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.

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 9, 2007 at 5:21pm

#507310

Music

I'm sitting here listening to some old Steely Dan ("Deacon Blues"). You gotta love a band that can put this in a song with a straight face:

I crawl like a viper
Through these suburban streets
Make love to these women
Languid and bittersweet

My wife has a power that I don't: she can decide, within about three seconds of listening to a song, whether it sucks or not. Me? I have to listen to it a few times, and maybe a few times more, before I can make up my mind about it.

I think music listeners are conveniently divided into two major categories: lyric listeners, and tune trackers. Of course, life isn't binary, and most people are some mixture of the two. I know where I fall on the spectrum, which is heavy on the lyric end of it. If the lyrics suck, I don't want to hear the song. A good example of this is Pretenders' "Brass in Pocket":

Got rhythm I cant miss a beat
Got new skank it's so reet
Got something Im winking at you
Gonna make you, make you, make you notice

I mean... what the ever-lovin' FUCK?? Y'know?

I'm not sure where my wife falls on the spectrum (though we share a deep, abiding hatred of that Pretenders song), but I suspect it's more on the "tune tracker" end of things - not that she doesn't appreciate good lyrics, and not that I don't appreciate a good tune. Still, there's a Golden Rule for me, and it's rarely been proven wrong:

If it's danceable, I can't listen to it.

It's not that I don't have rhythm; people say I'm a fair drummer. In fact, there's a picture somewhere of me drumming while Kirstin dances (thus distracting me). I'll post it one of these days, but like all pictures of me, I don't like it. I think it's more that when people are writing songs, they don't care what the words are if they're trying to make a dance tune. You know, like, "Come on baby, do the locomotion." Bleh.

There was a movie a while back called "Eddie and the Cruisers" (1983, for you sticklers). The title guy hires a guy (the narrator) to write the words. He puts two fingers together and says something like, "Words and music, man. They're like this."

Deep.

Now I'm listening to Brandi Carlile. There were a couple of Springsteen songs in there. I love "shuffle" buttons.

I always thought Bruce was a major influence on "Eddie and the Cruisers." The fictional band was from Jersey, too.

Anyway, this power of Kirstin's to declare within a few moments of hearing a song whether it's cool or it sucks (there's rarely an in between) really gets on me sometimes. I mean, it shouldn't, right? You like a song, or you don't, and what's the point in trying to change her mind? But sometimes it makes me feel like she's not even giving it a chance - especially when it's a song I really like, such as Neil Young's "After the Gold Rush" or Billy Joel's "Summer, Highland Falls."

And, of course, that lessens my own enjoyment of it. Illogical? Yeah, but there it is - it makes me feel like there must be something wrong with the song, when there's patently not.

Another thing I've noticed that makes me different from most people about music is that whether or not I like a song is irrespective of how old it is, how new it is, or how many times I've heard it. I'm of the firm opinion that good songs are timeless, and I never get sick of them. Oh, maybe if it got stuck on infinite repeat (I knew a guy who can't listen to Eagles' Hotel California because it kept playing while he was trapped in the wreckage of a car... and playing... and playing... and playing...)

The result of this is that there are songs from the 50s, 60s, 70s, 80s (very few), 90s and 00s that I like (and probably from before then if I happen to hear it), and I can't stand listening to Top 40 radio.

In fact, I've pretty much decided that ALL commercial radio sucks ass, and listen exclusively to WFUV from New York City. I can do this because of this thing called the Internet, upon which they stream their programming. If you're interested, check out http://wfuv.org/

And for the next few days I won't need the Internet to listen to it, because I'll be in NYC. Don't know if I'll have Web access from there - if I do, I might check in from time to time

Oh... and a happy 58th birthday to Billy Joel. Rock on!

It comes down to reality
And it's fine with me 'cause I've let it slide
Don't care if it's Chinatown or on Riverside
I don't have any reasons
I've left them all behind
I'm in a New York state of mind
-Billy Joel,
New York State Of Mind

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