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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Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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.

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

January 4, 2019 at 12:37am

#948837

Prognostication

Make a prediction for how technology will advance in the next 18 years.

Science fiction is the most important genre.

I don't mean space opera, though those can be fun. I mean actual science fiction. It helps us come to terms with both technological advances and social change.

A while back, I came across this bit:

https://www.bbc.com/news/technology-46736024

That wasn't the link I first found, though it describes it well enough, I think. To summarize, back in 1983, Asimov gazed into his crystal ball and came up with what 2019 might look like.

Being a science fiction writer and science communicator, his crystal ball was probably sharper than most. Hell, I distinctly remember thinking in 1983 that we'd have generation ships headed for Alpha Centauri or some other "nearby" system by now. And don't get me started again on flying cars. But Asimov got a surprising number of things right.

What's even more interesting than what they get right, in my opinion, was what prognosticators get wrong. For example, when I was a kid, a staple trope in science fiction was the video call. The way this was presented was invariably that it would be just like a land-line telephone, only with a video link included. As I recall, some folks actually tried to make that a reality, but the general public, who don't bother to read science fiction, could only think about how that would invade privacy, and it never went anywhere.

So instead, we developed video phone technology from the back end: through computers. By the time computers got powerful enough for things like skype, though, many of them didn't use phone lines. In fact, there wasn't the whole "dialing" thing at all - just a protocol that enabled voice/video transmission and receiving.

What's even more interesting now is that we all carry phones with us wherever we go, and we even call them phones (in the US anyway; the UK calls them "mobiles," which is a far better name), but we hardly ever make actual phone calls with them. It's all texting or email.

Incidentally, the video phone trope was far more prevalent in shows than in written fiction. This was mostly an easy way to view both actors' faces while they spoke remotely. Still happens, only there are a lot more holograms now. In the shows, I mean, not in reality.

Which leads to the question: how much does science fiction predict the future, and how much does it inspire it? But that's a whole 'nother blog post. Or series thereof. Not going there tonight.

So. To address the question. As most of you know, I follow science journalism sometimes, so I have some idea of the things they're working on: quantum computers, virtual reality, universal internet access, solar energy, stable controlled fusion power (which is about 20 years away, and has been 20 years away for the last half a century), gene editing, Mars colonies, nanotechnology, and more.

I'd like to think that some of these things will become reality 18 years from now. Proably not fusion - that will be 20 years away then. But maybe the others, along with autonomous vehicles and maybe even death rays.

But that's optimistic.

After all, we still have nukes. And it's entirely possible that, 18 years from now, those few of you who remain might rediscover how to carve a wheel out of stone.

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