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!

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

a day 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').

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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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 31, 2019 at 12:20am

#959930

The Magic that Actually Works

http://nautil.us/issue/69/patterns/the-math-that-takes-newton-into-the-quantum-w...

The mysterious symbols seemed like magic spells. And in a way, they are. Science is the magic that actually works.

Why, you may ask, the hell am I linking this here, on a site about writing (not math)?

Or you might note that the whole theme of this blog is centered around a math thing. It's not, really; but I'll get into that some other time.

It doesn't matter if you don't understand what this author is on about. Hell, it's right on the edge of my own understanding, and I think I have something of a background in this sort of thing, having gotten a degree in a math-adjacent field.

No, the reason I'm sharing this is this excerpt:

I have to think that I have some kind of inside track on this problem. That is, I have some sort of talent that the other guys aren’t using, or some way of looking, and they are being foolish not to notice this wonderful way to look at it. I have to think I have a little better chance than the other guys, for some reason. I know in my heart that it is likely that the reason is false, and likely the particular attitude I’m taking with it was thought of by others. I don’t care; I fool myself into thinking I have an extra chance.
-Richard Feynman

I noted in a recent entry my admiration for Feynman's towering intellect, asserting that I think he was even smarter than Einstein. Admittedly, that's arguable - once you get to that level of smart, IQ tests don't mean shit anymore (if they ever did) and it becomes a matter of opinion. Still, the point is, he was unarguably a genius, and this is a bit of a window into his thought process.

It has to do with making connections.

This isn't limited to math and physics, either. And it's not necessarily going to lead to world-shattering conclusions. I don't do physics or math (though I enjoy reading about those things); what I do is write, and the above quote can be applied to writing.

The best stories, I think, are the ones that look at things just a little bit differently - the way Feynman describes approaching physics problems. This is certainly true for nonfiction; a reporter is always trying to find an intriguing angle for a story, and a biographer seeks to make his or her subject relatable somehow, rather than simply listing facts about the person's life. But it's also true, in a different sense, for fiction.

When I write a story, it's never enough for me to have one idea. Oh, I've done it, but I've never been as satisfied with the results as I am when I'm combining two or more ideas, relating them. The article I linked reminded me of that. Sometimes an additional perspective is necessary, and when you can do that, magic happens: you gain a deeper understanding of your subject, and hopefully can pass that along to the reader.

And don't all of us writers like to think that we have that unique perspective, that we have that "inside track," that we can say something that, maybe, no one else has, at least not in that particular way?

Hell, I'm practicing this just by converting a quote about physics into something more general about what we create.

Anyway, it also struck me that the guy who wrote the article is Joan Baez's cousin, so even if you skipped all the sciencey and mathy stuff, here's a tune by Joan:

4 Like • Comment • View 2 comments

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