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

October 18, 2008 at 9:42am

#613460

Concert Review

When you're Elton John, and you and the band open the show (pretty close to on time, mind you, which even Bruce doesn't seem to be able to do) with "Love Lies Bleeding," with its extended, orchestral intro... you've got our attention.

The tickets, being free, weren't the best seats in the house, but we had little reason to complain: we were at stage right, all the way up in the nosebleeds, but we had a perfect, unobstructed view of the entire stage. We couldn't see the video screens, and had only a foreshortened, angular view of the backdrop screen - which meant I concentrated on the music, not the pretty lights.

I think my biggest hesitation with actually, you know, paying for Elton John tickets was that he'd play mostly newer stuff. I should have known better. The vast majority of the concert was songs from the 70s: Levon, Tiny Dancer, Madman Across the Water (one of my personal favorite songs of all time - originally about Richard Nixon, but just as appropriate today), Yellow Brick Road, Honky Cat, Philadelphia Freedom (a song that means exactly what you think it means), a 12-minute version of Rocket Man... basically, every one of the songs that made me like his music in the first place.

And yes, of course they played "Crocodile Rock" - with the audience doing the bridges.

Sadly, his voice isn't quite what it used to be - the higher notes that were signatures on many of his older songs elude him now, more due to the excesses of youth than age itself. But hey, he still sings better than I do (my CAT sings better than I do), and there is maybe one power on this Earth greater than Elton John on the piano - I will always give Billy Joel primacy there. And the DUDE can STILL ROCK.

For an encore, he started with his version of "Pinball Wizard" - a truly great rendition of the Who classic. And then, after dedicating the final song to "each and every one of you," he ended with the song that began his career: "Your Song."

All in all, the concert was one big tower of awesome. Not, mind you, as awesome as the Bruce show I saw in Richmond - no one can top that - but would have been worth every penny if I'd, you know, paid for it.

Oh, yes, and his actual middle name is "Hercules." Don't ask me why. That's what Wikipedia is for.

...But the years went by and the rock just died
Suzie went and left us for some foreign guy
Long nights crying by the record machine
dreaming of my Chevy and my old blue jeans
But they'll never kill the thrills we've got
burning up to the Crocodile Rock
Learning fast as the weeks went past
we really thought the Crocodile Rock would last...

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