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

Stik to My Own Beat

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 wanderlust-words-in-motion

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 wanderlust-words-in-motion

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 Surprised

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

September 17, 2010 at 10:06pm

#706382

Alcohol

Yeah o yeah you seen me walk
On burning bridges
Yeah o yeah you seen me fall
In love with witches
And you know my brain is held
Inside by stitches
Yet you know I did survive
All of your lovely sieges

Beer.

Some say that it is the defining characteristic of civilization; that the specialization and coordination necessary to take grain and ferment it in a certain way to produce a beverage that is nutritious, refreshing, and clean (as compared to some sources of water in a pre-scientific world) is what caused societies to form the hives of specialization we know as cities.

And you know that I'll pick up
Every time you call
Just to thank you one more time
Alcohol
And you know that I'll survive
Every time you come
Just to thank you one more time
For everything you've done

Alcohol
Alcohol

Today, I went to a beer tasting in a nearby town. The town, for all that it's tiny, has a claim to fame: it was featured in the old TV show, The Waltons. The Dew Drop Inn still exists.

There were ten brewers there today, featuring ten special craft brews. Most of these beers were porters, stouts and ales, the technical differences between which I'm still not all that clear on. It has something to do with what process is used to ferment them. What was notably lacking was tired, common lagers and weak pilsners.

I've mentioned before, in here, my affection for a particular beer, Old Rasputin Imperial Stout. "Imperial" is a style that harks back to the days of the Russian Empire, before the Bolshevik Revolution. It seems someone from the British Isles brought some Guinness to Moscow, and it was a great success - so much so that the Tsar ordered casks of the holy nectar to be shipped to them. Problem is, Moscow is cold in the winter, and the casks froze, ruining the beer. I call this "alcohol abuse."

In a flash of inspiration not seen in England since an apple fell on Isaac Newton's head, or pretty much ever since, the master brewers invented a brewing process that increased the alcohol content in the stout, while adjusting the other ingredients so that the increased ethanol didn't unbalance the beer. They shipped this to Moscow, and the increased alcohol content (about 10%, as compared to an ordinary brew's 4-5%) acted as a natural anti-freeze, keeping the sacred beverage from freezing.

Thus was born Imperial Stout, and the process has since been applied to other beers. I even had an Imperial style lager recently that was, without exception, the best beer I've ever tasted - and I'm not a lager guy.

And I'm sorry some of us
Given you bad name
yeah o yeah, cause without you
Nothing is the same
Yeah o yeah i miss you so
Every time we break up
Just to hit a higher note
Every time we make up

Anyway, this was a beer connoisseur's tasting. Some people there seemed to be fish out of water - one older guy said the beers were, in general, too dark for his taste.

That's okay. You find what you like and drink it. For instance, I don't like India Pale Ales (IPAs), because they're hoppier than a frog on meth. IPA, incidentally, has a similar backstory to Imperial style beers, but in this case they had to ramp up the hops content to ship the brew around the bottom of Africa, which took time - and hops are the preservatives of beer.

There was an IPA there today, and while it was an excellent example of an IPA, it's not a style I choose to drink.

The problem is... well, let me illustrate. Last night, at my poker game, I produced a large bottle of an Imperial stout (Allagash label, for those who care) and proceeded to drink it. I found it excellent, but hardly worth the $16 they charge for a large bottle of the stuff. So I switched to my old standby, Bass Ale...

And found I didn't like it.

So I'm forever ruined for cheap, mass-produced beer by the assholes who insist that quality matters - and with whom I agree.

Thing is, I'm not just a beer snob. I'm also a wine snob. And a tequila snob. Don't get me started on Cuervo Gold, for example: sure, in college, that was what I drank, but once I realized that just because the Mexican authorities consider anything with at least 51% agave that's distilled in a particular region of Mexico can be called tequila doesn't mean that anything called tequila is any good. Once I discovered real tequila - blanco, reposado, anejo; it doesn't matter - I could never go back to the cheap stuff.

Life is, after all, too damn short to drink the cheap stuff.

Who's crawlin' up my spine - alcohol
I've been waiting long long time - alcohol
Now you teach me how to rhyme - alcohol
Just don't stab me in the back with cartisol

Now we reunite - alcohol
And forever be divine - alcohol
Screw a light bulb in my head - alcohol
may that ceremony be happy or sad...

(Lyrics by Gogol Bordello)

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