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.

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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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
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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.

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October 16, 2023 at 9:50am

#1057492

The Trouble with Quibbles

This is pretty cool, recent, and informative, though of course I have some quibbles.

The true story of how humans are searching for intelligent alien life
Real scientists are searching for alien life. Don’t let the kooks distract you.

Quibble 1: "Intelligent." I've railed on this before, but, to summarize: What they really mean is "technology-using." The signals postulated in the article would be the signs of tech-using beings. It is possible to be intelligent and never invent radio, spaceships, lasers, or Tamagotchi. In fact, we did not, up until a few decades ago, which is an insignificant sliver of time compared to the age of the universe. And, finally, using "intelligence" just invites tired, outworn clichés like "we can't even find intelligent life on Earth," which is one of those rare statements that automatically disproves itself because one must have a minimum level of intelligence to utter it and have it be understood.

So. When the article says "intelligent," substitute "technological," and you'll be closer to what I believe the intent to be.

This summer, a stony-faced David Grusch, a former US Air Force intelligence officer, sat before a House Oversight subcommittee and made some extraordinary claims. Chief among them is that the American government has a clandestine program that locates then reverse engineers unidentified aerial phenomena (UAPs) — an ostensibly less-silly way of saying unidentified flying objects, or UFOs — and that US operatives were in possession of nonhuman biological matter.

Quibble 2: UAP may be more or less silly than UFO, but I believe it to be a better fit. Many phenomena formerly attributed to UFOs weren't "flying objects" at all, but mirages or electrical activity. At which point they weren't "unidentified" either, but I'm willing to bend on that one, as everything is unidentified until it's identified.

It has the added advantage of not yet having accumulated years of fringe.

Quibble 3: I am in possession of nonhuman biological matter, too. They're called cats.

Grusch didn’t provide an ounce of verifiable evidence, citing only anonymous sources telling him vague things. When pressed for confirmation, he said because this was all so exceedingly classified, he was unable to provide specific details while under oath.

Not-a-quibble: uh huh.

Let’s get something straight: Congressional hearings are not the way we are going to discover the existence of intelligent alien life. They are a distraction from the bona fide alien-hunting work — the sort that doesn’t involve grandstanding individuals and showy stunts, but scientists searching a sea of stars for the sounds or sights of extraterrestrial intelligence.

Quibble 4: We're also looking for signs of general life: byproducts of biological processes. There's difficulty there, because we only have one data point (Earf) for what to look for, but you gotta start somewhere.

Because space is inconveniently enormous and traversing it so intensely time-consuming (without bending the fabric of space-time to your will, anyway), it’s exceedingly more likely that humanity’s first brush with extraterrestrials (ETs) will come in the form of eavesdropping on radio transmissions they’ve sent, or seeing a sign of technological civilization with a telescope, than recovering a pancaked little green wayfarer from a crashed capsule.

Quibble 5: What's the first thing we did when we started exploring space? Sent robots, not people. No reason to assume hypothetical aliens wouldn't do the same. (No reason to assume they would, either; just gotta keep the possibility open.)

“If we detect a civilization, that means civilizations can exist for a reasonable amount of time and overcome their issues and problems,” says Ravi Kumar Kopparapu, a planetary habitability researcher at NASA. “That means there’s great hope for us.” (Or, if the grim history of colonization has anything to say about it, great peril.)

Quibble 6: This does not follow, logically, and rests on a biased premise. Aliens would be, by definition, alien, and their "issues and problems," if they have any, may not bear much resemblance to ours.

If they were to discover that there is life out there — intelligent life that has forged a civilization — it would first mean that biology is not a fluke. Instead, it is something that can take root on many worlds; something that does not merely arise but repeatedly produces thinking, technological, curious creatures, those that may wish to share their knowledge of the universe, and their way of traversing or surviving it, with others. And if this civilization existed on a world very different from Earth, it would demonstrate that the largely unlivable cosmos is populated by myriad different isles of habitability.

Quibble 7: There's a lot of assumption to unpack here. For starters, it is possible (I would even say likely) that we'd see signs of life first, not technology. As I noted above, on our world, the kind of technology that produces signs that we could, in theory, detect has only been going on for less than an eyeblink compared to how long life has existed. There is nothing about evolution that requires the eventual appearance of a curious species with the right combination of intelligence (using that word now in its general sense), manual dexterity, language, socialization and other factors to begin to develop complex tools. But as the article notes toward the end, we might not find anything, but if we don't make the attempt, we will definitely not find anything (unless of course it comes to us first).

Okay, that's all I'm going to quibble about. The rest of the article goes into great detail about the actual search, and it's probably worth at least skimming if you're interested in this sort of thing. Just keep in mind that, given the above quibbles, I'm not 100% on board with the speculative aspects.

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