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 13 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 15 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').

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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 27, 2020 at 12:08am

#984397

Amusing Musical Musings

There's a line in one of my favorite webcomics. I can't find it in a text search, but it went something like, "My life is so depressing I don't know whether I was born or sung into existence by Morrissey."

PROMPT May 27th

If your life were a song, 1) what genre would it be, 2) who would sing it, and 3) would it be a hit?

Well, my life isn't that depressing. I mean, the only thing more depressing than a Morrissey song is a Leonard Cohen song, so that's a pretty low bar to clear. Cohen is, sadly, no longer with us, or I'd be tempted to pick him in response to the prompt. But no, I clear that bar with plenty of room to spare, even if, when looking at it from the outside, it might seem pretty sad. But I'm not usually sad, even if I'm not exactly brimming with ecstatic bubbles. No, I like my life too much to bring it to Morrissey.

As it is, I'd consider three scenarios: ideal, pessimistic, and nightmare.

Ideal scenario: 1) rock; 2) Springsteen and 3) Of course; Bruce deserves it even if I don't.

Pessimistic scenario, based on Just The Way My Luck Goes: 1) country; 2) William Shatner; and 3) of course not; it's William Shatner.

Incidentally, no, I don't hate country music, but you know how that genre goes: Mama stole my pickup truck, got drunk and drove in front of a lonesome train. I mean, no, none of that happened, but that's never stopped country singers. And of course Shatner is one of the worst singers in existence.

Nightmare scenario: 1) jazz; 2) anyone, out of key (okay, you got me: I can't name any jazz singers off the top of my head) and 3) Why start having jazz hits now?

It would also be wrong to say I hate jazz. The genre is so diverse that it's impossible to hate all of it. But I hate most of it. Whatever the next note should be, they always hit the exact opposite one. It annoys me very quickly and makes me want to smoke the devil's lettuce, just like they warned it would.

You know how they say when you play rock backwards, you sell your soul to the devil? I've also heard the joke that if you play country music backwards, you get back your dog and your pickup and you sober up.

Well, what do you get when you play New Age music backwards?

New Age music.

What do you get when you play jazz backwards?

Music.

Seriously, though, my life is nowhere near interesting enough to write a song about. I mean, I've been through a couple of divorces, suitable material for any genre of music, but I ended up with the house and car both times. No one's interested in a happy divorce story unless it involves some lady going to a foreign country to find herself in the arms of some tall exotic dude with a deep-voiced accent and a fifteen-inch schlong. I mean, really, how many movies have been made about that crap? I call it Divorce Porn, because it does for lonely, middle-aged housewives the same thing that porn does for perverts. And no one wants a story or song about a dude doing something similar, schlong or not.

Still, I'd bet Springsteen would find an angle or two in my life to write about, like maybe the times I made holy pilgrimages to Asbury Park, or drove my car fast on an empty road.

It'd never actually be a hit, though.

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