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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Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

about 2 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

June 4, 2024 at 9:27am

#1072131

Drakes and Pains

There are topics I've never discussed before in here. This isn't one of them. From Gizmodo:

Updated Formula on Alien Intelligence Suggests We Really Are Alone in the Galaxy
An adjustment to the famous Drake Equation could radically refine estimates of intelligent civilizations in our galaxy.

Sigh.

Using the adjective "intelligent" there just begs people to make tiresome "no intelligent life here, either" jokes. In an effort to forestall this, I'll use "tech-capable," as that's really what we're looking for, and there's little debate over whether we fit that description (any such debate that takes place on the internet would be the dictionary definition of irony).

Astronomer Frank Drake...

Unclear if he's related to Sir Francis Drake.

...formulated his influential equation in 1961 to estimate the number of civilizations in the Milky Way capable of communicating with us.

Eh... not really. It's more subtle than that. Look, this article fails on many levels, but the very top level is that it doesn't explain the Drake Equation, which isn't really an equation but a way to think about these things systematically.

I'm not going to copy it here, either, because I can't be arsed to do the formatting. But at least I'm providing a link to the Wiki page on it.

I also want to emphasize that the search for tech-capable life (or, by extension, any autonomous machines they might have created) isn't the same thing as the search for extraterrestrial life. The latter doesn't necessarily result in the former; as I've noted before, there's nothing inherent to evolution that requires the development of tool-using species who go on to build spacecraft. All we know for sure is that it can happen, as it did happen here. On the other hand, I think it's a safe assumption that tech-capable life requires non-tech-capable predecessors.

Fortunately, the Drake Equation explicitly separates those two.

Since Drake thought up this way of looking at things, we've gotten a better handle on a lot of the parameters. For one thing, back in 1961, we had absolutely no evidence that other stars had planets orbiting them. Now, based on the sample set of our relatively close stellar neighborhood, it seems likely that the vast majority do.

Anyway, back to the article.

Our understanding of planetary science has changed a lot since then, leading a team of scientists to propose a pair of important adjustments that produce an answer that could explain the Great Silence.

This is, as the article notes in the next paragraph, related to the Fermi Paradox, which I mentioned last week.

However, as we'll see, these "adjustments" don't need to be incorporated into the Drake Equation, as they're inherent to one of its factors.

Planetary scientists Robert Stern from the University of Texas at Dallas and Taras Gerya from ETH-Zurich, the two co-authors on the study, suggest that the presence of both continents and oceans, along with long-term plate tectonics, is critical for the emergence of advanced civilizations.

With all due respect to the credentials of the scientists in question, I, though admittedly an amateur, feel like it really just puts a lower limit on the f_i term in the Equation, the one that represents the fraction of life-bearing planets that go on to produce tech-capable species.

In other words, as we have absolutely no idea what f_i actually is, having only a sample set of 1 to work with, it's just as valid to incorporate the ideas of continents and continental drift into that term, which probably makes it lower. That fraction might be 1/1. But it also might be 1/10¹⁰⁰. Or lower. Or anything in between. As I've noted before, we might have won the planetary lottery, and once you've won the lottery, the prior chance of having won it is irrelevant.

They consequently propose the addition of two factors into the equation: the fraction of habitable planets with significant continents and oceans and the fraction of those planets with plate tectonics operating for at least 500 million years. This adjustment, however, significantly reduces the value of N in the Drake Equation.

So does plugging in arbitrarily low values. I've seen online Drake Equation calculators, and none of them that I've found allow for what I consider to be realistically low guesses. And "guesses" is what they would be.

“Our work suggests that both our planet Earth with continents, oceans, plate tectonics, and life and our active, communicative, technological human civilization are extremely rare and unique in the entire galaxy,” Gerya told Gizmodo.

To be clear, I have little doubt that they're right about the continental drift thing, and a slightly higher but not significant doubt that they're right about this guess. What I object to is further muddying the waters by adding terms to the DrEq (I'm calling it that from now on), which is already confusing to a large number of humans, thus adding fuel to the "no intelligent life down here, either" crowd.

The researchers argue that the presence of large oceans, plus Earth’s shift from single-lid tectonics (a stable surface layer) to modern plate tectonics about 1 billion years ago, were critical to the rapid development of complex life.

Again, not arguing that point. But it's just part of f_i.

According to the new study, plate tectonics are crucial for developing complex life and advanced civilizations. Earth’s plate movements create diverse habitats, recycle nutrients, and regulate climate—all vital for life. It’s important for plate tectonics to last for 500 million years, Gerya explained, because biological evolution of complex multicellular life is extremely slow. “On Earth, it took more than 500 million years to develop humans from the first animals, which appeared around 800 million years ago,” he said.

Something about that math doesn't work out, as humans branched off from other apes something like 6 million years ago, admittedly with a large error bar.

I should take this opportunity to point out that the other reason I use "tech-capable" instead of "intelligent" is that there's more and more evidence that "intelligence" is not unique to humans. Nor is tool use. But the idea of making a tool to make other tools, and so on recursively until we get spaceships, well, that seems to be unique to humans on this planet. Dolphins don't seem to show any interest in creating computers. Crows (see yesterday's entry) just seem to use their intelligence to fuck with us.

The modified Drake Equation suggests that advanced civilizations are extremely rare, with the chance of planets having the right conditions being between 0.0034% and 0.17%. This means there could be anywhere from as few as 0.006 to as many as 100,000 active, communicative civilizations in our galaxy, with the actual number likely being on the lower end, considering the limited time these civilizations might communicate due to potential societal collapse or extinction.

Oh, Thor's balls. 0.006 is obviously wrong, because we'd make it 1. And that's still a damn wide range. Not to mention the "limited time" bit is already in the DrEq; it's represented by L.

Because the low estimate is really close to zero, it means there’s a good chance there might not be any other civilizations in our galaxy. This would help explain why we haven’t detected any signals from other civilizations yet.

Well, DUH. That's what I've been trying to say.

In the past, the Drake Equation gave a much higher low-end estimate, suggesting that it was almost certain we weren’t alone and that there should be at least 200 civilizations trying to communicate with us.

That's not a failure of the equation. That's a failure of humans to comprehend the possibility of really large denominators.

Stern and Gerya aren’t the first to propose the idea that suitable planets for advanced life are few and far between. This suggestion, known as the Rare Earth Hypothesis, was first articulated in the 2003 book Rare Earth: Why Complex Life is Uncommon in the Universe, written by scientists Peter Ward and Donald Brownlee. Interestingly, Ward and Brownlee were likewise fixated on plate tectonics as a factor.

I don't think this is an ad for that book. But, full disclosure here, I've read it. Some of its hypotheses, such as the requirement to have a relatively big moon to help stabilize polar shifts, have been pretty much ruled out, from what I understand. Not surprising; a lot has been discovered in the last 21 years. But that doesn't negate the whole book.

The Rare Earth Hypothesis, while seductive, fails to account for the adaptability of life and the potential diversity of habitable environments.

Neither of which guarantees the development of tech-capable species. It's been a while since I've read that book, but as I recall, their guess was that life is pretty common, but tech-capable life, not so much.

Another limitation of this study, and this is no fault of the researchers, is that we’re still far from knowing which values to plug into the equation.

Exactly. Though, as I've said, we continue to refine some of the estimates.

Part of the problem with adding more terms is that each term magnifies the possible range of the result. Say you have a=b*c. Each term on the right of the equation has a range of possible values. By multiplying them together, you're also multiplying the range. Now incorporate d and e into the equation, both ranged, and those ranges multiply, too. All of which is to say that the "we just don't know" factor increases substantially with each term. And that's pretty much the reason Occam shaved with his Razor.

None of this gets past the "we don't know" stage. I know some people are dead certain, one way or the other. What's important, to me anyway, is that we're prepared to adjust our prior beliefs in the face of evidence. If aliens landed in my street, for example, I'd suddenly believe in them (provided, of course, someone hasn't slipped me some psychedelics).

What I really object to is lazy science reporting. Especially when the article, at base, does make some good points.

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