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
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

October 6, 2022 at 12:02am

#1038722

That's Cold, Man.

As it gets colder here, sometimes I have to be reminded that, objectively, 60F isn't really all that cold.

What’s the Temperature in Space? The Weirdly Nihilistic Reason Why It’s So Cold
Because it's all a bunch of nothing, baby.

When I was in college, with access to the proto-internet, I was able to look up temperatures in cities around the world. Whenever it got really cold around here, I used to make myself feel better by looking up the temperature in Nome, Alaska, a town (if you can call it that) barely south of the Arctic Circle.

That stopped when, one day, by some climatic fluke, Nome was actually warmer than Charlottesville.

Even looking up the temperature on Mars didn't always help. Sometimes it gets warm enough there to go out in shorts and a t-shirt. If, you know, it had about 100 times more air pressure and any oxygen.

But of course, some places are even colder. The South Pole in winter, for example. Or most of outer space.

Far outside our solar system and out past the distant reachers of our galaxy—in the vast nothingness of space—the distance between gas and dust particles grows, limiting their ability to transfer heat. Temperatures in these vacuous regions can plummet to about -455 degrees Fahrenheit (2.7 kelvin). Are you shivering yet?

Nope. I'm sitting on my deck on a mid-fifties (F) night, with the infrared heater on, and feeling better about things because at least it's not as cold as intergalactic space.

But understanding how cold is space, and why the vacuum of space is this cold, is complicated.

Because of course it is.

Vacuum doesn't technically have a temperature. There's nothing to have temperature. Vacuum is a remarkably good insulator, which is the science behind advanced Thermos technology. Of course, outer space isn't really a complete vacuum, just close enough for most practical purposes.

For physicists, knowing what the temperature in space is is all about velocity and motion. “When we talk about the temperature in a room, that’s not the way a scientist would talk about it,” Jim Sowell, an astronomer at the Georgia Institute of Technology, tells Popular Mechanics. “We would use the expression ‘heat’ to define the speeds of all the particles in a given volume.”

Yeah, that's a bit misleading. It's more like the average speed of all the particles. Some are zipping right along, while others are in the slow lane getting in other particles' way.

Coincidentally, the lowest temperature ever recorded in our solar system was clocked much closer to home. In 2009, scientists measured the depths of a dark crater on the surface of our moon and found that temperatures dropped to about 33 kelvin, according to New Scientist.

Just a bit colder than Nome in winter.

Well, that’s where things get tricky. Within near and distant galaxies, the mesh of dust and clouds that weaves between the stars has been observed at temperatures between between 10 and 20 kelvin. The sparse pockets of space that contain little but cosmic background radiation, leftover energy from the formation of the universe, hover in at around 2.7 kelvin.

That's where the 2.7K comes from, incidentally: it's the leftover radiation from the Horrendous Space Kablooie.

These temperatures dip perilously close to an elusive measurement: absolute zero. At absolute zero, which to -459.67 degrees Fahrenheit—no motion or heat is transferred between particles, even on the quantum level.

Now, I don't claim to be a quantum mechanic. Hell, I can't even change the oil in a neutron. But that wasn't my understanding; even at absolute zero, some quantum fluctuations occur. From Wikipedia: "Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion."

"Minimum" isn't the same as "none."

Back here on Earth, we have it easy. “You can have high-speed particles zipping by us outside the Earth’s atmosphere, but if you took off your space suit, you would feel cold because there aren't that many particles hitting you,” says Sowell.

"Cold" wouldn't be the only thing to worry about in that situation.

Were you to weave between galaxies in the vacuum of space without a spacesuit, the heat from your body—about 100 watts, according to Space.com—would start to radiate away from you because conduction and convection don't work in space. This would be a slow, frigid way to go, and, eventually, you'd freeze to death. But ... it's likely you'd asphyxiate first.

You know, I'm usually the first one to get annoyed at unscientific scenes in SF movies and shows. Like when a spaceship zooms by and you hear it; you wouldn't, really, because there aren't enough particles to transfer the sound. Or when they portray zero-g doinking; that wouldn't be nearly as much fun, or as easy, as you might think. But one thing they usually get close to right: you can, hypothetically, survive limited exposure to vacuum. I've heard "thirty seconds" bandied about, though I don't think anyone's had the balls to test that. But you can probably hold your breath for more than 30 seconds, so it's not the lack of air. Nor do you get cold enough fast enough for a 30 second exposure to vacuum to kill you. You won't enjoy the experience, sure, but those shows where someone leaps out an airlock and makes it to the next ship over in a few seconds? Sure, probably. The biggest thing you'd have to worry about, I think, is the pressure differential; if you take a deep breath first, your lungs can explode. And the jury's still out on whether your eyeballs would freeze before they exploded, or vice-versa.

That's why the common method of execution on a fictional starship is inhumane. You throw someone out an airlock, and they get to flail around in near-total vacuum for maybe up to a minute before they pass out and/or explode and/or freeze and/or boil (at low pressures, you can freeze and boil at the same time, fun fact). Hell, even beheading is more humane than that; the severed head might stay conscious for "only" up to about ten seconds.

No, if I were running the judicial system on a starship, and we had the death penalty, I'd take advantage of there being airtight compartments on the thing, and pump in some knockout gas before evacuating the chamber. Painless, quick, not cold.

Why am I thinking about these things? Well, science fiction writers have to. I may not write much fiction these days, but when I do, I want to get the science (mostly) right.

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