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.

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September 23, 2022 at 12:03am

#1038019

Dark Sun

Now that autumn is here (dammit), it seems appropriate that this article about fusion came up tonight. The equinox, which occurred yesterday, marks the point where the sun appears to cross the Earth's equator, and of course the sun is powered by fusion.

Yeah, yeah, it's a stretch, I know.

The frontrunners in the trillion-dollar race for limitless fusion power
Nuclear, believe it or not, is booming again. And with a serious pile of private and public funding behind them, these companies say they’re even getting closer to fusion.

I see even Fast Company isn't above making puns in its headlines. Nuclear? Booming?

Anyway, as the article points out, Earthbound fusion reactors seem to be just a few years away—just as they've been for at least my entire life.

With energy prices on the rise, along with demands for energy independence and an urgent need for carbon-free power, plans to walk away from nuclear energy are now being revised in Japan, South Korea, and even Germany. Last month, Europe announced green bonds for nuclear, and the U.S., thanks to the Inflation Reduction Act, will soon devote millions to new nuclear designs, incentives for nuclear production and domestic uranium mining, and, after years of paucity in funding, cash for fusion.

It's important to note that while fission and fusion are both technically "nuclear power," there's a huge difference; fusion (at least theoretically) isn't anywhere close to as dangerous, despite the incredible temperatures involved.

You’ve likely heard this one before. The running joke is that economically harnessing fusion power, which is what a star or hydrogen bomb does, is about 30 years away. What’s not a joke is that we have about zero years to stop powering our civilization with earth-warming energy.

Thirty years? I've been hearing "twenty" for at least fifty.

One milestone came quietly this month, when a team of researchers at the National Ignition Facility at Lawrence Livermore National Lab in California announced that an experiment last year had yielded over 1.3 megajoules (MJ) of energy, setting a new world record for energy yield for a nuclear fusion experiment. The experiment also achieved scientific ignition for the first time in history: after applying enough heat using an arsenal of lasers, the plasma became self-heating.

There's a very good brewery right across the street from that lab. I'm pretty sure I drank with some of the people working there.

And you’d be forgiven for missing another milestone in July, when the Energy Dept. announced awards of between $50,000 and $500,000, to ten fusion companies working on projects with universities and national labs. Here are a few of the awardees, who include some of the industry’s leading companies, and whose projects offer a sampling of the opportunities—and hard problems—in fusion.

Those awards seem lame in comparison to the total investment in fusion research, but I'm sure every little bit helps.

The rest of the article talks about said awardees (understandable, considering the source), and it's all very interesting, but I want to point out a more philosophical angle on fusion.

Pretty much every source of energy we have right now is, when it comes right down to it, solar.

Hydroelectric systems work by harnessing the energy of surface runoff, which got to where it was because the sun evaporates water, which turns into rain, which turns into runoff. Wind is largely driven by heat differences; the heat source is the sun. Fossil fuels such as coal or natural gas were the result of living things, which also ultimately draw their power from the sun. Solar is, of course, direct from the accursed daystar. And while fission isn't "solar" per se, the fissile material (uranium or whatever) was forged in the incomprehensible power of a supernova.

Fusion, at least in principle, doesn't rely on the sun. Oh, sure, the daystar is composed primarily of hydrogen, but most hydrogen was created in the Big Bang itself. (Though I should note that the article points out that tritium, which is actually a hydrogen atom with two neutrons instead of zero, is derived from lithium, which wasn't a huge proportion of the primordial elements.)

In any case, fusion is kind of the Holy Grail of power. While I can't speak to the monetary costs involved (the article does that to some extent), it doesn't seem to directly cause global warming the way fossil fuels do, or result in radioactive waste like fission reactors, or fuck up fish habitats like hydroelectric, or take over a landscape like wind farms.

I'm sure it has its drawbacks, too. Water use, maybe? I don't know. The only really clean alternative would be to not use electricity at all, and that ain't gonna happen.

At least until the inevitable end of civilization.

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