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Friday, Nov. 1, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

tj: Tenebrous Jekyll

about 11 hours ago
In response to "Phases"

Perhaps, if the full moon drives one to madness, the half moon will only make one half-crazy.

1 Like

Battywyn🎶

Oct 29, 2024 at 12:18pm
In response to "The Numbers Don't Lie"

I hear ya. I like to think I'm discerning enough to detect fakes, but that might just be my ego talking. When it comes to local businesses, I also go to social media and request recommendations. With both reviews from strangers and recommendations from friends, I give more credence when more (anecdotal, heh) details are provided. It's more an art than science, but liars either have such flawless details that their story sounds fabricated, or they get embarrassed when they trip themselves up. Truthtellers make errors, because memories are fallible, but they get more confused than embarrassed. Ultimately, the only way to really know is to try the product/service yourself. Which is why you'll hear about my Cancun vacation next month, but you don't have to believe me.

1 Like

Robert Waltz

Oct 29, 2024 at 11:07am
In response to "The Numbers Don't Lie"

Battywyn🎶

- I, too, look at the 1-star reviews, though again, reviews aren't my sole criteria. But sometimes it's blatant fake reviewing from the competition: "Don't stay here! It has rats, bedbugs, and roaches! Stay at this other place instead!" where "this other place" is clearly owned by the reviewer.

On the other paw, five-star reviews are often salted by the establishment in question.

Used to be, you could look at the four- and two- star reviews to get more honest, nuanced opinions, but the establishments, and their competition, have figured out that trick.

Which is one reason why reviews aren't my sole criteria.

(None of this applies to WDC reviews, or to the artificially inflated Uber rating system)

Battywyn🎶

Oct 29, 2024 at 11:00am
In response to "The Numbers Don't Lie"

And I was going to conclude by saying, when evaluating Caribbean vacation destinations, even if 5,000 people gave it a 4.8 rating, if the one star reviews were from seven people who encountered rats in their rooms, I'm very likely to not choose this destination.

2 Like

Battywyn🎶

Oct 29, 2024 at 10:58am
In response to "The Numbers Don't Lie"

Well, that was annoying. I was still in the process of editing, so please excuse the multitude of errors. I also intended to italicize the quotes from your post. Stupid f****** Parkinson's.

1 Like

Battywyn🎶

Oct 29, 2024 at 10:56am
In response to "The Numbers Don't Lie"

You know what sometimes lies? Anecdotes.

Also, dogs, I've heard.

Oh, hell no.

🤣 I love a good inside joke, but my marketing guru would tell you to cross link here

The fallacy is named after Robert McNamara, the U.S. Secretary of Defense during the Vietnam War, where his over-reliance on measurable data led to several misguided strategies

Bet he's proud of his claim to fame.

What's the point in data, then?

The point in data is a decimal separator, unless here in non-english speaking countries, in which case it's more likely a thousands separator.

So, whenever I'm evaluating online reviews for pretty much anything, and for the record I check them for almost everything - I always consider three things: 1. The average rating (eg, number of stars); 2. The The number of contributors to the rating; and three. What the one star ratings say.

1 Like

Annette

Oct 28, 2024 at 2:47pm
In response to "Waste Not"

In Southern California, every single family residence has a green bin. Every trash day, three different trucks make the rounds.

1 for composting
1 for recycling
1 for garbage

Trader Joe's sells almost exclusively packaged foods, even perishables. They have contracts with college food pantries. In many colleges, there are now pantries that are open to the students 4 or 5 days a week plus 1 day when people from the community can drive up and receive a bag with a mix of fresh fruits and vegetables along with non-perishable foods and donated sundries.

Churches are huge food redistribution centers also.

The "too hard to set up" systems have created good jobs here. Instead of seeing food donations as a liability, they have created a supervised and refrigerated second market.

The cost for employees is small because, of course, many of the positions are volunteers. But the supervisors and some of the staff are full time employees with benefits.

The article has thrown up obstacles that are easy to overcome.

3 Like

AnotherDreamer

Oct 28, 2024 at 11:33am
In response to "Waste Not"

https://cdn.prod.website-files.com/6156cebc4c605e281289a232/6156cebc4c605eb8bc89... e%20for%20beginners%2024%2009%2018.pdf

1 Like

Robert Waltz

Oct 28, 2024 at 11:33am
In response to "Waste Not"

AnotherDreamer

- Hey, it's a replicator from Star Trek!

Now let's invent the holodeck.

2 Like

AnotherDreamer

Oct 28, 2024 at 11:27am
In response to "Waste Not"

We the people of the world could use those products before they are compost as free sources for 3D printed foods that could be used for emergency food, or if tasty enough, re sold.https://builtin.com/articles/3d-printed-food

2 Like

February 2, 2020 at 12:30am

#974712

Paradoxxed

It's Groundhog Day, and later I'm going to a screening of the eponymous movie. In it, Bill Murray relives the same day over and over. It's not a trope that originated with that movie - I eventually read the book that inspired it, though it has an entirely different premise and cast of characters. Besides, ST:TNG did a time-loop episode first. Not by much; "Cause and Effect" aired in 1992, while GHD The Movie came out in 1993. Why, yes, I am a nerd; why do you ask?

Since then, the time-loop trope has become a staple of science fiction and superhero shows. There seems to be an unwritten rule (which is now written) that every show gets to do that exactly once, unless the show is about time travel.

In my life, though, there are times I seem to be looped, myself, like when I have to keep explaining the true meaning of the "blue moon," or, as illustrated by today's linked article, the question of alien life.

http://nautil.us/issue/63/horizons/our-attitude-toward-aliens-proves-we-still-th...

Our Attitude Toward Aliens Proves We Still Think We’re Special
Why we downplay Fermi’s paradox.

Um, the fact that we can conceptualize extraterrestrial life and write about it using picture-sound-symbols demonstrates that we are special.

One summer’s day in 1950, the great Italian-American physicist Enrico Fermi was having lunch with the physicists Edward Teller, Emil Konopinski, and Herbert York at Los Alamos... Then, after some delay—and, one might imagine, in the midst of some tasty dish—Fermi allegedly asked his famous question. Where, indeed, is everybody? Where are the extraterrestrials?

I'm not imagining "some tasty dish;" I'm imagining copious quantities of booze.

The Milky Way galaxy is about 100,000 light-years from edge to edge, Fermi reasoned, which means that a star-faring species would need about 10 million years to traverse it, even if moving at a very modest velocity of 1 percent of the speed of light. Since the galaxy is more than a thousand times older than this, any technological civilization will have had a lot of time in which to expand and colonize the whole galaxy. If one species were to fail in this endeavour, another wouldn’t. Consequently, if intelligent species were out there in any appreciable numbers, they would have been here already. And yet, we do not see them on Earth or in the solar system. For Fermi and many thinkers since, this constituted a paradox.

Most philosophical paradoxes are failures of using picture-sound-symbols, and not reflective of reality. Consider the ancient Greek Zeno's "paradox," which, to grossly simplify, asks the question: To get from Point A to Point B, you have to traverse half the distance. And then half again. And then half again, and so on ad infinitum. One can, consequently, never get there. If I recall the story, Aristotle at that point got up and walked out the door, disproving the "paradox." There are sound mathematical reasons why it's not a paradox, anyway, but they're irrelevant to my discussion. What is relevant is that just because we can formulate a brain-teaser doesn't mean it's a philosophical brick wall, any more than an optical illusion shows that all of reality is an illusion.

When you consider fiction and movies, it’s clear that Fermi’s paradox has become an important part of contemporary culture, challenging us to think more deeply about our place in the cosmos.

I referenced Star Trek above; that franchise can be viewed as an answer to Fermi's Paradox: its basis is that aliens quarantined us until we developed warp drive on our own, a kind of coming-of-age ritual necessary to become a galactic adult. I love Star Trek, but that doesn't mean I don't think that premise is utter crap, or that we can expect to find Vulcans and Klingons and whatnot when we do go out there.

If we believe that humans and the planet we live on are not particularly special compared to other civilizations on other planets, we would conclude that the stage of the biosphere and technology on other occupied planets must be, on average, older than the corresponding stages we see on Earth. If we humans are now on the cusp of colonizing our solar system, and we are not much faster than other civilizations, those civilizations should have completed this colonization long ago and spread to other parts of the galaxy.

And here we get to both an explicit assumption and an implicit assumption. The explicit assumption is the "If..." clause in the above quote. What hubris can make us think we're not special?

The implicit assumption is that life, necessarily, produces technological civilizations.

I'll go into a bit more detail on this. First of all, my standard definition every time I get Groundhog Dayed into a discussion like this; this definition is explicitly to forestall smartasses like me from saying "hurr durr but there's no intelligent life on Earth either." So for our purposes, I'll simplify "intelligence" to mean something like "the ability of some members of the species in question to build spacecraft." By that definition, we're intelligent. Any discussion about detecting traces of alien life implies that said life is intelligent by this definition; an otherwise sentient species who might be swimming in an ocean somewhere won't leave telltale traces for us to find with SETI or whatever.

I bring this up to segue from cosmology to biology, to wit: there is nothing inherent about the process of evolution that requires the development of intelligence; intelligence is not the end product of evolution; and there are plenty of other modes of being that facilitate species survival without the need to develop technology, as one can see by noting that, given the vast number of species on this planet, the number of intelligent species is approximately 0.

Yes, I've just said there's no intelligent life on Earth, contradicting my above definition. No, not really; I'm just saying that if there are 8.7 million species on the planet, and we're one of them, the difference between 1 and 0 is a rounding error. Point is, we haven't been around that long compared to the 4+ billion years of life on Earth, and there are other ways to survive and propagate a species than through developing bigger brains, opposable thumbs, and Netflix.

In other words, from an evolutionary standpoint, pigeons are just as successful as we are. Civilization, technology, rocket science, etc. may well be a fluke. And I'm not even going to go into the argument about how eukaryotic life was almost certainly a fluke, and how unlikely it would be for prokaryotic life to build spaceships; that would take me all night.

Basically, the idea that if life exists, then it must necessarily produce a spacefaring civilization, is spurious and about as egocentric as it gets. It's entirely possible that we won a lottery. What were the odds of the lottery? No one knows, but it's irrelevant; even if the odds are one in 10¹⁰⁰, once you win the lottery, your chance of having won it becomes unity. We exist, obviously, and even the most navel-gazing of philosophers would be hard-pressed to argue otherwise.

The consequences are rather dramatic: If life is quick to form after its host planet has formed, we get good probabilistic support for the existence of simple life on many planets in the Milky Way, and potentially complex life on some of them.

And here we come to another implicit assumption, in addition to the "complex life must produce intelligence" assumption: the conflation, ubiquitous in popular culture, of "life" with "intelligent life." I don't doubt that other planets harbor life, though we have no evidence for any besides on our own planet. But, again, life could very well stay simple.

I'm not going to rehash the author's other points here, but if you're at all interested in this sort of thing, it's worth it to read the article. I mean, yeah, I don't believe its premise or its arguments, but I've never let a little thing like vehement disagreement stop me from hearing others' opinions.

Just one more quote, from near the end:

We should not miss that opportunity by fighting for an outdated vision of ourselves as pinnacles of complexity in the universe.

As Douglas Adams noted, "Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is." I'd be surprised if there wasn't intelligent life somewhere else in the universe, if not in our own galaxy. I'd be really, vastly, hugely, mind-bogglingly surprised if there wasn't non-intelligent life, however simple or complex, in our own galaxy. But the further out you go, in intergalactic terms, the farther back in time you see - so the "argument from time duration" our author puts forth starts to fall away there. Besides, we have absolutely no evidence that superluminal travel is achievable, and overwhelming evidence that it is not, so any meaningful discussion about extraterrestrial intelligences must be limited to our local neighborhood. And let me reiterate for emphasis: "life" should not be confused or conflated with "intelligent life" (as I defined it above).

Fermi's paradox isn't a paradox at all. It's an argument against the ubiquity of alien rocket scientists.

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