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:

Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

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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