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 5 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 7 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 29, 2023 at 7:44am

#1061544

I Don't Know, But Alaska

Ever read something just to remind yourself how good you've got it? For me, that's pretty much everything from Outside, especially this article:

The Joys of Cabin Living in Alaska
Want to know what domestic bliss looks like? A rundown cabin with no electricity on the edge of rain-soaked Alaskan wilderness.

Huh... I didn't know that "domestic bliss" was a synonym for "literal Hell," but in retrospect, it makes sense.

Article is from 2013... not enough time for climate change to significantly affect the subject matter or my feelings about it. Yet.

My two brothers and I, along with a buddy of ours... own a shack at a place called Saltery Cove on Southeast Alaska’s Prince of Wales Island.

Morbidly curious, I looked it up. It's almost as south as you can get and still be in Alaska. The northern boundary of British Columbia is quite a ways north.

The place is still way too north.

The shack is about 36 feet long and 12 feet wide, with the warped shape and discoloration of a cardboard shoe box that’s been soaked in the rain.

Luxury!

The appeal of our shack isn’t so much the structure itself, but rather the bare-bones nature of its locality. Surrounded largely by the Tongass National Forest, it’s a place where black bears gnaw mussels from the rocks in what might be described as our yard and killer whales pass by so close that you can hear them even with the door closed.

Okay, fine, that would be kinda cool to experience. For, like, a day or two.

But in truth that’s only half the answer. The other half is more difficult to explain and also a bit masochistic: Saltery Cove is a place where everything—the weather, the ocean, the mountains, the people, the trees, the animals, even the buildings—seems capable of kicking your ass in a very physical way. And in today’s increasingly tame and virtual world, where our primary sensations tend to be delivered by our Wi-Fi connections, a good old-fashioned ass kicking is something worth paying for.

Bullshit.

I mean, okay, you feel that way, fine. We gotta have differences of opinion, or else one spot would get too crowded. But to me, this is like saying, "I built myself a beautiful mansion, but I prefer the feeling of living in a drainage ditch."

I prefer the simple life. By which I mean, if I'm cold, I press a couple of buttons, and then I'm not cold. If I'm too hot, I press the same buttons in a different order. Hungry? Buttons. Thirsty? Okay, no button for that, but there's a lever. Dirty? Knobs for that. Entertainment? Back to the buttons. Couldn't possibly be simpler, at least not until we further refine AI. No, I don't yearn for the complicated lives of my ancestors; they worked their asses off so I could sit on mine all day, and I'm ever so grateful and appreciative of them for that.

Another way in which the cabin kicks my ass is through my wife, Katie. She often regards my purchase of the shack with that eye-rolling sense of dismissal that people will use when confronted with the subject of their spouse’s past girlfriends or boyfriends.

And yet she puts up with you. I've never really understood people, and articles like this only add to my confusion.

I went there with my brother Danny to fish salmon and halibut with one of Saltery Cove’s eight full-time residents, Ron Leighton, a man of mixed Native Alaskan and Irish descent who’ll tear your head off for tangling an anchor line and then send your kid a birthday present even though the nearest mailbox is an hour’s boat ride from his house. Ron’s résumé includes a tour of duty as a door gunner in Vietnam, a career as a detective with the police force in Ketchikan, Alaska, and a parallel career as a halibut long-liner.

Gotta admit, though, that's the kind of person whose life seems tailor-made for writers.

The article goes on to describe why it's such a wonderful idea to buy a cabin on the coast of nowhere, sight-unseen: in summary, the place was utterly trashed and they spent months cleaning up.

Oh, well. It works for them. More space in semi-civilized areas for me, then.

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