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.

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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.

3 Like

January 28, 2007 at 7:07pm

#484155

When does loss become gain?

One of my great revelations in life was when it came to me that those things we call "creation" and "destruction" are not antonyms; that, in point of fact, they are one and the same thing. Certainly I'd heard this expressed before, but it's one thing to hear somebody say it in relation to some form of mysticism, and another to have its reality punch you in the gut. They're not just "two sides of the same coin," I realized; they ARE the coin, yin and yang, two heads, two tails.

So much of our worldview is dictated by language. Did what we call "intelligence" in humans predate the development of language, or was it the other way around? Or, alternatively, was it all part of the same evolutionary development? In any event, we insist on calling this event a "creation" and that other event a "destruction." But I can't imagine an instance of creation that does not also involve destruction. If I write on a piece of paper, creating a poem, I destroy the configuration of the paper and whatever I use to write it. If I got my hands on high explosives and blew something up, I would create a mess (and light, heat, noise, and legal trouble). Even if I only create something in my mind - an idea, or whatever - it destroys whatever neural configuration was there before (or however memory works).

Whether we call some event a creation or a destruction is a matter of the value we place on whatever is being created or destroyed.

Today has just been one of those days when my mind refuses to stop dwelling on the past, on all that I've lost along the way. But can I apply the semantic gymnastics above to the loss/gain relationship? Can we really lose or gain anything without gaining or losing something else? I suppose it's all in how you look at it. "Loss" has negative connotations - except, maybe, when it comes to losing weight. "Gain" has positive connotations, but certainly it's possible to gain something undesirable, like a tumor or, somewhat less malignantly, a traffic citation.

Can it really be all in how we look at things? Certainly there are absolutes, but so much of our experience is polarized through this half-empty vs. half-full filter. The optimists have been telling me this all along, but I don't listen to optimists; I don't trust anyone who smiles all the time.

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