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 20, 2008 at 10:34pm

#569023

Dark Side of the Moon

http://antwrp.gsfc.nasa.gov/apod/ap080220.html

I was concerned, today, that the cloud cover would preclude my viewing of the total eclipse of the moon this evening. As with much in life, I hoped for the best but prepared for the worst... but it's a wonderfully clear night, and the moon around 8:30 looked like it had a big chunk bitten out of it.

It's at the times of eclipse that I can best visualize how the solar system is put together. In this case, we're treated to a view of Saturn just a few moon-widths away (the moon subtends an arc of 0.5 degrees of sky; thus, two moon-widths is about one degree of arc. There are, of course, about 180 degrees of arc across the visible celestial equator, and a total of 360 degrees, corresponding to 360 degrees of longitude. Why 360? Largely because there are 365 days in a year, and 360 is the closest number with many factors.

They came up with that system in ancient Babylon, or perhaps inherited it from an even older culture. The ancients were pretty good with numbers, but fractions sometimes stumped them; hence, they looked for numbers that were easily divided. That's why there's 24 hours in a day instead of some multiple of 10; it's divisible by 1, 2, 3, 4, 6, 8, 12 and, of course, itself. 24 itself goes neatly into 360. And then there's the mystery of why there are seven days in a "week" and not six or eight; I think that calendar evolved separately from the solar calendar, as weeks of seven days work relatively well in marking off lunar cycles. Perhaps, long ago, weeks alternated between seven and eight days, before some bright caveperson noticed: seven days in a week; seven moving celestial bodies. Coincidence? He or she didn't think so. And so the days of the week took on the attributes of the Sun, Moon, Mars, Mercury, Jupiter, Venus and Saturn (in that order), disengaging the week from its relationship to the moon. Just a thought I had; I don't know if there's any reality to it.

On a more personal note, things are starting to turn around a bit, for me. My back continues to improve. We're starting to get some jobs dribbling in at the office. My dad's going to be released from the hospital soon, perhaps even tomorrow, to go back to his assisted-living place - though he'll require more nursing care.

Well, time to go check on the moon again - it should be at its darkest right about now. It can only get brighter from here.

Comment • View 4 comments

February 20, 2008 at 7:03pm

#568984

Hot Dog Pun

Wienermobile wipes out
Famed vehicle spins, crashes on snow-covered Pa. highway; 'hotdoggers' OK.

Be sure to read the entire article.

http://www.star-gazette.com/apps/pbcs.dll/article?AID=/20080211/NEWS01/802110323...

Comment • View 6 comments

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