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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about an hour 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

about 3 hours ago
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.

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.

May 24, 2023 at 7:52am

#1050033

Machine Learning

In which The New Yorker discovers science fiction.

What a Sixty-Five-Year-Old Book Teaches Us About A.I.
Rereading an oddly resonant—and prescient—consideration of how computation affects learning.

Pick a random selection of science fiction from the past—I don't mean space opera, which is fine in its own right, but actual attempts to write about the intersection of society and advancing technology—and you'll find that some of them are, in hindsight, "oddly... prescient," while the vast majority are like "well, that never happened."

Neural networks have become shockingly good at generating natural-sounding text, on almost any subject. If I were a student, I’d be thrilled—let a chatbot write that five-page paper on Hamlet’s indecision!—but if I were a teacher I’d have mixed feelings.

If I were a teacher, I'd be like, "hey [chatbot], write a syllabus for a sophomore English class aimed at mid-range students."

On the one hand, the quality of student essays is about to go through the roof.

Yeah, except, well, see JayNaNoOhNo's rant about that sort of thing, here: "AI Detectors are Horseshit"

Luckily for us, thoughtful people long ago anticipated the rise of artificial intelligence and wrestled with some of the thornier issues.

Substitute pretty much any technological advancement for "artificial intelligence," and "thoughtful people long ago" anticipated it and thought about some of the possible consequences. That's basically the definition of science fiction.

But I don't expect anyone at TNY to understand that.

Their book—the third in what was eventually a fifteen-part series—is “Danny Dunn and the Homework Machine.”

Just including this bit so we all can see what they're talking about.

There follows several paragraphs of meanderings about the setup, plot, and characterization, before getting to the stuff that is really relevant to the topic. Typical TNY. Still, it's worth glancing over so you know what he's talking about later.

I hesitate to give away too much of the plot, but (spoiler alert!) two mean boys in their class, one of whom is jealous of Irene’s interest in Danny, watch them through a window and tattle to Miss Arnold.

Oh, no, wouldn't want to spoil a plotline from 65 years ago. (Spoiler alert: the author pretty much gives away too much of the plot.)

Incidentally, Rosebud was the sled.

He points out that Danny, in order to program Minny to do his homework, had to do the equivalent of even more homework, much of it quite advanced. (“Gosh, it—it somehow doesn’t seem fair,” Danny says.)

I must have read these books as a kid. I have a vague memory of them, anyway. But it may explain why, whenever I have to do anything more than once, I search for a way to automate it, and often spend more time crafting an Excel spreadsheet or some code than I would have spent on the projects.

“Danny Dunn and the Homework Machine” is ostensibly about computers, but it also makes an argument about homework.

And yet, there is still homework. I'm of the considered opinion that grade-school level homework has the primary purpose of making the kids leave their parents alone for a few precious minutes in the evenings.

The article spends an inordinate (to me) amount of time arguing about homework in general and not on the ethical implications of AI, but the main point remains: people have been discussing this sort of thing since long before it was technologically feasible.

Just like with the rest of science fiction.

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