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

December 19, 2010 at 7:50pm

#713855

Eclipse

There is no dark side in the moon, really. Matter of fact it's all dark.
-Gerry O'Driscoll (as recorded by Pink Floyd)

As most of you know, I think astronomy's cool. I feel lucky to live in an age where we're finding extrasolar planets, and even, in some cases, being able to analyze their atmospheres. I really hope I'm still around when they first find extraterrestrial life (by which I mean probably microbes or something similar, not being taken over by space aliens).

But sometimes it's the stuff close to home that's really cool. Case in point: Tuesday's total lunar eclipse.

Lunar eclipses aren't all that rare; they come around even more frequently than a blue moon (for which see some October blog entries here and "Fantasy Newsletter (November 23, 2010)"

) but with different patterns. Most of us have seen one. One time I didn't even know there was going to be an eclipse, which is rare for me since I usually keep up with such things, and I saw the shadow on the Moon and freaked out for about three seconds until I realized, "eclipse."

While not rare, the event can be striking. The moon generally turns a copper or blood-red color, the result of selective bending of the red end of the sun's spectrum through the edges of the Earth's atmosphere. It's every sunrise and sunset on Earth, projected onto the Moon, which is pretty awesome to contemplate. If you were standing on the Moon, you'd see our planet, dark but for scattered lights of civilization, surrounded by a halo of fire.

Cool.

Anyway, the result of that is the sunlight bends through our atmosphere, hits the dark rocks of the Moon, and bounces back to our night side. Result: Red. Or sometimes copper. Depends on how much dust is in the atmosphere when it happens.

The truly cool thing about this eclipse, though, is that it occurs on the date of the (northern hemisphere) Winter Solstice, December 21. Which doesn't really mean anything - but it's still cool.

Here's a Wikipedia article

about it, complete with animated simulation of the Moon traversing the Earth's shadow:

That article in turn links to this,

which contains the following note:

This lunar eclipse falls on the date of the northern winter solstice. How rare is that? Total lunar eclipses in northern winter are fairly common. There have been three of them in the past ten years alone. A lunar eclipse smack-dab on the date of the solstice, however, is unusual. Geoff Chester of the US Naval Observatory inspected a list of eclipses going back 2000 years. "Since Year 1, I can only find one previous instance of an eclipse matching the same calendar date as the solstice, and that is 1638 DEC 21," says Chester. "Fortunately we won't have to wait 372 years for the next one...that will be on 2094 DEC 21."

Rare enough. The chance of me being alive in 2094 approaches zero, so I don't want to miss this one - even though it's bloody damn cold out there, and I'll have to be outside at 3 in the morning.

My plan? If it's not cloudy Monday night (if it is, I will be five kinds of pissed off), I'm going to wear fourteen layers of clothing, grab my camera and binoculars (they have to have SOME use besides perving on the neighbors), drive up to the Blue Ridge, and find a spot with a good view of the western sky. There are plenty.

I won't be worth a damn on Tuesday, but what the hell - it's the solstice.

And if the cloud bursts, thunder in your ear
You shout and no one seems to hear.
And if the band you're in starts playing different tunes
I'll see you on the dark side of the moon.

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