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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Turkey DrumStik

about 7 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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.

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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

November 1, 2018 at 1:29am

#944617

Labyrinth

Ever wonder if knowledge inhibits creativity?

I have.

I remember once when I was a young child, there was an illustration in a children's book that featured a night sky through a window. I don't remember the context or anything else about the book, but I do remember a crescent moon and stars against a dark blue background seen through the classic rectangular grid of a double-hung window.

Or I might be remembering some details wrong. Memory is like that. But what I am certain of is that there was a stylized star between the horns of the crescent moon.

"That can't happen," I told the teacher, or babysitter, or caretaker, or whatever. "The moon is still there; it's just dark. It would block the star." Okay, I probably didn't use a semicolon, but you can't know that for sure, because I spoke it, and it's not like I would have vocalized the punctuation.

"It's just a drawing," the adult explained. Or something to that effect.

That's the first time I remember using knowledge I'd gained to revoke someone's artistic license. I've done it countless times since then, most often when some science fiction show depicts the asteroid belt as a dense field of tumbling rocks. In reality, the view from any given place in the asteroid belt might feature one or two extra points of light, but is otherwise identical to the view from, say, the other side of the moon. Asteroids are widely separated. If they're not, they turn into smaller asteroids which then become widely separated. The result is barely even a hazard to astrogation, and you're not going to be able to use it to lose the bad guys who are chasing you.

And yet, such a depiction immediately conveys "asteroid belt" to the viewer, just as stars - misplaced or not - and a crescent moon on a dark blue or black background says "night."

Knowledge is like a labyrinth, in a way. A really big and convoluted one. Fixed, but with dark, twisty corridors and branches that lead nowhere, but must be traversed anyway. It has a floor, and walls, and a ceiling, and even adjoining spaces might be miles apart as one traces one's path. It occurred to me recently that the idea of such a maze might have even been inspired by the naked human brain, with its lobes and crevices - the brain, of course, being the container of knowledge.

The classical labyrinth housed the Minotaur, a bull-man chimera with a penchant for munching on humans. The Minotaur never made sense to me, either. Even if you allow for divine intervention or genetic engineering, the circulatory, muscular, nervous, lymphatic, and other bodily systems of human and bovine are simply incompatible. Not to mention that, in its classic depiction as a human figure with a bull head, its teeth would be that of a ruminant: dull, flat, and entirely unsuitable for chewing on any kind of meat - long pig or otherwise.

If the labyrinth represents knowledge, the Minotaur would have to represent ignorance. Fake news. Unfounded fears; the monster in the shadows. Illusions.

Theseus had to a) defeat the Minotaur and b) escape the labyrinth. For the first task, he had a sword. Swords are extremely useful for such things, but only if you know how to use them. For the second task, he had a ball of string helpfully provided by some chick who probably realized that, while Theseus could have simply marked his path with the sword so that he could follow the marks back out, doing so would reduce the sword's effectiveness at its primary task; to wit, turning the Minotaur into prime rib.

I think that part of the story was put in there to illustrate that Cretan women were smarter than Athenian men. But I digress.

It occurs to me that Theseus could have accomplished both tasks if, instead of a sword, he'd carried a sledgehammer. Go in, find the Minotaur, bash it between the horns, and then knock down some walls. But I suppose that would be cheating.

Hang on, though - isn't creativity all about cheating? Did Alexander the Great cheat when he used his own sword to cut through the tangle of the Gordian Knot? Did Captain America cheat when he knocked down the flagpole to earn a ride in the Jeep?

As I mentioned above, two adjacent spaces in a labyrinth could be very far apart if one traverses the maze like a good little lab rat. But let's give the lab rat some supplies - say: a) a chocolate-chip cookie; b) a paper clip; and c) a toy truck. Rats are both smart and creative, so can our Theseus-rat MacGyver his way out of the labyrinth?

This is where creativity comes in. It turns out that each chocolate chip is actually a concentrated block of C-4. Go to a wall, press one of the chips to it, stick one end of the paper clip in the chip, attach the truck's ignition to the other end, and use the starter motor on the truck to ignite the explosive. Boom. No more wall. Step through, see another wall. Pull out another chocolate chip. Repeat as necessary, and you're out. As a bonus, you can even eat the cookie when you're done. And maybe even the truck.

There, now - just because something is absurd doesn't make it impossible, you know. Especially if you're a rat - or a writer.

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