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 2 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

about 4 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.

December 4, 2022 at 12:02am

#1041364

Revisited: "Aren't you a little short for a stormtrooper?"

In my ongoing quest to find random old blog entries so I can express how I might have changed over time (among other reasons), I hit on this one from almost exactly 15 years ago: "Aren't you a little short for a stormtrooper?"

The entry contains two links and, remarkably, the first one hasn't rotted away. For your amusement, in case you can't be arsed to visit the original post, I present it as an xlink here.

I'm pretty sure those are the original images from over 15 years ago.

So, what's changed? Well, back then, I wrote, by way of introducing the link:

These days, it's hard to find just one, solitary thing that you can point to and say, "There it is. There is the Sign that the end is near; that we've reached the Cosmic Cul-de-sac and some asshole's U-Haul is blocking the turnaround." But here's a candidate

And oh, my, I bet I thought I was clever as hell when I wrote that. The photos are now mildly amusing to me and in no way a sign of the impending apocalypse. We've been through so much in the last 15 years that I guess the needle moved further in the direction of Armageddon, making these pictures of a couple of young people doing stuff in Stormtrooper helmets a refreshing chunk of normal. And maybe even slightly day-brightening.

Worse, though: Okay, so I spent my career designing, among other things, subdivisions, and most of those subdivisions included roads with a cul-de-sac. Not once in the 20 or so years I spent doing that did I ever bother to find out what the English translation of cul-de-sac is. It's "ass of the bag." Okay, no, not really, but 'cul' is, from what I hear, slang for ass (buttocks, not animal); the more polite translation would be "bottom of the bag." The point being that these days, I wouldn't have juxtaposed it with the English word "asshole" meaning "unbearable person."

Worser and worser, it turns out that the French translation of "cul-de-sac" is "voie sans issue," probably because 'cul' has that slang meaning.

Okay. Anyway.

The second link in the original entry has, unfortunately, expired, surviving only in a couple of lines I quoted back then. Here they are again:

A man who called himself "Papa Pilgrim" and took his family far from civilization to raise them according to his interpretation of the Bible was sentenced to 14 years in prison for sexually assaulting a daughter.

...

Hale insisted that he had a perfect spiritual understanding, his wife, Kurina Rose Hale, testified Monday.

Which was fucked up enough that I had to do a search on this fine, upstanding gentleman, and I almost wish I hadn't, but since I did, here it is.

I won't paste stuff from that link here. It's dark as shit. It details charges against a man who tortured and abused his family on religious principles. I'm not too worried about that link going invalid at some point, because this good Christian man has his own Wikipedia page too—according to which, fortunately for everyone involved, this outstanding specimen of humanity croaked just a few months later.

Back then, I discussed at moderate length how there are decent people and horrible people in every religion, and ended with "judge by the deed, not by the creed."

Today, I'd add:

There are none so evil as those who proclaim themselves righteous.

Not even Stormtroopers.

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