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

October 14, 2009 at 6:53pm

#671757

No, seriously, it really DOES suck.

Yesterday, Lorien

quoted a song, as she is wont to do on occasion, for a blog entry: "Invalid Entry"

.

My first comment was "That song sucks."

Lynn McKenzie

disagreed, even going so far as to impugn my favorite psychotherapist, Jung.

Well, we can argue about Jung, but there is no doubt whatsoever that the Eurythmics' song "Sweet Dreams (Are Made of This)" sucks massive rat turds.

And I can prove it.

First of all, I'm not saying the Eurythmics suck. Annie Lennox has a beautiful voice, and their music was polished and not too overproduced compared to some of the other crap the industry was pumping into a tone-deaf public in the early 1980s, pre-Born in the USA. I hear the video was pretty well done, too, but the video is not the song and I would no more judge the song by the video than say The Matrix was a good movie just because it had killer special effects.

That said, "Sweet Dreams (Are Made Of This)" sucks.

Let's start with the lyrics, because to be good, a song has to have good, meaningful lyrics. Otherwise, why bother making it a song? It can be an instrumental.

The lyrics:

Sweet dreams are made of this
Who am I to disagree?
I travel the world
And the seven seas--
Everybody's looking for something.
Some of them want to use you
Some of them want to get used by you
Some of them want to abuse you
Some of them want to be abused.

That's all. That's it for the lyrics. It's all chorus. Or, looked at another way, it's all verse. Oh, except for the almost-unintelligible part:

(Hold your head up--Keep your head up--MOVIN' ON)

Which before I could look up lyrics on the internet, I could swear went, "Powderhead (movin' on) Chowderhead (movin' on)" - but that's me, and that doesn't contribute to the song's massive vacuum pull. More, that was, in fact, the only saving grace of the song, and to find out that it was as much of a lie as "'scuze me while I kiss this guy" was a massive disappointment.

Okay. Okay. So the lyrics are spare. Haiku gets by with only 17 syllables; maybe they were going for that same kind of feel?

Well... No. Look at the lyrics again. Is there any imagery there? Is there a metaphor, an onomatopoeia, ANY kind of poetic trick?

Um... no.

In fact, the second line is a cliché: "Who am I to disagree?"

Oh, wait, I lied. There is one poetic device used: Rhyme.

But wait - what's the first line? "...this / ...disagree."

Do. WHAT?

They get around this by singing, "Sweet dreams are made of THESE."

That isn't a mishearing; that's no "There's a bathroom on the right." That's a LONG E and Z sound, when the lyrics - and the song title - and the album title all say THIS, which is a SHORT I and HARD S sound. You don't get those two mixed up, even if you are British.

If they'd said "Sweet dreams are made of CHEESE" and wrote that in the lyrics and song title and album title, then I'd change my opinion of the song. Really. Surrealism is always a good thing.

But it's not cheese - not even Wensleydale - it's THIS.

Okay, you say, you're wrong, idiot. "Disagree" is meant to rhyme with "seas." And yes, I'm willing to forgive the absence of the sibilant at the end of the first word. But my point remains - they don't say "this;" they say "these," creating an internal rhyme. So, no, I'm not wrong, especially if you consider "Sweet" through "seas" to be FOUR lines, which is how I've always seen it written.

Moving on: "Everybody's looking for something." So? You might as well say "Everybody needs air to breathe" or "My cat likes to lick her asshole."

I'm not even going to go into the next four lines, except to say that they'd make a set theory mathematician cry. The categories are neither overlapping nor mutually exclusive nor, as with the previous line, do they really mean anything.

As for "Hold your head up," well... Argent did it better, and first, with a song that's only marginally longer, lyrics-wise, than "Sweet Dreams:"

...And if they stare
Just let them burn their eyes on you moving
And if they shout
Don't let them change a thing what you're doing... Hold your head up, etc, etc.

Overall, if I encountered those Eurythmics lyrics written out here on WDC, I'd probably have to rate them

- assuming everything was spelled correctly.

Now, if you want a Eurythmics song that was popular AND good, you need to look no farther than "Here Comes The Rain Again."

But hey - at least it's not "Brass In Pocket" by the Pretenders. So "Sweet Dreams" does have one thing going for it.

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