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

August 14, 2023 at 10:00am

#1054078

What's the Big Idea?

Today's article, presented after languishing in my queue for several months, is from The New Yorker, so it's long and rambling and to be honest, I didn't read the whole thing. I'm linking it anyway.

Dimming the Sun to Cool the Planet Is a Desperate Idea, Yet We’re Inching Toward It
The scientists who study solar geoengineering don’t want anyone to try it. But climate inaction is making it more likely.

For starters, the headline is a bit misleading. It makes it sound like people are actually going to turn down the dimmer switch on the accursed daystar itself. We don't have that technology yet.

If we decide to “solar geoengineer” the Earth—to spray highly reflective particles of a material, such as sulfur, into the stratosphere in order to deflect sunlight and so cool the planet—it will be the second most expansive project that humans have ever undertaken. (The first, obviously, is the ongoing emission of carbon and other heat-trapping gases into the atmosphere.)

We do have that technology, or are very close to it, so it's a matter of "should we" not "could we."

I would also like to point out the logical fallacy right there in the first paragraph: the idea that our "most expansive project" is "the ongoing emission of... heat-trapping gases into the atmosphere." Normally, I'd quit reading right there, from a combination of mistaking a result for an intention, and, well, it being in The New Yorker. Belching out greenhouse gases wasn't what we set out to do. It would be like saying you set out to purposely damage your liver by drinking booze and popping Tylenol.

The idea behind solar geoengineering is essentially to mimic what happens when volcanoes push particles into the atmosphere; a large eruption, such as that of Mt. Pinatubo, in the Philippines, in 1992, can measurably cool the world for a year or two.

You know what else would work? Nuclear winter. That doesn't mean it's a good idea, either, but it would work, at least temporarily.

It is being taken seriously because of something else that’s speeding up: the horrors that come with an overheating world and now regularly threaten its most densely populated places.

Nothing wrong with thinking about it. That's just science. Implementing it would be a different story.

Even before 2030, we may, at least temporarily, pass the 1.5-degree mark.

Ah, the naîvité of last November. We've already crossed that line,

snowflake.

It’s likely, in other words, that conditions may force a reckoning with the idea of solar geoengineering—of blocking from the Earth some of the sunlight that has always nurtured it.

Like I said, thinking about it is fine. When it comes to actually doing it, though, you have to deal with the possibility of overcorrection, which could make things worse in the other direction.

Everyone studying solar geoengineering seems to agree that it’s a terrible thing. “The idea is outlandish,” Parker told me. Mohammed Mofizur Rahman, a Bangladeshi scientist who is one of Degrees Initiatives’ grantees, noted, “It’s crazy stuff.” So did the veteran Hungarian diplomat Janos Pasztor, who runs the Carnegie initiative on geoengineering governance, and said, “People should be suspicious.” Pascal Lamy, a former head of the World Trade Organization (W.T.O.), who is the president of the Paris Peace Forum, agreed, saying, “It would represent a failure.”

I'd take the experts' opinions over that of ordinary people (including me) any day. I can't help but feel that these quotes were cherry-picked, however.

(Sulfur dioxide is the most commonly discussed candidate, but aluminum, calcium carbonate, and, most poetically, diamond dust, have also been proposed.)

The poetry about using diamond dust isn't that it's diamond dust. Diamond isn't nearly as rare or precious as people are led to believe. We can make it easily in laboratories. No, the irony is that diamond is carbon, and carbon (in other molecular forms) is part of the problem.

The question is more: what else would it do? On a global scale it could, at least temporarily, turn the sky hazy or milky (hence the title of Kolbert’s book); it could alter “the quality of the light plants use for photosynthesis” (no small thing on a planet basically built on chlorophyll—studies have shown that U.S. corn production increased as polluting aerosols went down in the wake of amendments to the Clean Air Act); and it might damage the ozone layer, which is only now repairing itself from our recent assault with fluorocarbons.

And those are just the known unknowns.

Worse, if they did this (and I'm not saying they will, or should), then any random weather disaster will be turned into the dust cloud's fault by ignorant hicks. "A tornado hit my house. Goddamn weather machines!" "My garage fell into the river! That geoengineering caused a flood!" Don't believe me? Look at any random weather disaster now and see how fast someone screams "climate change" before the scientists even have a chance to weigh in on it. Or, in another field, the people who are absolutely certain that a certain vaccine is worse than the disease.

Never underestimate the power of human stupidity.

The article touches on that, a bit, and I won't quote from it further; as I said, it's long, though not nearly as convoluted as a lot of the stuff that rag puts out.

I will, however, point out that there are other possible mitigation efforts under thought. One of the most interesting, to me, is enormous carbon capture plants, designed to essentially filter the atmosphere to directly reduce the proportion of greenhouse gases. (Sure, vegetation does this too, but this could potentially be faster than trying to plant billions of trees.) It reminds me of the massive atmosphere plants in Burroughs' Mars books, a last-ditch effort of his fictional Barsoomians to prevent the ultimate climate disaster.

In other words, think big. Just remember that not every idea is going to be a good one.

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