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 11 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 13 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 17, 2022 at 12:01am

#1032516

Industrious

"Invalid Item"

The industry is/was your town/area noted for?

Brains.

Also, tourism.

Those two industries are incompatible.

As I noted a couple of weeks ago in "Charlottesville"

, this town probably started as a rest area. Not a destination, but a place to spend the night and drink while on the way from an actual place to another actual place. Tourists still come and go.

As for "brains," I'm making a pun. A lame pun, but still a pun.

These days, people only care about college for two things: 1) job preparation and 2) sportsball. But in the good old days, a college -- or, in this case, a university, the distinction between which is still a bit hazy to me -- was a place to expand one's mind. After UVA was founded in the early 1800s, Charlottesville became more of a destination. Now, instead of tourists coming and going, students came for four or more years (or in the case of Poe, one semester) before leaving.

Me, I came and stayed. I just like it here.

Now, I'm not saying university makes a person smarter. More knowledgeable isn't the same thing as smarter. It's the difference between being able to win a trivia contest, and developing a Grand Unified Field Theory or whatever. But one of the original purposes of a university was to expose a person to a wider range of culture, knowledge and experience than they'd have gotten in one farm or town.

Times change, and thanks to faster transportation and the internet, you can get that bit anywhere. Not that either invention is without its downsides. Now, like I said, people seem to only do higher education for vocational training (I'm guilty of that myself) and sportsball. And maybe partying, which we were known for in the days before they foolishly raised the drinking age to 21.

In any case, as I noted before, UVA is the largest employer in town, so it's the "industry" that drives our economy. But it's not only the college; there's a hospital affiliated with it. As with any hospital, opinions vary on it, but I consider them world-class. (They certainly do charge world-class fees. Or, rather, America-class, because we have to pay for that shit ourselves). In particular, there are only three places I'd want to go for neurological problems: Mayo, Johns Hopkins, and UVA. One of these is two miles from my house.

It's where they flew Superman to when he broke his neck.

Hence, "brains."

As with most college towns, there are certainly other big businesses here as well, though we don't have much of the hot, sexy tech industry. Which suits me just fine; the last thing we need is a SpaceX launch facility or an Amazon whorehouse. There used to be a clothing factory here, and there are some defense-related employers as well, making us a target. And the usual assortment of retail outlets, including at least one that's unique to us. But it wouldn't be Charlottesville without UVA.

And purveyors of fine fermented and distilled beverages.

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