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').

1 Like

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

1 Like

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

October 17, 2024 at 10:19am

#1078457

Pros and Contrails

In a sane world, this PopSci article would put an end to one of the stupidest conspiracy "theories" immediately.

Why airplanes leave white streaks in the sky
Contrails are a problem, but not how you might think

Of course, in a sane world, the "theory" would never have gotten traction in the first place.

If you’ve ever looked up at a mostly blue sky and seen straight white lines criss-crossing the horizon, or watched a plane puff out a plume as it passed above, you may have wondered, “what causes that?”

Yeah, back when I was a kid. This was before the internet, so I used the version of Wikipedia we had back then: I asked my dad, who launched into a long-winded explanation of what he called vapor trails.

I think that was his way of keeping me from asking too many questions: answer so thoroughly that I'd get bored and stop asking questions.

And no, they’re also definitely not “chemtrails.”

I'd like to think that, even without the influence of my father's scientific mind, I still wouldn't have bought into that idiocy. But I can never be sure.

What they are, instead, is frozen water vapor crystallized on soot particles, both of which are standard byproducts of a jet’s combustion engine.

Well, there you go: the answer.

And though conspiracy theories about airplanes distributing mind control chemicals aren’t correct, contrails are having a negative planetary effect.

Because of course they are. Nothing we do is allowed to actually be good for the environment.

Atmospheric conditions have to be just right to enable the jet vapor to crystallize: moist enough that the water doesn’t evaporate, and cool enough that it freezes.

But as everyone knows, thanks to the internet, nothing that's described as "moist" can ever be cool.

A growing body of scientific research, including studies conducted by both Barrett and Stettler, indicate that contrails cumulatively have a warming effect, contributing to human-caused climate change.

These studies are important, because the leading hypothesis that I'd heard was that, being basically cirrus clouds, they'd have the effect of reflecting some sunlight, reducing the "heat in" side of the climate warming equation.

Perhaps they do, but other things about them overshadow (pun intended) that.

Since contrails, like other clouds, are bright white and insulating, they reflect light and heat, and also trap it. On a sunny day, a contrail does two things simultaneously. It “acts like a blanket,” preventing heat radiating from Earth’s surface from escaping to space, says Barrett–this is the warming effect. Simultaneously, it also reflects sunlight from space away from Earth’s surface, in a cooling effect. Unfortunately, even when the sun is shining, generally the blanket effect outweighs the reflector effect, says Barrett.

At least according to this guy.

Anyway, the article goes into what can be done about them, and, surprisingly, the answer isn't "just stop flying" or "stop using plastic straws." In other words, this is one contributor to climate change that can be mitigated fairly easily, and it's not even about us doing penance for how sinful humans are (just corporations).

And yet, some of us are just going to blithely keep on believing that they're mind-control chemicals.

1 Like • Comment

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