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 11 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 13 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 21, 2022 at 12:01am

#1029258

Isn't It Ironic?

Sometimes I actually get to talk about writing in here.

What Irony Is Not
A handy guide to distinguishing the notoriously slippery concept from its distant cousins coincidence, satire, parody, and paradox.

It's a book excerpt, but a fairly long one, and I'm only going to share a few highlights here. The short summary is right there in the subheading; to determine what irony truly is, it's helpful to define what it is not.

“Irony” is a term that everyone uses and seems to understand. It is also a concept that is notoriously difficult to define. Much like Winona Ryder’s character in the 1994 rom-com “Reality Bites,” whose inability to describe irony costs her a job interview, we know it when we see it, but nonetheless have trouble articulating it.

I never saw that movie, but I doubt most of us would face utter failure for getting the concept wrong. At worst, we'd get dinged by pedants.

Since I'm a pedant, though, I want to know.

And following your mother’s advice — to look it up in the dictionary — is liable to leave you even more confused than before.

Looking something up in the dictionary is a child's way of gaining basic understanding. There's nothing wrong with it, but as I've noted before, dictionaries are descriptive, not prescriptive. A more complete, nuanced view requires wider readings. Like this one.

A song about irony is mocked because its lyrics contain non-ironic examples.

I'm assuming they're talking about the infamous Alanis Morissette "Isn't It Ironic?" which would be better titled "Doesn't it suck?" Because none of her examples that I can recall are actual irony.

Coincidence

In everyday conversation, the term “coincidence” is often used as a synonym for situational irony. For example, someone might rhetorically exclaim, “Isn’t it ironic that the rain stopped just as I was finishing my morning run?” In many such instances, “coincidence” would probably be a better descriptor, particularly when no greater meaning or import connects the two events.

And even if you believe that the rain stopped right after your run because the Universe is clearly out to get you, that's still not irony; that's paranoia.

Doesn't mean you're wrong.

A good example of this subjectivity can be found in “Brain Droppings,” by the comedian George Carlin. He argues that the concept of irony has to do with opposites and has “nothing to do with coincidence.” He provides a number of examples to make his point, including the following: “A diabetic, on his way to buy insulin, is killed by a runaway truck. He is the victim of an accident. If the truck was delivering sugar, he is the victim of an oddly poetic coincidence. But if the truck was delivering insulin, ah! Then he is the victim of an irony.”

The older I get, the more I realize that comedians should be running the world. And not just one country of it.

Paradox

As with coincidence, irony is often conflated with the concept of paradox.

I'm not sure I've ever made that connection. The coincidence one, sure -- I'm hardly an expert, which is why I read articles like this one. But to me, paradox is pretty obvious.

The crucial issue that separates verbal irony from paradox is that the apparent contradiction can, in fact, be resolved. If someone mutters, “What lovely weather we’re having!” as they crouch in a tornado shelter, the statement would appear to stand in stark contrast to the true state of affairs.

See, I always figured that was sarcasm, not irony or paradox.

Satire

Satire may feel like a relatively new and subversive form of humor, but in reality it has existed for millennia.

Well, yeah, anyone who knows anything about Roman history and literature knows that. Also, I insist that Shakespeare's Romeo and Juliet was intended as satire.

The satirist has many weapons at her disposal, including parody, a form that I discuss separately. She may also employ outlandish exaggeration, blatant falsehoods, and sly innuendo. And virtually all satirists use verbal irony and sarcasm extensively, since these devices are especially well suited to the twin goals of being humorous and critical at the same time.

Satire is something I should really work on. I have next to no practice with it, even though one of the first writings I got published (without pay) was a satirical article in the April Fools' issue of the student paper. I've mentioned that one in here before.

Hm. April Fools' is coming up. I wonder where I'm going to hide this year?

A prime example can be seen in “The Colbert Report,” in which Stephen Colbert played the role of a bloviating conservative pundit, rarely breaking character. Progressive audiences enjoyed his outlandish claims and proposals as a sophisticated rebuke of perceived excesses by pundits of the far right. However, some conservatives also enjoyed the program and believed that Colbert genuinely meant what he was saying.

This is a perfect example of Poe's Law

, which has nothing to do with that Poe. (This article does mention Poe's Law, later on in the text.)

Parody

As we have seen, pretense is an essential component of many forms of irony, and this holds true for parody as well. At its heart, a parody is an intentional imitation of something else, although the goals of the parodist vary. At one extreme, the intent may be to gently poke fun at the original work or its creator. At the other extreme, however, it may involve full-throated condemnation. The more aggressive forms of parody liberally use exaggeration, satire, and sarcasm to ridicule the authentic work and, by extension, the person who produced it.

Parody, on the other hand, is something I've done quite a bit of. I could still work on upping my parody game.

Some of my favorite movies and songs are parodies.

The lack of shared context in online communities makes users less likely to recognize nonliteral language as such.

Someone once observed that "there is great need for a sarcasm font." This is because, without visual or auditory cues, it becomes difficult - not impossible, but difficult - to state or understand a sarcastic comment.

I've proposed the use of Comic Sans for that purpose, because everyone loves Comic Sans and sarcasm.

(That was sarcasm.)

Parody can also be difficult to express online, since it requires an audience to recognize that some prior work is being imitated.

It's not just online. I had occasion, recently, to re-read Alice's Adventures in Wonderland, which is in the public domain and available online (just don't try to pirate Disney's version or you face the Wrath of the Mouse). In it, Carroll included several poetic parodies of works that would have been well-known when he wrote them -- but few today recognize them as such, because the fame of Alice has far surpassed that of the works that he parodied.

Is that irony? I don't think so, not technically. But it certainly makes me smile.

So, this is going to require a bit of background. Yesterday, I went to Alamo to see an old movie. I mean really old -- from 1924 -- and it was a black and white silent film made in the newly formed USSR. The film had been restored, apparently frame-by-frame, and a modern soundtrack added. I wanted to see it because I love science fiction, movies, and the perspectives of other cultures.

One-Sentence Movie Review: Aelita, Queen of Mars (Scored by Chris Bullock):

A fascinating insight into 1920s-era Soviet Russia, wherein, true to form, you invade Mars; it features remarkable sets and costumes, and a twist ending that, really, we all should have seen coming considering the movie is nearly 100 years old.

Rating: 4/5

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