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# Complex Numbers

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.
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September 19, 2024 at 11:15am
September 19, 2024 at 11:15am
#1076986
September 18, 2024 at 9:31am
September 18, 2024 at 9:31am
#1076947
 It Came From Outer Space Today's article, from Popular Mechanics, is a few years old, but I'm not aware of any related developments since then.     Some Scientists Believe All Life Started on Mars   Maybe it's not as crazy as it sounds. Let me state right off the bat that I have multiple issues with the way the headline is phrased. I'll give it a pass for now, though, because what's really important is the actual article. Could life as we know it have begun on Mars instead of Earth? Could it? Sure. Now, how about some evidence? A handful of scientists believe so, and even more think we should at least consider the possibility. One of my issues with the headline, and this sentence, is the use of the word "believe." It's as out of place in science as a clown at a black-tie ball. This special case of the overall theory of panspermia, where life on Earth began somewhere else and traveled or was planted here, has some prominent supporters. "Traveled or was planted" could imply that some sentience or conscious guidance was involved, so I don't like that in science either. As I've said before, if aliens planted life on Earth, where did said aliens come from? There are some good hypotheses as to how life emerged from non-life (though the distinction isn't a bright bold line) right here on Earth; where are the competing hypotheses for extraterrestrial origin? I've described it before as "kicking the can down the road," and I suspect the underlying impetus for wanting to "believe" this is either a) an attempt to bring religious beliefs into science, or b) a conviction that life on Earth is not "special." Or maybe both. In a new Salon article, these proponents say the theory makes intuitive sense based on what the two planets are like. If there's one thing you need to know about science, it's this: the whole point is to question intuition. Lots of solid scientific theories and discoveries very much contradict intuition. On the other talon, there's some evidence (as the article later points out) that Mars cooled off to the point where it had liquid water—essential for life as we know it—way before Earth did, so, okay, perhaps the chemical reactions that turned non-life into life took place there and not in our own oceans, only getting kicked off that planet by asteroid impacts. There's also some evidence that simple life can survive such a journey. So I can accept the possibility. But again: evidence? "Can" isn't the same thing as "did." There is, of course, at least one other possibility: life started on Mars, and also started on Earth, perhaps later, but separately. BUT. If you're going to assert that sentient aliens did it, then you also have to ask when and how said aliens got their start. Or, of course, you can hand-wave it and say "God did it," but then it's not science, but religion. And I have one more quibble with the article: Either way, Mars’s once-molten core slowed and solidified, reducing the planet’s gravity and atmosphere to nearly nothing and removing essential protections for any life form of which we know. Solidifying a molten core would do exactly jack and squat to the planet's gravity, which is correlated to its mass. I guess maybe a tiny fraction of a bit, due to energy/mass equivalence, but come on. I think they meant "magnetic field," not gravity. It's the magnetic field (which on Earth is probably generated by processes in the molten interior) that keeps the worst of the dangerous emanations of the accursed daystar from knocking complex molecules (like proteins and DNA) apart. Now, don't get me wrong, here; I'd be as excited as anyone to hear that real evidence for life, now or in the past, on Mars has been confirmed (by which I mean microbes or their equivalent, not pop-fiction sentient Martians). And maybe then we can compare it to Earth life and have a better idea if the one came from the other. But right now, it's all speculation, and, again, we have some very compelling, evidence-based ideas for how inert molecules might have come together to initiate the self-replicating process we call life, right here on this planet.
September 17, 2024 at 10:00am
September 17, 2024 at 10:00am
#1076900
September 16, 2024 at 9:42am
September 16, 2024 at 9:42am
#1076872
September 15, 2024 at 10:44am
September 15, 2024 at 10:44am
#1076824
September 14, 2024 at 10:14am
September 14, 2024 at 10:14am
#1076783
 One Thing In Common Most Saturdays, for the last couple of months, I've done a travel plans update. This will be the last one: I leave on Friday, and I'm still not ready. That is all. ...okay, maybe not all. And I did get a bunch of stuff taken care of. I got more suitable luggage (pun intended. Suitable? Suitcase? No? Damn.) I acquired a relatively cheap travel laptop so that, hopefully, I can make everyone here jealous by posting what I'm doing (I said this was the last plan update, not the last you'll hear of it). I checked with my mobile provider to ensure I'd still have connectivity over there (of course, I don't really trust them). I even made a few clothing purchases specifically for the trip. Still, I have a few more things to do and less than a week to do them. Then there's my utter lack of confidence in actually trying to speak French, combined with knowing that I'll be spending a few days in the Dutch-speaking part of Belgium. I suppose I could bully my way through like a stereotypical American tourist, but I'd really rather not. Oh, well. "Beer" is pronounced pretty close to the same way in English, Dutch, and French, so I'm sure I'll be fine.
September 13, 2024 at 11:21am
September 13, 2024 at 11:21am
#1076742
September 12, 2024 at 10:13am
September 12, 2024 at 10:13am
#1076648
September 11, 2024 at 11:48am
September 11, 2024 at 11:48am
#1076615
September 10, 2024 at 9:16am
September 10, 2024 at 9:16am
#1076559
September 9, 2024 at 9:39am
September 9, 2024 at 9:39am
#1076518
September 8, 2024 at 9:02am
September 8, 2024 at 9:02am
#1076471
September 7, 2024 at 7:20am
September 7, 2024 at 7:20am
#1076413
 Spit in the Ocean "Blog Week Birthday Bastion 2024"   [13+] by WakeUpAndLive️~Happiness Prompt 7. Sept 7. The world is not an ideal place for lots of people. What can YOU do to make it better? Well, the one most overwhelmingly important thing I have already done is: avoid bringing another human into a far-from-ideal world. The second most important thing is that I attempt to bring knowledge and comedy into an ignorant and largely humorless world. In that, I fail most of the time, but at least it makes me feel good to try. And there are other things I do. I recycle, to the extent that I'm able. I help out with causes, when I can be sure that it's going more to the cause than to the organizer. I try to minimize my water use, and refrain from littering. All of that amounts to spit in the ocean, and it doesn't even make me feel good to brag about it. And for every infinitesimal improvement we can make individually, millions of times worse stuff gets done by other people, governments, and big corporations. "Every little bit helps," my ass. You think it makes a difference if you give a dollar to a billionaire? No, they only notice if you try to take one. If there's anything that the pandemic years drove home to me, it's that any effort we make has to be a collective effort—and that it will never be a collective effort. There will always be those who deliberately make things worse. If an asteroid were about to hit the ocean, and the only way to avoid it would be for everyone to, I don't know, jump up and down at the same time, at least half the population will continue to sit on their asses, out of ignorance, apathy, a death wish, or maybe just plain spite. "The other political side wants me to do this, so I absolutely refuse to do it." So you're jumping up and down... for what? Some marginal improvement in your fitness? So you can say you're at least trying to help? I mean, in that particular case, I'd probably do it, just on the off-chance that everyone else will, too. But they won't. Yeah, some people are in a position to make improvements now, and I wish them the best. But I'm not in that position. What we need are big, systemic changes, not that one extra dollar. I can travel less, reducing my carbon footprint. I can move to a plant-based diet, helping the environment while reducing animal suffering. I can drive less, walk more. (I know this because I spent over a year without a car.) I can volunteer my time to causes I care about. I can mentor youth, sharing my vast intelligence and infinite wisdom with the impressionable younger generation. Yes, I can do all these things. But I won't. FINAL DAY! We get to give out a free Merit Badge every day this week. Want one? Anyone who comments here before 11:30 pm WDC time today could get today's. (I'll need that extra half-hour to pick a winner and send the badge before midnight.) To clarify: When I say "comment," I mean comment. Not review. Though reviews are always welcome. I also mean "here," not on the newsfeed post. MB recipient will be chosen at random. Maximum of one MB per commenter for the week. If I don't get comments, I'll pick a previous commenter, and maybe not at random. The MB will be the one I commissioned two years ago, "Complexity," which is a publicly available MB. I appreciate all comments; this is just a little incentive.
September 6, 2024 at 1:47am
September 6, 2024 at 1:47am
#1076349
September 5, 2024 at 9:04am
September 5, 2024 at 9:04am
#1076299
September 4, 2024 at 10:43am
September 4, 2024 at 10:43am
#1076238
September 3, 2024 at 10:05am
September 3, 2024 at 10:05am
#1076171
September 2, 2024 at 1:35am
September 2, 2024 at 1:35am
#1076064
September 1, 2024 at 8:25am
September 1, 2024 at 8:25am
#1076001