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 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.

September 22, 2020 at 12:02am

#993899

Déjà Vu All Over Again

Ah, yes, the old "save game" scenario.

PROMPT September 22nd

If you could live the next 24 hours and then erase it and start over just once, what would you do?

Drugs.

Okay, no, probably not.

I'm going to make some assumptions here -- this is obviously an entirely speculative scenario, and there aren't enough details. Basically, I'm figuring it's like saving a video game, proceeding with the game, and then reloading the save. Or, perhaps, the Groundhog Day situation.

I would have memory of what happened. (Otherwise, what's the point?)

No one else would have any memory of what happened.

No consequences upon reload.

No objects come back upon reload (that is, no profit from, for example, robbing a bank).

Sometimes when I'm playing Skyrim, I do this sort of thing on purpose (usually the save point is just in case my character bites it). I'll save a game, then go into a town and pick fights with the guards, just to see if my character is strong enough to take on an entire town's worth of guards. Or to see how much I can get away with stealing from a shop, or finally taking out my frustrations on a particularly annoying but important NPC.

Of course, that's just a game, with no real-world consequences anyway (except that I've once again successfully procrastinated chores). Reset or not, I have no desire to commit mass murder (or even, you know, non-mass murder), or steal a bunch of stuff that I'm not going to get to keep anyway, or even beat anyone up. Has nothing to do with the penalties if caught, just don't wanna.

That leaves experiences. Ever have a place you wanted to explore, but there are all kinds of "do not enter" signs up? Like, for instance, Area 51. Now, that one in particular wouldn't do me much good because I'd get shot before I even got ten paces inside the fence, but you know... that sort of thing. A bell tower or something. They never let anyone up in those things, and I always like to see the view from up high. Wouldn't risk getting caught under normal circumstances, but with the reset thing? I might give it a shot.

There is, however, one thing I'd definitely do. Today's Tuesday. If I know I'm going to relive Tuesday, I'd pay really, really close attention to what the various stock markets are doing on Tuesday. Then, after the "reload," I'd place enormous bets, using margin, on the stocks I know are going to go way up, and short the ones I know are going to crash. Yep, I'd totally do that... but the problem is, the stock market follows chaos theory. That is, even minuscule differences in starting conditions can have outsized effects on events. So if I shorted a stock on Tuesday 2 that I didn't on Tuesday 1, that very well might be enough to change the course of trading that day.

But, you know. Probably not. My input would truly be minuscule, so it'd be worth taking the chance.

So you see? I don't even need to rob a bank to get rich. I just need to see the future.

11 Like • Comment • View 7 comments

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