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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Brandiwyn🎶

a day ago
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.

1 Like

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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

September 1, 2022 at 12:34am

#1037153

Intelligence Gathering

You know how sometimes you find something that agrees with your own ideas, but the way they present it leaves something to be desired?

Intelligent Life Really Can't Exist Anywhere Else
Hell, our own evolution on Earth was pure luck.

Problem #1: using the word "intelligent."

That's just begging some amateur comedian to go "Well, there's no intelligent life here, either ha ha heh ha hurr."

I prefer to use "technological." There's no question that we're a technology-using species. While the vast majority of us couldn't invent a tin can string telephone, we are really, really good at copying what smarter humans do, and smarter humans have built, among other things, airplanes, spaceships (albeit primitive ones), a vast communications network, and refrigerators, to name but a tiny fraction.

The reason this matters is that, in the search for extraterrestrial civilizations, what we're looking for isn't signs of "intelligence" but evidence of alien technology: communications, a Dyson sphere, robot infestations of other worlds (like we've done with Mars), maybe obvious signs of stellar engineering, etc. Those would, in principle at least, be easier to find than signs that some other planet has flowers.

Corvids are "intelligent." So are octopodes, dolphins, cats, one or two breeds of dogs, and the mold that used to be a container of leftovers in my fridge. None of them have developed radio or interplanetary travel.

Okay. Enough about that. I've beaten that deceased equine in here before.

Problem #2: using the word "can't."

Jury's out on whether the Universe is infinite or not. What is known is that the observable universe is finite, but extremely large, larger than our puny, unintelligent minds can comprehend. I find it extremely unlikely that there "can't" be another technology-using species in its vastness.

The problem is, the farther out you look, the farther back in time you go. Technology requires access to relatively heavy elements, and those are only produced over vast cosmic timescales, in supernova explosions and the like. In practice, we can only look at our own neighborhood in our own galaxy, and even there, the distances involved are (currently at least) insurmountable even if we could detect, say, a Ringworld.

And that's just the headline.

In newly published research from Oxford University's Future of Humanity Institute, scientists study the likelihood of key times for evolution of life on Earth and conclude that it would be virtually impossible for that life to evolve the same way somewhere else.

While I tend to accept this as it agrees with other findings in the past...

Problem #3: misconceptions about evolution

Humans are not the inevitable end-product of evolution. It's absolutely not inevitable that evolution will produce a technology-using species. Millions of other species right here on Earth are highly successful without having sent rockets to the Moon or spending time doomscrolling when they should be trying to sleep. Further, we're not the end product, but only one of many ways populations have adapted.

Problem #4: "virtually impossible."

It's "virtually impossible" that Halle Berry will show up at my door, wearing the catsuit and carrying two growlers full of delicious craft beer. But as long as she's alive, there's a chance.

For decades, scientists and even philosophers have chased many explanations for the Fermi paradox.

Problem #5: the Fermi paradox is not a paradox.

I don't have time tonight to go into the various kinds of paradoxes, but I assert that the Fermi paradox doesn't fit into any of those categories. The "paradox," as stated, is (based on Wiki): "The Fermi paradox is the conflict between the lack of clear, obvious evidence for extraterrestrial life and various high estimates for their existence."

First, like I said, we only have a relatively small sample size to find such evidence; and, second, those who make "high estimates" of the chance of it existing could be very, very wrong. Also:

Problem #6: Finding life is not the same thing as finding species we can communicate with.

This is related to #3 above, but we can't effectively communicate with other intelligent species here on Earth, beyond some very basic signals, and they have the same origin as we do; what makes us think we'd be able to communicate with a species that doesn't share our evolutionary history?

Additionally, there's a very good chance that there's life under the ice covering Jupiter's moon Europa. What I mean by that is there might be some bacteria analogues there, not that they've built radio towers.

How, in an infinitely big universe, can we be the only intelligent life we’ve ever encountered?

See above discussion of "infinitely." And "intelligent."

Even on Earth itself, they wonder, how are we the only species that ever has evolved advanced intelligence?

Using the "i" word again. We are not the only intelligent species; we're just the only one stupid enough to dig up billions of years' worth of concentrated energy to power an industrial society.

There are countless naturally occurring, but extremely lucky ways in which Earth is special, sheltered, protected, and encouraged to have evolved life.

Problem #7: Not explaining this very well.

I absolutely agree, but I don't have time to give examples, and apparently neither did this writer. There are hotlinks at the article, though; maybe they shed some additional light. And this also seems to equate "life" with "technological civilization," which I've already complained about. Basically, "life" can exist in far worse conditions than our own, but not necessarily a (mostly) cooperative civilization that can send robots to neighboring planets.

“The fact that eukaryotic life took over a billion years to emerge from prokaryotic precursors suggests it is a far less probable event than the development of multicellular life, which is thought to have originated independently over 40 times,” the researchers explain.

Problem #8: Not explaining this very well, either.

This article is from Popular Mechanics, not some science publication where the audience can be expected to know these terms.

Basically, prokaryotic life is simple cells: nucleus, cell wall. Like bacteria. Eukaryotic life, like us and most multicellular organisms on Earth, adds mitochondria to the cells, making them better able to produce the kind of energy one would need to become a rocket scientist.

But again, agreed. Of course, we won't know until we find some hard evidence of evolved life elsewhere, which we have not.

In this case, they’ve used a Bayesian model of factors related to evolutionary transitions, which are the key points where life on Earth has turned from ooze to eukaryotes, for example, and from fission and other asexual reproduction to sexual reproduction, which greatly accelerates the rate of mutation and development of species by mixing DNA as a matter of course.

Problem #9: "True believers" will argue with this, saying that there might be other biological avenues to producing Chee-Tos-eating apes or their equivalents. They're not necessarily wrong, but their arguments need to be addressed.

And using their model, these scientists say that Earth’s series of Goldilocks lottery tickets are more likely to have taken far longer than they really did on Earth.

Problem #10: "If technological life is so unlikely, how is it we're here?"

I've said this before, but I'll say it again: The chance of winning a lottery is absolutely irrelevant after you've already won it. Once you've won the lottery, the chance of having won it is unity.

Anyway. Enough. None of this is anything I haven't banged on about in here before, but it gave me a chance to look at it from another direction.

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