be an instance of the original idea being twisted, and that twist becoming the cliché.

4. The Emperor’s New Clothes Was About Fear of Being Disinherited

That story isn't so much trope or cliché as it is an important literary reference that everyone should know.

In the Hans Christian Andersen fairy tale, the emperor walks around naked, and no one’s willing to point this out, having been told that only smart people can see the clothes. This offers lessons for everyday life. Sometimes, people are just pretending to know what they’re talking about. Sometimes, the stuff that wins awards is garbage. Also: A whole lot of people are afraid of looking stupid.

One might even suggest it's a ready-made metaphor.

3. The First Man Who Asked for Three Wishes Was a Dick Joke

A man finds a lamp, and when he rubs it, out comes a genie. “I will grant you three wishes,” says the genie. “Great,” says the man. “I will wish for two normal things, and for my third wish, I will trigger the punchline.”

The "rule of three" doesn't only apply to comedy. That's only where it's most apparent.

The header here is perhaps misleading, as the original format (from the One Thousand and One Nights) was less joke than fable.

2. The First Bad Boy Was Pretty Lame

We use the phrase “bad boy” to describe not just actual children but excitingly rebellious men. We consider that normal, but it’s a little odd. There are only a handful of arenas in which it’s considered okay to call a man a boy.

And there's at least one situation where one should never, ever do that. But apart from that, clearly, the author of this piece isn't Southern.

We call rebels “bad boys” because of an 1870 novel called The Story of a Bad Boy. The main character, Tom Bailey, is a rebel but is also very much a boy. His antics include scaring people by setting off a cannon and pushing a car into a fire.

As the "car" as we know it wasn't invented for almost another 20 years, this must be one of the other definitions of "car."

1. “The Butler Did It” Wasn’t a Cliché But a Cheat

Another one that makes great joke fodder, but is cheating in detective stories. Apparently, it's always been a cheat.

You might think that The Butler Did It became an unforgivable trope after tons of murder mysteries pulled that trick, until it got old. That’s not really what happened. Instead, in 1928, an author of detective novels published a set of rules that he claimed mysteries should follow, and among them was a rule saying the culprit must not be a servant.

I'm just going to pause here and bask in the elitism of it all.

Moving on...

That set of rules was written by S.S. Van Dine, who went by “Willard Huntington Wright” when he wasn’t writing detective fiction. The rules start out reasonable enough, talking about how the author must play fair and provide all necessary clues. Then it makes some questionable blanket statements about what all mysteries must do — there must be just one detective, the crime must always be murder and “there must be no love interest.” By Rule 16, Dine is insisting that mysteries must have “no long descriptive passages, no literary dallying with side-issues, no subtly worked-out character analyses, no ‘atmospheric’ preoccupations.”

I don't usually subscribe to the notion that rules were made to be broken. In this case, though, I'll make an exception. Breaking my own rule, as it were.

Rule number 11 says, “A servant must not be chosen by the author as the culprit. This is begging a noble question. It is a too easy solution. The culprit must be a decidedly worth-while person — one that wouldn’t ordinarily come under suspicion.”

Elitism intensifies...

Dine’s rules laid out a few other solutions that he says stories should never use. The death should never be revealed to be an accident, he said, or a suicide. The detective must never be the culprit. There must never be multiple culprits. The solution must not involve the killer committing the murder after the police have broken into the crime scene.

Yeah, those aren't rules. Those are story ideas.

In other news, I'm going to attempt to make this my last hyphen-pun-titled entry for a while. It's becoming cliché.

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