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 6, 2007 at 7:12pm

#547356

Turning West Virginian

So this toothache I've been having, which I thought - because the dentist told me so - would be fixed with a relatively simple crown, turns out to now have a major crack in it. This, apparently, is why I've had to overdose on Advil and paint my gums with whatever-caine.

The crack, according to this same dentist, means now I have to do one of the following:

a) have the tooth extracted, perhaps getting an implant to take its place.
b) get a root canal AND a crown, or
c) do (b) and then if it doesn't work do (a)

I'm glad I have dental insurance, because (b) could run $2000 and (a) could be as much as $3000. And that would be one less tooth - I've already lost a lower molar in addition to my 'wisdom' teeth. It's too bad my 'wisdom' teeth didn't make me 'wise' enough to go to the dentist sooner, when I first started having temperature problems with the tooth.

He gave me a referral to an endodontist for further evaluation - but since endodontists specialize in doing root canals, I have a feeling that's in my future. I've had them before. They're not as painful as people say. What they are is expensive and boring (for once, no pun intended)

I'd almost rather have pain.

What has 37 legs and three teeth?
The front row at a Willie Nelson concert.

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November 6, 2007 at 6:39pm

#547347

B minus 5 Days

My town has become a Police state.

Yes, tonight, Sting's old band has a concert at a local venue. This has made the streets almost as clogged as my arteries - except with dumber cholesterol.

When I got to college at 17, I found myself rooming with a guy who should have been compatible. We were both from northern Virginia, both virgins, and we were both younger than most of the other first-years (UVA doesn't have freshmen, sophomores, etc.; it has first-years, second-years, and so on and, just for me, ninth-years).

Should have been, but he worshipped reggae music.

Now, I like reggae as I like most music that isn't rock: I'll listen to it, and it's okay, but it's not my drug of choice. So at first it was cool; I was expanding my musical horizons, and I found out more about Rastafarianism than I ever wanted to.

Of course, mostly he'd listen to Bob Marley and Peter Tosh. But he wore out the grooves of Synchronicity.

The Police is reggae by musical definition, if not by culture. For that matter, Hotel California (song, not album) has a reggae beat, and it was the one song we could agree on - but I digress.

So after hearing The Police a few dozen times, I OD'd on Bruce as an antidote. This was, for those of you who don't do math well, the year before BitUSA came out, so I only had up to Nebraska.

Steve, my roommate, liked Bruce about as much as I liked reggae: in small doses.

Still, everything was fine until the day I came "home" and Darkness on the Edge of Town was on the floor, unwrapped, and Synchronicity was on the turntable - not turning, because he'd stepped out and it had finished playing.

LPs, remember? So it's not like a CD where you can leave it around for a little while and it'll still play, and it's not like an iPod by any stretch. LPs were pretty fragile, and I always handled them with care and respect.

So that's why I'm not going to go see the Police tonight. It has nothing to do with not liking them - I grew to like that album again after first year, and everything to do with Steve's lack of respect for the Bruce.

The midnight gang's assembled and picked a rendezvous for the night
They'll meet 'neath that giant Exxon sign that brings this fair city light
Man there's an opera out on the Turnpike
There's a ballet being fought out in the alley
Until the local cops, Cherry Tops, rips this holy night
The street's alive as secret debts are paid
Contacts made, they vanished unseen
Kids flash guitars just like switch-blades hustling for the record machine
The hungry and the hunted explode into rock'n'roll bands
That face off against each other out in the street down in Jungleland

Comment

November 6, 2007 at 1:06pm

#547294

Erection Day

So, I voted.

Big, fat, hairy deal. There were elections for state senator and representative - each of which ballots included exactly one candidate, both Democrats. Not that there's anything wrong with that.

The state representative candidate is a guy I know personally: David Toscano. Good guy, even allowing for him being a lawyer. So I voted for him rather than trying to write in Dave Matthews or something.

The senate candidate was Creigh Deeds, and his name was cool enough that I voted for him rather than trying to write in Dave Matthews.

There were also elections for City Council and the school board. Every last City Council candidate was a Democrat, and I was tempted to write in Dave Matthews, but figured it wouldn't make a difference - they'd still try to stonewall every one of my projects.

School Board was another matter. I didn't recognize a single name. Now, you might say that as a childfree individual, I should have no say in the school board. If that were the case, though, why would my taxes be going to schools? I have a vested interest in an educated crop of kids. Still, recognizing no names, I pretty much picked four people at random.

Thus was my vote counted. Thus was my citizenship validated. Thus another few years of enabling politics.

Incidentally, I wouldn't write in Dave Matthews because I think he'd do a better job. I'd write him in so he'd be too busy to put out another album.

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