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

May 7, 2007 at 5:40pm

#506773

Careful what you wish for.

Robert, this is going to be a whiny entry. I don't know if you noticed, but I mentioned you as one of my favorite writers on wdc when I was "Center Stage.". I love your writing and these one sentence entries make me want to whine. Do you want to be responsible for that? I didn't think so! -susanL

The item to which Susan refers is another in a long line of strokes of genius from my good friend and loyal Minion, terryjroo:

Invalid Item

This item number is not valid.
#1231865 by Not Available.

Now, Susan's glowing praise of me has been archived for posterity at "Invalid Entry"

: "His fiction is not always "people friendly" but he writes so very well; no reader could have trouble picturing what he writes because it's so descriptively clear, yet he doesn't spend too many words doing it."

Fortunately, that's exactly the effect I strive for with my writing. I'll point out here that susanL

is no slouch, herself, and is a far more accomplished writer than I (hell, she's been published and I haven't, at least not paid). Still, I appreciate the compliment, even though it made it so my hat didn't fit for a few weeks.

Well, I told Susan by way of email reply that I'd "write something pithier (or at least pissier) soon." I'm not in a pissy mood, so I'll be pithing you off instead. Incidentally, be sure to check out this week's upcoming Comedy newsletter; all my writing energy went there and not here, this week.

So, yeah, I've been pretty busy, which has served to keep my blog entries to quick "weird news" stuff - which I find amusing, anyway. Here's a couple links I found today, in fact. The first is an in-depth article about how a Christian right-wing extremist actually thought about the whole homosexuality thing, rather than reflexively thumping the Bible; the second, from a whole different philosophical basis, is a story about how some scientists have managed to convince themselves that people can see the future:

http://www.alternet.org/sex/49912/

and

http://www.redorbit.com/news/science/925987/many_scientists_are_convinced_that_m...

Why the hell are people still using "man" to stand for the entire human race, anyway? I thought that kind of went out with free sex and chivalry. I mean, I wouldn't like it if people used "woman" as the collective noun for humanity. Besides, China's birth-control and post-natal abortion policies notwithstanding, there are still more women than men on Earth - by a wide margin. So, single girls, if you're wondering where all the men are? They're not. And yet we keep sending them off to get killed in wars. Come on, folks, we're a diminishing commodity. Stop the war so my single female friends can get laid, mmkay?

Now, where was I? Oh, yes. Taking a pith.

I had a long, philosophical discussion with the office cleaning lady the weekend before last.

I was in the office, supposed to be working - and I was, but then Mindy came in and started asking me about religion.

Mindy's a short, waifish woman who's struggling with like three jobs because she's trying to save money for an education. Her family has disowned her - they are, she said, members of a fringe religious group (a cult, really) that had her completely brainwashed, cloistered and secluded from the rest of humanity. Home-schooled, even. Strict dress code, learned to believe that the world was divided into the Group (who simply called themselves "Christians") and Everyone Else, who were all going to Hell. And, of course, since she started questioning everything she'd been taught, she's now in the latter category.

Well, her mind opened up, and she's eager to learn about different philosophies, different religious practices, different beliefs. So she asked me - and this is what made me take an hour break from work - "How do you know what's true, what's fact, when there's so much conflicting information out there?"

"Why, you look on Wikipedia, of course!"

(Not really. I just threw that in there to see if Lorien

is reading.)

What I really said was, "It's hard, but here's some ways to spot bullshit..."

Which is what took me so long. Being quite a bit older, and having grown upspent my childhood in the completely opposite environment (that is, an atmosphere that fostered learning and curiosity, rather than dogma and rigidity), I like to think I have a reasonably well-tuned bullshit detector. Besides, I spent my childhood on a farm; while we didn't raise much cattle ourselves, the next farm over did, and I got to be able to spot bovine feces a mile away. But, faced with someone who had grown up with a certain set of beliefs forced into her like they were facts... well, I'm amazed she's as open-minded as she is, but sometimes when you open your mind, it's possible that your brains fall out... and I was kind of at a loss.

I was faced with the knowledge that, when you don't have the comforting arms of faith, you encounter a lot of uncertainty. Me, I thrive on the uncertainty. I love it. It helps my writing. It lets me consider things from all different perspectives. Some people can't handle the uncertainty; it's chaos, and they'd rather get all their "facts" from one source, like the Bible or Wikipedia.

Once, I tried to track down information about the artificial sweetener aspartame. I found out a lot more than I ever wanted to know about it, including the utter certainty that aspartame isn't especially good for you. But - here's the problem - there's a lot of conflicting information out there as to whether it's just run-of-the-mill crap, or if it's actively bad for you, as in causes cancer or Republicanism. Many of the websites I found asserted that, while it's non-nutritive, it's better for people to eat or drink aspartame than sugar - obesity, diabetes, tooth decay, etc. Others, however, asserted the "truth" that aspartame is carcinogenic, unsafe in any quantity, and makes your penis shrink.

Now, I strongly suspect that the latter category of web page is, if you really do your research, the product of shills for the sugar, saccharine and Splenda manufacturers. Still, how do you KNOW?

Well, you could drink a whole lot of aspartame, and when you don't get cancer, you can thumb your nose at the shills. Or you can just eschew the stuff altogether; it's not like it's especially good for you, like I said (the problem with this is that I require caffeine to get through the day, and I've never acquired a taste for coffee, so it's Coke Zero for me).

Point is, it's hard to know what the real truth is. Kennedy assassination? 9/11? The missing Watergate transcript? There's a lot of speculation out there, and believing the government line is usually a lot harder, taking more suspension of disbelief, than believing the fringe conspiracy theorists.

In the end, I find it doesn't really matter what the truth is.

Oh, put down your eyebrows.

No, it usually doesn't matter. What matters, what affects the world around us, what affects our own lives, is what people believe the truth to be. People can spend a lot of time and money and energy (all the same thing, really) tracking down facts, but no one really has the big picture - except, theoretically, God. In fact, the best definition of God I've ever come up with has been "that which knows all the relevant facts." Actually, I just came up with that, but it works for me. And speaking of God, God only proves what I just asserted about truth: some people don't believe in God. Most people do, in some form or another. The people who do tend to be the ones who set policy, and said policy is often set based on their belief (or lack thereof) in a supreme being. God may or may not affect us directly - I'm not going there today - but it can be demonstrated that an individual's belief in God does affect you directly.

If that's not enough, consider a child who accepts the existence of Santa Claus. That child's reactions near Christmastime revolve around her belief in Santa Claus. She's good because Santa wants her to be; when she's not, she feels regret at letting Santa down or that she won't get as many presents. Santa may or may not exist, but in that child's world, he is the motivator, and his existence is what defines, in part, her reality.

When, as most children do, she finds that Santa is a myth, she will likely go through all the stages of grief: disbelief, anger, etc., etc.

My cleaning lady has a perfectly understandable distrust of Christianity. I spent much of the time I talked to her convincing her that most Christians aren't like the cultists who raised her. I honestly don't know if that's true or not, but she was going on as if it weren't, as if Christianity itself is suspect, as opposed to the fringe group that was the only home she knew, and against whom she had to rebel. And anyone who knows me knows I'm about the most unlikely apologist for Christianity out there.

I've said before: reality is that which affects other people as well as yourself; imagination, in contrast, affects only yourself. But the point there is that imagination - beliefs, assumptions, and so on being included in imagination - does affect you.

And that's enough philosophy for one day.

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