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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 23, 2021 at 12:01am

#1019922

Birth of a Spaceship

The headline may be a bit misleading, but as a movie fan, I find this stuff fascinating. Also because it's Cracked, it's funny.

How A Jammed Camera Helped Create All Of Science-Fiction

Like I said, the title's misleading -- science fiction predates movies by nearly a hundred years. I suppose "helped create science fiction movies" would be more accurate.

The movie camera was the internet of its day. As in, it was touted as this great technical achievement that would revolutionize science and usher in a new era of progress and enlightenment. But, instead, people almost immediately started using it to make dirty movies and goof around.

Someone has probably enshrined this into a law already, but it seems to me that the first thing anyone does with any new technology is figure out how to make porn with, or of, it.

Hence the old observation that the holodeck will be humankind's last invention.

Born in the mid-19th century, Méliès was an illusionist who wanted to use the camera to film plays and his magic acts.

Turns out magic acts don't translate well to recorded media, for the simple reason that it's too easy for cameras to do tricks, and audiences will think it's all film effects. This is why magic acts are usually filmed in front of studio audiences.

Then one day, a weird thing happened. When filming scenes of city life, Méliès’ camera jammed. It took him a few seconds to fix, and when he later developed the film, he noticed that he accidentally invented the jump cut/stop trick. While the camera was malfunctioning, the scenery around the amateur filmmaker continued to change, none of which was being captured on film until Méliès started rolling again. And because the angle and position of the camera didn’t change, the resulting film seemed to show objects disappearing, a man turning into a woman, or a carriage turning into a hearse, etc.

While it's inevitable that someone would have thought of this sort of thing eventually, accidents can absolutely drive invention.

Méliès immediately realized he had stumbled onto something big, and he continued to experiment with other camera tricks until he was ready to put them all into a short film: A Trip to the Moon.

That's the one with the space capsule in the man in the moon's eye. Yeah, you know the one. It's iconic, and for a reason. Go to the original article; the video is embedded there, so I won't reproduce it again here.

Released in 1902, it was heavily influenced by the writings of Jules Verne and depicted the thing from the title. It also employed “every trick had learned or invented” to create what some consider the first science-fiction movie ever.

It is my unshakable opinion that the first science fiction book was Frankenstein. Whether this movie was the first SF movie or not, I'm not as sure of. But like I said, it's iconic.

And great job, Méliès, since A Trip to the Moon apparently convinced the world that movies could be full of fantastical scenes, flights of imagination, and, eventually, pants-shitting horror. Seriously, watching some of those old-timey movies feels like snorting bath salts off a heavily decomposed clown corpse dressed in your mother’s wedding gown.

Just leaving this here so I'm sure you read that imagery.

The point is, Méliès ushered in an era of fantasy and sci-fi movies, which officially ended in 2015 when Mila Kunis said out loud that she wanted to bone the half-human half-dog Channing Tatum in Jupiter Ascending.

Come on now, Jupiter Ascending wasn't that bad. I mean, it was bad, but it certainly didn't end SF movies. Hell, Dune just came out. I'm not seeing it until Sunday. Which reminds me, I keep forgetting to do a review of the movie I saw earlier this week (which was definitely not science fiction for once), so now's as good a time as any.

One-Sentence Movie Review: The Last Duel

I keep going back to Ridley Scott movies in the vain hope that at some point, one of them will come close to the greatness that is Blade Runner: The Director's Cut; this hope is inevitably dashed, but The Last Duel, unlike some of his other historical fiction movies, doesn't completely suck -- it's longer than it needs to be; Matt Damon has already been in France once before this year; and Kylo Ren has already gratuitously taken his shirt off approximately 143 times, but the acting is superb and the movie is surprisingly topical for modern audiences despite being set in medieval France (which the director doesn't let you forget even though everyone speaks American English, because you keep seeing shots of Notre Dame de Paris under construction).

Rating: 3.5/5

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