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:

🌝 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.

1 Like

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 11, 2007 at 10:19am

#548372

The Day

And so we're finally here.

In a rare example of good timing, Kirstin and I finished watching The Sopranos last night, ending the last episode at midnight. Then I went to the DVD's special features and clicked on the "music" link, because, well, I like music and I thought the Sopranos soundtrack was excellent.

Fortunately, one of the guys talking about it was cast member and E Street Band guitarist Steve Van Zandt. Now, I always liked Miami Steve (or Little Steven, or just plain Steve) - his guitar work was always spot on, and the interaction between him and Bruce on stage is part of the dynamics that make Bruce concerts great. Now, in the early 80s, Steve left the Band - amicably, by all reports - to pursue a solo career; he was "replaced" by Nils Lofgren for the BitUSA tour (now, they're both in the band). He did solo work during that time, as well as working with other musicians.

He's also a radio show host, and is program director for some satellite radio channels, and has been involved in social and political activism throughout his career.

But in the Sopranos interview, he said something to the effect of, "No one comes up to me and says, 'I'm a fan.' 90 percent - no, 100 percent of the time, they come up to me and say, 'Hey, I really like your work; I'm a big fan of Bruce.'" He didn't say it with anger or regret, just kind of matter-of-fact, but it made me think - especially since I would have been one of those people, had I ever met him.

I was thinking, what must it be like, to always live in the shadow of someone else, like that? And it's not just him - Clarence Clemons, who is probably more associated with E Street Band than anyone else, has his own solo work out. I have one of his albums, and it's pretty amazing stuff. Roy Bittan, the band's keyboard player - well, Bittan has played with some of my favorite artists, including Bob Seger, Dire Straits and Meat Loaf, and is the only E Streeter to work with Bruce when there was no E Street Band; Roy seems to be one of those guys who's always in the background, though he's probably the best rock and roll keyboardist ever (with the possible exception of Billy Joel, though it's a stretch to call most of Joel's music "rock," however great it is). And surely you know - hell, even I know this and I don't watch TV - about Max Weinberg's work with Conan O'Brien on late night television. Let's not forget to mention Bruce's wife, Patti Scialfa, who has her own solo career (how she finds time for one is beyond me). These people, and the rest of the Band - Garry Tallent, Nils Lofgren, Soozie Tyrell, Danny Federici - are all accomplished musicians in their own right, and yet they live - perhaps even choose to live - in the shadow of Bruce.

I don't know; they're just people, like you and me, like everyone, and they have their own talents, their own heartbreaks, their own joys, triumphs, losses and insecurites. As much as we like to put celebrities and sports figures up on pedestals, that's something I always try to remember. I can joke around about Bruce being God, but I know it's a joke. What it comes down to, what's really important, is quite simple: I like the music. That's all. I have no musical talent myself, though I have had some training - enough to be impressed.

Life can change quickly. I have few regrets, and mostly they involve music: that I never got a good recording of Boyd Tinsley's pre-Dave Matthews Band work; that I never saw Pink Floyd live, back when they were all getting along; that I never saw the original Led Zeppelin in concert; and that I never had any musical talent of my own. Part of the spice of life is that it may change suddenly, so you have to appreciate what you have when you have it, seize opportunities when they come your way, and, mostly, make choices that you don't regret later. You make your own music, and you put it down on vinyl or plastic, and then it's frozen for all time, warts and all, except when sometimes you can drag it out and play it for a few thousand people. For some of us, those opportunities come only rarely; for others, it's a living. And if you can look back at your body of work and say, "I touched a few lives for the better," then you've achieved a kind of immortality.

As Neil Young pointed out, rock and roll will never die.

I want a thousand guitars
I want pounding drums
I want a million different voices speaking in tongues

This is radio nowhere, is there anybody alive out there?
This is radio nowhere, is there anybody alive out there?
Is there anybody alive out there?

I was driving through the misty rain
Searchin' for a mystery train
Boppin' through the wild blue
Tryin' to make a connection to you

This is radio nowhere, is there anybody alive out there?
This is radio nowhere, is there anybody alive out there?
Is there anybody alive out there?

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