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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 24 hours 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').

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.

December 25, 2019 at 12:13am

#972005

Here's Your Sign

Merry Christmas to those who celebrate it. As for me, well, a few days ago Drunk Me got into my Amazon account and ordered the final book in a fantasy series I've been following (Brent Weeks, if you're curious, or even if you're not), so I'll be spending the day reading the massive tome. Thanks for the holiday gift, Drunk Me! You shouldn't have!

Nah, I'd have bought it sober eventually.

But then, seeing articles like this one kinda makes me want to drink:

https://www.newyorker.com/magazine/2019/10/28/astrology-in-the-age-of-uncertaint...

Astrology in the Age of Uncertainty

Millennials who see no contradiction between using astrology and believing in science are fuelling a resurgence of the practice.

Fuck your stupid generation demographical nonsense again, but still, the article has some interesting stuff in it.

“That is one of my favorite things, as a Leo and as a person—building community,” she said. It was a little before eight-thirty, and some of the fifty-two participants—who had paid between $19.99 and $39.99 each—were typing hellos; one woman, in Europe, had set her alarm for 2:30 A.M., to log in.

You know, I can't even work up outrage over people making money from astrology. Gotta keep the economy going somehow, and I don't know many astrologers with private jets (likely, the planets never told them when would be the optimal time to buy one).

In its penetration into our shared lexicon, astrology is a little like psychoanalysis once was. At mid-century, you might have heard talk of id, ego, or superego at a party; now it’s common to hear someone explain herself by way of sun, moon, and rising signs.

Psychoanalysis has had - probably still has - its problems, to be sure. Freud is all but dethroned as anything except the guy who started it all, and many closely-held beliefs of early psychology have been turned on their head - "id, ego or superego" among them. But you know, that's how science works: Make a theory. Test the theory. Revise or replace the theory. (That's an oversimplification of the scientific method, but bear with me here.) Just as medicine moved away from balancing the humors and whatnot, psychology advances, in fits and starts, as more studies are done and more data collected. They learn from their failures - most times.

There is no science to astrology. I mean, okay, there's math, but as this article points out, that's all done by computer programs now; I'd wager few of these younger astrologers could compute charts by hand as it used to be done. But computing the chart is only the first step; next up is interpreting it. And there's no science to that; only the accumulated guesswork of generations of astrologers.

Sure, sometimes it seems to be right, even scary right. There's a lot of stuff at work there, not the least of which is confirmation bias.

Back when I was genuinely interested in the practice, a friend of mine who's an accomplished astrologer wanted to do my natal chart. Thing is, I have no idea what time I was born (as the article points out, this is essential). I know the day, and the place, but not the time, and obviously over the course of the 24 hour day any one of the 12 star signs could be "rising." And supposedly the rising sign (the zodiac sign subtended by the eastern horizon) is third in importance only to sun sign and moon sign - which, in my case, are the same thing.

So. If there's any science to it at all, we ought to be able to work backwards. I had a pretty good idea of my personality and, being my friend, so did he. So, in theory, if you subtract the influence of sun-in-Aquarius and moon-in-Aquarius, the dominant aspect that remains would be my rising sign, et voilà, I'd know to within a couple of hours the time I was born.

It shouldn't come as a surprise that every one of the 12 possible rising signs had near-fits to my actual personality.

That's the thing about us Aquarians: we're highly skeptical of astrology.

Anyway, I'm not really feeling like quoting more from the article. I've never been a fan of The New Yorker's preferred highbrow, rambling style that usually circles the point like a dog about to take a nap - if it ever gets to the point in the first place. This doesn't, really; it just goes on and on about how different people approach astrology.

But, as unscientific as it is, I just can't work up to the point where I get mad about it. Energy crystals? Hogwash. Homeopathy? Dangerous hogwash. UFOlogy? Balderdash. Prosperity gospel? Fraudulent balderdash.

Can it be dangerous? Sure. As the article points out, that utter asshat Reagan was a fan; imagine consulting an astrologer to decide when would be the optimum time to launch nukes. But that's an edge case; most of us don't have access to the national weaponry. The way I see it, in the world we live in, we're overwhelmed with choices, and it can be paralyzing. Everything from which laundry detergent to buy to which restaurant to visit is fraught with possibilities that even our recent ancestors would have boggled at. If some people want to turn to the skies and those who interpret them to help them make such choices? I get it. Sometimes I roll the dice, like I do when I decide which of my many waiting articles to post here; that's as good as astrology.

I just wish more people actually looked at the stars. Maybe that'd get 'em out of the gutter.

In parting, since it's Christmas, enjoy my favorite holiday song:

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