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 10 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 12 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 2, 2022 at 12:01am

#1031786

Charlottesville

Some of the entries this month will be from "Invalid Item"

. This is one of them.

Prompt # 1 May, 1st. '22 — A bit about the history of your town/area.

Every place has a history. Some towns were founded to support gold rushes, others were close to oceans.

What's the History of where you live? What was the early industry that made your town the place to be? Who founded your town and when?

Before the Europeans invaded, it was the Monacan people who lived here. As I understand it, the Monacan claimed most of the Piedmont region as their territory, and they didn't get along with the Powhatan Confederacy in the Tidewater area of Virginia, who are much more famous thanks to Disney movies.

So it probably didn't take much for the Powhatan to team up with the English to venture into enemy territory and start setting up shop here. Once that was all settled and the Monacan had dispersed -- mostly being absorbed into other tribes, which bought them a few years at least -- the English were perfectly content to backstab their erstwhile allies.

Meanwhile, though, they'd established trade routes across the Blue Ridge into the Shenandoah Valley, which had mostly been settled from the north, by people of German descent (think Pennsylvania "Dutch.")

It was one one of these trade routes (called Three Notch'd Road) that, at some point, Europeans started living around here. A lot of that early history is quite vague; my personal feeling is that someone set up a brewpub, and after a long day's travel from Richmond (a trip that now takes about an hour), it must have been very inviting to stop in, have a few, pass out, and then continue on across the mountain pass the following day. But I may be biased.

I should point out here that the Wikipedia page for Charlottesville includes a whole section devoted to our current breweries. No, I didn't edit that part in. I was just pleased to see one of our greatest features so prominently displayed on the webpage.

The thing about Charlottesville's history, though, is that we don't really have any other excuse. There are no significant natural resources around (apart from good agricultural and timber land, which isn't exactly sparse in Virginia); no navigable waterway extends this far up the drainage basin (though the James River is partially navigable, it's about 30 miles from here); and it took a while for any real industry to develop in the area.

So... we don't really have a founder, not like other cities. The "Charlotte" in the Ville's name was King George III's queen. You may remember King George III as the guy who got his ass handed to him by a bunch of Indian-trained colonists, and who had a bit part in Hamilton. I don't think she ever set foot around here. So the town's not named after its founder. No, in 1762, before the Hand-George's-Ass-To-Him War, the city was chartered to serve as the county seat of Albemarle -- which, of course, implies that there was already a fairly significant settlement here, and probably more than one brewpub. (It's still technically the county seat, even though it's a separate political entity. Don't ask; it's a Virginia thing.)

Therefore, the closest thing we have to a founder is Thomas Jefferson, who you might also remember from Hamilton.

Among the many other hats he wore, Jefferson was a land surveyor. As part of my job, I used to have to dig through City and county records to determine the history of a piece of land my clients were destroyingdeveloping. Inevitably, if I searched back far enough, I'd find surveys signed in Jefferson's distinctive fuck-you-George handwriting.

Problem is, some of those surveys were dated during the time Jefferson was in France doinking French women. My personal theory is he had a slave who did all the work and then forged his signature. Typical.

Really, while the town has a really flimsy excuse for existing, it was Jefferson who gave it a purpose by building the University of Virginia here. While there was certainly more traditional "industry" around -- the Woolen Mills come to mind -- Charlottesville got its boost as a college town. UVA is still the biggest game in town, though technically most of it isn't in the City.

The next most important historical event was when I moved here in 1983 from Powhatan territory to attend the university. And I do what I can to keep the local brewpubs in business.

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