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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🌝 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.

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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.

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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.

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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.

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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.

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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"

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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!

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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.

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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.

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February 14, 2009 at 7:09pm

#635824

Thirteen

13. I chuckle whenever I see a building that doesn't have a 13th floor.

By which I mean, of course, a building that has 13 or more floors anyway, which is relatively rare.

I found this article on Wikipedia, for what it's worth:

http://en.wikipedia.org/wiki/13th_floor

Even landlords who are not themselves superstitious realize that the desirability of suites on a floor 13 might be compromised because of superstitious tenants, or commercial tenants who worry about losing superstitious customers. Based on an internal review of records, Dilip Rangnekar of Otis Elevators estimates that 85% of the buildings with elevators did not have a floor named the 13th floor.

Of course, there's also the Chinese practice of omitting the fourth floor, which I mentioned in this blog a few weeks ago.

A superstition, as the word is used, often means, simply, a practice based on a belief other than that of the person using the word. The identification of something as a superstition generally follows the form, "Do X, or don't do Y, or Z will happen," where Z is usually (but not always) something Bad.

Now, there are a lot of situations where "Don't do Y or Z will happen" has a causal basis; for example, "Don't get caught driving drunk, or you'll have your license taken away." That's not a superstition; that's cause and effect, for most people in most jurisdictions. The easiest way to avoid getting caught driving drunk is to simply not drive when you're drunk. However, "Step on a crack and you break your mother's back" has no logical or causal basis, and can safely be considered a superstition. But if you go out and purposely step on sidewalk cracks, and you find out later that your mother broke her back... you notice that, more than you notice when you've stepped on cracks and you later find your mom doing her yoga exercises.

But I think that few superstitions translate to real-world results; the thirteenth floor thing is one of the few.

(There are good, sound reasons not to walk under a ladder; however, "bad luck" isn't one of them.)

I've said for a while now that the main difference between imagination and reality is that imagination affects only the imaginer, while reality affects others, too. The fear of the number 13 (triskaidekaphobia) falls into that gray zone between imagination and reality, where, while it's obviously a superstition without logical cause, enough people believe it that it affects reality; hence, no 13th floor, no Gate 13, etc. Also, a person who puts any stock whatsoever into the superstition is more likely to behave differently on Friday the 13th, thus, perhaps, causing anomalies. In that sense, it's real; it affects others.

There are three Fridays the 13th this year; as in every non-leap year that contains Friday, February 13th, there's also one in March and again in November. That's the maximum; in every calendar year, there is at least one Friday the 13th.

Here's an article about it, on "How Stuff Works:"

http://www.howstuffworks.com/friday-thirteenth.htm

Ultimately, the complex folklore of Friday the 13th doesn't have much to do with people's fears today. The fear has much more to do with personal experience. People learn at a young age that Friday the 13th is supposed to be unlucky, for whatever reason, and then they look for evidence that the legend is true. The evidence isn't hard to come by, of course. If you get in a car wreck on one Friday the 13th, lose your wallet, or even spill your coffee, that day will probably stay with you. But if you think about it, bad things, big and small, happen all the time. If you're looking for bad luck on Friday the 13th, you'll probably find it.

In any event, I've long maintained that Saturday the 14th is far scarier than Friday the 13th... because the paraskavedekatriaphobics let their guard down on the day after.

Sometimes, you don't have to understand superstitions - you just have to be aware that they're there, and plan accordingly.

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