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

July 2, 2008 at 4:02pm

#594239

4th of July, Asbury Park

Sandy, the angels have lost our desire for us
I spoke to 'em just last night and they said they won't set themselves on fire for us anymore
Every summer when the weather gets hot they ride that road down from heaven on their Harleys they come and they go
And you can see 'em dressed like stars in all the cheap little seashore bars parked making love with their babies out on the Kokomo
Well the cops finally busted Madam Marie for tellin' fortunes better than they do
This boardwalk life for me is through
You know you ought to quit this scene too

Sandy the aurora's rising behind us, the pier lights our carnival life forever
Oh love me tonight and I promise I'll love you forever

Well, it seems she couldn't wait for the actual 4th of July, but the famed Madam Marie died a few days ago, aged 93.

http://www.app.com/apps/pbcs.dll/article?AID=/20080701/NEWS/80701031

Madam Marie bragged that she had told the fortunes of everyone from Judy Garland to Springsteen himself. Legend has it that she told Springsteen he was going to be a success. Springsteen later joked that she told all the musicians that.

And Springsteen never forgot Madam Marie.

"He always comes by to say hello," she told Press columnist Bill Handleman in May. "He knows where he came from."

The song - parenthetically titled "Sandy" - came out on Bruce's second album, a compilation largely ahead of its time in musical composition and lyrical athletics. But "4th of July, Asbury Park (Sandy)" itself is a straightforward tune about being young and then becoming disillusioned with the trappings of youth, of wanting to grow up and do something with real meaning.

Interestingly enough, the lyrics he sings today (quoted above) aren't the original lyrics on the album. They began:

Sandy, that waitress I was seeing lost her desire for me
I spoke with her last night, she said she won't set herself on fire for me anymore
She worked that joint under the boardwalk, she was always the girl you saw boppin' down on the beach with the radio
The kids say last night she was dressed like a star in one of them cheap little seaside bars, and I saw her parked with Loverboy out on the Kokomo...

I have no idea why he changed them. I liked the original better. But either way, I always thought the Madam Marie line (which was artistic license on his part; I understand she was never actually arrested) was one of the best in the song. To me, it described something so forlorn, so end-of-an-era - in the song, it's the straw that breaks the camel's back, the thing that finally makes the narrator go, "You know what? Forget it. I don't want this anymore. It's just not the same."

So what did Bruce have to say about the end of this era? From his website:

Back in the day when I was a fixture on the Asbury Park Boardwalk, I'd often stop and talk to Madam Marie as she sat on her folding chair outside the Temple of Knowledge.

I'd sit across from her on the metal guard rail bordering the beach, and watched as she led the day trippers into the small back room where she would unlock a few of the mysteries of their future. She always told me mine looked pretty good - she was right. The world has lost enough mystery as it is - we need our fortunetellers. We send our condolences out to her family who've carried on her tradition. Over here on E Street, we will miss her.

Kind of makes me want to take a road trip on Friday. I probably won't - but I've always wanted to walk the Boardwalk on the 4th of July. Kind of a holy pilgrimage.

And maybe one day I'll find out what in the hell the Kokomo is.

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