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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🌝 HuntersMoon

Sep 20, 2025 at 12:12am
In response to "Numbers End"

Boy, am I behind thee times. I thought you just put this on hiatus. Of course, in March I wasn't a complete devotee of your brilliance yet.

I'm more than happy with Carrion Luggage, however.

1 Like

Elisa, Snowman Stik

May 1, 2025 at 7:09pm
In response to "Shhhhhhh"

Anytime I hear/read about the Quiet Zone, I am always reminded of this video.

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:16pm
In response to "Penultimatum"

As my Grandmother used to say, "Everything changes except that which stays the same." I never did understand why she said it, but it sounds good.

1 Like

TJ says, "keep on keeping on!"

Mar 12, 2025 at 8:13pm
In response to "Numbers End"

Looking forward to the new book and hoping you don't encounter a meteor.

1 Like

Lazy Writer est 4/24/2008

Mar 12, 2025 at 4:19pm
In response to "Numbers End"

Sorry this one is closing..I will look for the new one tomorrow! Looking forward to it.

2 Like

Aiva Raine

Mar 12, 2025 at 3:45pm
In response to "Numbers End"

Thanks for a great 3000 entries.

Looking forward to the new blog, whatever it might be.

2 Like

Wordsmitty ✍️

Mar 12, 2025 at 1:36pm
In response to "Numbers End"

OK. What have you done with the real Robert Waltz?

Also, for the scientist engineer in me, have you actually tried to make entry 3001 and been rejected. Until there's proof, it's not real.

Thank you for your indulgence and sharing of wisdom.

R.I.P. "Complex Numbers"

3 Like

Anni Pon

Mar 12, 2025 at 11:20am
In response to "Numbers End"

Sorry that this one is coming to an end, but I'm excited to see what you have in store for the new one!

2 Like

Wordsmitty ✍️

Mar 11, 2025 at 1:42pm
In response to "Penultimatum"

It's been an enlightening run. We wish you well in your next adventure.

It will be interesting to see if after the next entry that WDC actually does stop you from adding another.

We're all waiting on pens and pencils for the unveiling of the new blog and to discover what it will be called.

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BIG BAD WOLF is Merry

Mar 11, 2025 at 10:46am
In response to "Penultimatum"

I'm reminded of this scene from The Princess Bride.

(Vizzini has just cut the rope The Dread Pirate Roberts is climbing up)
Vizzini: HE DIDN'T FALL? INCONCEIVABLE.
Inigo Montoya: You keep using that word. I do not think it means what you think it means.

3 Like

November 10, 2007 at 7:53pm

#548273

B minus 1 Day!

At this time tomorrow, I will be at Verizon Center in DC, most likely waiting for Bruce to take the stage (everyone takes the stage late these days, including Bruce).

This is getting to be a familiar feeling for me. After the Richmond concert and the Philly Vote for Change set, Bruce came out with a new album, Devils and Dust. It was a solo album, pretty much just Bruce and a few instruments, a departure from the E Street Band sound to something more spare and raw.

He'd done this before, of course: Nebraska and The Ghost of Tom Joad. The former was a demo tape recorded in his basement, and was meant to be the beginnings of another Band album, but apparently the producers decided it sounded just fine like it was and released it with minimal edits. The latter was an album of social commentary released after the popular flops of his early 90s collections, and (in my opinion) it suffered from shitty engineering and lousy production: on the title track, for instance, you can't hear anything until the harmonica part, and then the harmonica splits your ears open.

So Devils and Dust got past all that, and moreover deals with more varied themes. The concerts for that album were intended to be run at smaller venues, for a more intimate, almost nightclub kind of feel. Of course, we're still dealing with Bruce here, so it's the difference between playing 50,000 seat arenas and 20,000 seat arenas.

I got tickets to the Fairfax show - managed to just get two, this time. And the atmosphere was totally different: darker, almost like the set of Phantom of the Opera. With just Bruce on stage, there was no interaction with the rest of the band: just Bruce and his audience.

And he ended the set with something odd: a cover of a song by an obscure New York punk group called Suicide, a haunting melody called, "Dream Baby Dream." He started out on the Hammond organ, but partway through the song got up while the instrument played itself, and walked slowly off the stage singing.

For some reason, that image will always stick with me.

Especially since I got to see it again, when they announced more tour dates and we ended up back in Richmond for another show.

But then, something different: eschewing the E Street Band, solo work AND original material, Bruce produced an entire album of traditional folk songs, in the spirit of Pete Seeger. And it was the polar opposite of D&D as he took the stage with over a dozen sessions band musicians - with only two from E Street, one of them his wife.

Saw that concert in Northern Virginia, and wrote about it in my previous blog. Here's a link: "It Came from New Jersey"

(someone challenged me to write a blog entry in verse...)

We won't have awesome seats tomorrow like we did then - but those weren't really "seats," more like "stands," and by the time the concert was over all either of us wanted to do was sit the hell down. Still, I wouldn't have traded that experience for anything.

Well I sought gold and diamond rings
My own drug to ease the pain that living brings
Walked from the mountain to the valley floor
Searching for my beautiful reward
Searching for my beautiful reward

From a house on a hill a sacred light shines
I walk through these rooms but none of them are mine
Down empty hallways I went from door to door
Searching for my beautiful reward
Searching for my beautiful reward

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