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 15 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 17 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.

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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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

April 13, 2020 at 12:35am

#980936

The Truth is Out There

I'm not the only one who riffs off of articles I find online. Meta-riff today!

https://freethoughtblogs.com/pharyngula/2020/03/03/nee-ned-zb-6tnn-deibedh-siefi...

NEE NED ZB 6TNN DEIBEDH SIEFI EBEEE SSIEI ESEE SEEE!!

I've been following this guy, PZ Myers, for a while now, and while I don't always agree, I appreciate his point of view. Except about movies. Guy has no taste in movies, except occasionally and by accident. Fortunately, this isn't about movies, but science, and Myers is a scientist, not a movie producer.

Wired tries to defend SETI and UFOlogy. They argue that there are 3 branches of inquiry — exobiology, the search for extraterrestrial intelligence, and the study of UFOs — and each has their place in our battery of methods.

For reference, here's the article he links:

https://www.wired.com/story/what-scientists-can-learn-from-alien-hunters/

I've been by the place in the article's header photo. It also gained notoriety last year, as it was mentioned as a gathering place for the Assault on Area 51, or whatever it was called, that internet hoax that blew up. Simpler times...

Anyway, it's worth reading the original Wired article, even if it's not up to Myers' standards.

Speaking of which...

There’s a continuum of legitimacy [the Wired article argues], and it’s entirely arbitrary that we place UFOs in pseudoscience, and don’t fund SETI, and think exobiology is valid and interesting. That is a good point, except that I think there is a solid criterion that is rooted in how we do science.

Arbitrary, maybe, but if you can defend something, is it truly arbitrary? To give an example, it's entirely arbitrary that we start the year on January 1, which in turn is about 10 days after the winter solstice. There's no defense of that other than "it's the way we've done it for centuries," which is no defense at all. With science, on the other hand, you can set boundaries and then see where shit falls with regards to the boundaries.

Here’s the deal: early in our training, we’re taught to keep an open mind — you use hypotheses to guide a line of research, but we must be prepared to find unexpected results and alternative explanations. We’re adapted to thinking, “My experiment to test my hypothesis should find X, but if it finds Y we’ll have to modify the hypothesis, and if the answer is Z, well, back to the drawing board, but gosh, that would be exciting.”Here’s the deal: early in our training, we’re taught to keep an open mind — you use hypotheses to guide a line of research, but we must be prepared to find unexpected results and alternative explanations. We’re adapted to thinking, “My experiment to test my hypothesis should find X, but if it finds Y we’ll have to modify the hypothesis, and if the answer is Z, well, back to the drawing board, but gosh, that would be exciting.”

In case you're wondering how science is meant to work, well, that sums it up well.

Myers goes on to explain the difference between exobiology, SETI, and UFOlogy, as they relate to science.

It’s as simple as asking, “What will we learn from doing the observation/experiment?” SETI’s answer is nothing, unless we find a one in a trillion possibility, then it’s the jackpot. UFOlogy’s answer is that they already know little green men exist, so we just have to photograph thrown pie plates until we’ve persuaded the establishment. Neither is good science.

I haven't had much experience with SETI, but I was always under the impression that whenever they found something interesting but not clear evidence of ETI, the data could be used by astronomers. Seems to me that there's been at least one case of that. So I wouldn't call their negative results, or lack of results, entirely useless.

Both SETI and UFOlogy are strongly susceptible to apophenia as well. They are trying to fit complex data to a prior expectation, so there’s a tendency to impose patterns on noise. Here’s a classic example: NASA has observed complex sand dune formations on Mars.

Thanks, Myers, you made me look up a word for the first time in years. Apophenia: the tendency to mistakenly perceive connections and meaning between unrelated things. I tend to think of writing as a kind of imposed apophenia (without, I admit, using that word); when I write, I look for metaphorical connections. But I don't labor under the delusion that these connections have significance outside of what I'm writing.

The sand dunes he mentions here lead to the title of Myers' post. To me, it's a bit like looking for patterns in a Jackson Pollack painting. We're pattern-seeking animals; when we see a pattern, we try to interpret it, to make sense out of it, when sometimes it's just a meaningless pattern. Might as well look for meaning in the coastline of Great Britain. Sure, if you squint just right, maybe you can see something else, but it's coincidental. The only "meaning" involves land formation and shoreline erosion, or, in the case of Mars, natural wind sculpting.

In fiction, everything has a meaning. That's why we like fiction. This isn't the case for observed phenomena.

Right. So someone, probably as a bit of lark, tried to interpret them as dots and dashes, and then translated them into Morse code (why ancient Martians would have used a code devised by a 19th century American is left as an exercise for the reader).

Because Samuel Morse didn't die; he was transported bodily to Mars, along with John Carter, only Morse was too geeky for Burroughs to have written about. Duh. What? You can't disprove that.

That’s the problem with SETI, though. The universe produces patterns all the time, and human brains strain to impose interpretable derivations on them — SETI will milk that for all the news and attention they can get, even if it is ultimately meaningless.

Meanwhile, UFOlogists already know that the aliens are living on Mars, and have trained Bigfoots raking the dunes to send secret messages to the fleet hovering invisibly in our atmosphere, and you ignore it at your peril, you fools.

Is it Bigfoots or Bigfeet? When I see one, I'll be sure to ask him. Or her.

So my point here is the difference between being open-minded, and being certain. As I pointed out in my last entry, we can't be absolutely certain about anything, even if we think we are. The open-minded person may have ideas, and then looks for ways to refute the ideas. That, ideally, is how science is supposed to work. Whereas having a conclusion already set in one's mind and then looking for evidence to support it (and usually ignoring or downplaying evidence to the contrary), well, that doesn't get anyone anywhere, except in their own minds.

Now, I've said this before, but it bears repeating: the universe is a big place, and I would be very surprised if there wasn't other machine-building life out there... somewhere. But I'm not certain of it, and I haven't seen any evidence to support it. Consequently, I don't think SETI is utterly useless. But the odds, I think, favor the existence of non-sentient life at least in our galactic neighborhood, and so I'd have to put my money on the exobiologists to get there first.

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