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

February 18, 2009 at 11:10pm

#636619

Pink Floyd

17. One of my biggest regrets in life is that I never saw Pink Floyd in concert.

By which I mean Waters, Gilmour, Mason and Wright, all on the same stage. I was but a toddler when Syd Barrett went off the barmy end, though I came to appreciate his... whatever... later in life.

The definitive Pink Floyd albums are, in my world, The Wall, Wish you were Here, and Dark Side of the Moon. Back when I had LPs, those were the three Floyd albums in my collection, plus The Final Cut, which I think me and one other person liked. But it was really Disc 3 of The Wall, in a lot of ways.

Dark Side, of course, is one of the best selling, and most influential, albums of all time.

Pink Floyd, in its prime (the 70s, and a few years before and after) were known variously as psychedelic rock, concept rock, and progressive rock. Me, I don't care much for labels; besides, what they play isn't - technically - rock and roll.

What they did was explore insanity. Not drugs, as some would have it, but the human mind (okay, yeah, drugs played a role insofar as they illuminated the mind). Insanity in all its forms.

In part, this was a reaction to Syd Barrett's descent into madness, but I get the feeling Roger Waters was a few cards shy of a full deck. Psychologically, that is; musically, he certainly had a full deck - of Tarot.

The Dark Side album cover - a prism breaking a beam of light into its component wavelengths, creating a rainbow - is probably the second most recognizable symbol in popular music, after the Rolling Stones tongue (which is just weird and dumb-looking).

The strange thing for me is, after my album collection got lost in the flood in a basement apartment I was renting not far from where I live now, I never got a replacement for my Dark Side album. I picked up a bunch of their albums on CD, including some of the post-Waters material - which is excellent, but not quite the same - but for whatever reason, I never got around to replacing Dark Side.

Until today.

On Sunday, I said "what the hell," and bought myself a nice birthday present - a CD of Dark Side of the Moon, which I've been without for twenty years. I'm listening to it now.

Of course, you all know me as a Bruce Springsteen fan, and really, Bruce and Floyd are about as different as they can be and still be played on the same radio stations. I never claimed to be consistent. But while you can go to a Bruce concert this year and get the same kind of feeling and energy you'd have gotten in 1975, Pink Floyd ain't what it used to be. Oh, I wouldn't turn down tickets, but it would be like seeing Zeppelin with a different drummer. Or Paul and Ringo without George and John.

I could have, in high school. I was just old enough, and The Wall was one of the most popular things on the radio. If I'd tried, I might have gotten tickets, before they came out with The Final Cut. Before Roger Waters struck out on his own. I never did, but I feel like if I had, it'd have been worth it. I could say, "Yeah, I saw them."

Which is why I'm going to see Leonard Cohen tomorrow - because if I didn't, I'd kick myself for the rest of my life.

Nothing lasts forever. Nothing but the echoes.

Ticking away the moments that make up a dull day
You fritter and waste the hours in an off hand way
Kicking around on a piece of ground in your home town
Waiting for someone or something to show you the way

Tired of lying in the sunshine staying home to watch the rain
You are young and life is long and there is time to kill today
And then one day you find ten years have got behind you
No one told you when to run, you missed the starting gun

And you run and you run to catch up with the sun, but its sinking
And racing around to come up behind you again
The sun is the same in a relative way, but you're older
Shorter of breath and one day closer to death

Every year is getting shorter, never seem to find the time
Plans that either come to naught or half a page of scribbled lines
Hanging on in quiet desperation is the English way
The time is gone, the song is over, thought I'd something more to say...

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