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

January 23, 2007 at 4:17pm

#483170

Well, THAT was fun.

Looks like a lot of people have strong feelings about Hillary Clinton. Okay, I can understand that, I suppose. Of all the folks who would be King, she's probably best known. Of course, yesterday's blog was really meant to be more about "the media" than about Clinton.

I can't help but wonder if she would be so popular with "the media" if she were a man. Her husband's admitted infidelity was not, as some would have it, a reason to label him a bad President. There were plenty of other reasons for that label, but they don't matter now. She's not her husband. But let's consider for a minute a role reversal. Let's pretend it was Hillary, and not Bill, who was president eight years ago. Better yet, let's not put familiar names to it: say it was Joan Brown who was President, and her well-spoken, also politically powerful husband, Jim Brown, was First Gentleman. There's a scandal, and it turns out that Joan Brown had a brief affair with an aide. Now, I know some people would have trouble accepting Jim Brown as a viable candidate, ten years later. After all, he couldn't even inspire loyalty in his wife!

Is there a double standard? Would there be a double standard, more appropriately, if there were more women in powerful positions? Are we still so culturally ignorant that we can accept infidelity from men more than we can from women - or are there some of you out there who think it's perfectly understandable in women, but not in men?

Or is the whole cheating thing becoming an outmoded concept? I know when Bill Clinton was caught with his cigar out, my first thought was, "Naughty boy was cheating on his wife." Same as it would be if any other guy had done it, basically. Yes, there's some moralistic overtones there; I was raised to believe that's wrong - not for any mystical reason, but because it's not fair to the spouse. My second thought was, "This is between him and his wife," because, really, it's not our business. (Lying under oath WAS our business, but even there, everybody lies about sex; it's axiomatic.)

Would my reaction - would your reaction, whatever it was - have been the same if the sexual dynamics were reversed?

And what bearing does all this have on the 2008 primaries and elections? Pretty much nothing. Just some thoughts I had on gender dynamics and double standards. I'm far more concerned, as I said, with what any of the candidates might do for (or to) our country.

Anyway, no, I don't have strong feelings about it. Not yet. My wife asked me last night why I'm so "threatened" by her candidacy. I don't have an answer for that - it's like asking me when I quit beating my kids. I did have strong feelings about, and felt threatened by, the candidacy of W, both times. I didn't want him to win. I very, very badly didn't want him to win, under any circumstances, even if it meant possibly electing lame-ass Democrats. Of course, he did, and life went on - after a fashion, and without a Constitution. So I guess my answer would be: I'm concerned (gods know why; I have no stake in it) about the future of this country, and the rights of its citizens, and whether there will be an environment for your children to inherit.

I think the core issue for me isn't the plumbing of the candidates, but how the process is treated, and received by the people. The 2000 elections proved to me that few people in this country have a good idea of how a presidential election is supposed to work. The Supreme Court, for example, is not supposed to be allowed to decide on any aspect of an election. The 2004 elections emphasized that ignorance once again to me, what with all the folks who were actually surprised that a majority of votes doesn't automatically translate into an electoral college victory. And I'm not even going to touch the hot potato of election fraud.

It's turned into a goddamn sporting event, is the problem. Go Red Team! Go Blue Team! Hey, look, the coaches are shouting obscenities at each other! "We just couldn't get control of the ball. Next time, we'll focus on controlling the ball." People are Democrats or Republicans for the same reason they're Steelers fans or Baptists: Because Daddy or Mommy was.

I shouldn't be surprised, and really, I'm not. Disappointed would be a better word. Democracy is predicated on the assumption that a well-informed, educated populace knows what's good for it. But a misinformed, ill-educated populace also knows what's good for it: bread and circuses.

We're not lacking for bread, and the whole bloody thing is a circus. What sucks is that we are the clowns.

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