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

January 4, 2009 at 2:42pm

#627763

That didn't take long.

We are people of this generation, bred in the most hedonistic society the world has ever known, housed now in over-mortgaged tract homes, and looking uncomfortably to the world we leave to our children. We were born in the days when the Soviet Union was the Evil Empire, a college degree was the ticket to the middle class, and being born American meant that unlimited opportunity was your birthright.

http://corporatemofo.com/society_and_antisocial_tendenc/reality_bites.html

You know, I follow financial discussion boards and websites, and you can probably imagine the kind of freaking-out that's going on at said boards these days. People with portfolios down 30, 40, 50%. People who have lost or are about to lose their jobs. People wondering why they should keep their home when it's worth 3/4 of what they paid for it - without interest...

[Allow me a venture into digression; as this is my blog, I don't feel too out of line doing this: I don't understand the "We're upside-down on our mortgage, so we're just going to default and let the bank foreclose" kind of mentality. You bought a house, you signed the papers - whether you fully understood all the Ferengi print is another discussion entirely, but let's assume "yes" - and you entered into an agreement to faithfully pay off the mortgage under the terms provided. Not ONE of those terms included a clause guaranteeing that your house would always appreciate in value; you just assumed that it was despite all historical evidence to the contrary. And now you want to default on a contractual obligation? There's a word for you, dude, and it's not a nice one.

[To continue my digression, consider a house that you may have purchased for, say, $300,000 - "modest" by the standards of the real estate boom, depending upon your area. Assuming a $50,000 down-payment and a 6.5% fixed interest rate on a 30 year loan, you're going to pay the bank a total of $568,861.22 over 30 years. (That's in today's dollars. Since you got a fixed rate mortgage, your monthly payment of $1580.17 is effectively less in 2039 dollars, when the mortgage will be paid off) (I never said there would be no math; quite the contrary). Anyway, the point is, you're already upside-down to the tune of $568,861.22 - $300,000 = $268,861.22 right from the get-go. So if your home value drops to $225,000 because of the current housing crisis, your upside-downedness is $343,861.22 instead of $268,861.22. Big effing deal.

[Besides, what are you going to do, rent? Fixed-rate mortgages never increase, except for any escrow portion. Rent will go up ever year. End of digression.]

Getting back to the essay at hand,

We grow into middle age not surrounded by prosperity and security, but by our doubts and fears. Even as the rich have gotten richer, we have seen our standard of living fall. The middle class is barely reproducing itself, bifurcated into those barely treading water and those on an endless paper chase after useless honors. Our hopes have been dashed, our dreams sold for firewood to keep warm and hold back the wolves for one more night.

That, fellow writers, is how to mix metaphors. This guy uses a Cuisinart.

Somewhere along the way, someone might have tricked us into caring or having hope, but we have come to realize that the current "crisis" is not the result of a great country hijacked by a cabal of free-market capitalists: It is, in fact, the new baseline.

In other words, in the immortal spirit of Queen Victoria, perform your duties faithfully. It will be unpleasant, but just lie back and think of England.

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