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 24, 2007 at 2:03pm

#483376

The Pursuit of Happiness

First of all, no, I haven't seen the Will Smith movie with that (misspelled) title.

It's axiomatic - or perhaps clichéd; sometimes I can't tell the difference - that "ignorance is bliss." Mavis Moog

once wrote, "Knowledge is blisser," but I'm not so sure about that.

Human mythology is rife with examples of knowledge being Bad. The best-known myth of that sort is the one involving Adam and Eve. They were living in a state of shining happiness when they "ate of the fruit of the Tree of Knowledge" - which God told them would kill them, but admitted in private that it would make them "become like us" (the antecedent of "us" still being hotly debated). Running a close second, and a favorite of industrial-age writers, is Prometheus. In this case, Prometheus is a god already (well, technically, a Titan) who defies mean old Zeus by giving fire (a metaphor for knowledge, technology and civilization) to the poor, shuddering humans.

At some point I'm going to explore in some depth the central importance of the "apple" imagery among all these knowledge myths - and its use even in myths of science, as an inspiration for Newton among other things. But not today.

Point is, God sentences Adam and Eve to lifetimes of tedium, and Zeus sentences Prometheus to having his liver torn out on a daily basis. Nothing good comes to those who explore knowledge at the expense of a deity's wishes.

All of which makes me wonder what my neighbor, Thomas Jefferson, was thinking when he co-opted Locke's idea of property rights, replacing it with this silly "pursuit of happiness" nonsense. I mean, Tom was an educated man and a great thinker; he had little use for organized religion, and owned quite a lot of property (I'm talking about land here, not slaves. Don't give me crap about Jefferson's owning slaves. I've heard it all.) Still, it's right there, in one of my country's founding documents: the right to (among other things) the pursuit of happiness.

But happiness is anathema to knowledge. This idea is supported by deep-rooted cultural memes, and is one of the few philosophical ideas at home in both eastern and western spirituality (Buddhism's destruction of the ego as a means to enlightenment comes to mind; one must know nothing to experience Nirvana (or be its lead singer (nootch))). Knowledge is elusive; the more one knows, the more one wants to know. Happiness, too, is elusive (and we have no explicit right to it, anyway; all we have is the right to pursue it) but is quite simply not found in the assumption of knowledge. Clearly, some people can find satisfaction in the pursuit of knowledge, but the very force that drives scientific investigation is by its nature unquenchable. There is always the drive to find out more, gain more knowledge; a discontent with what is already known. And discontent is not happiness.

So I've decided, today, that this whole pursuit of happiness thing is crap. It's a semantic paradox, for starters; and the clincher for me was when I realized that I'd rather know stuff, if necessary in defiance of the gods and mortal authorities, than fit myself into some lotus-eating state of ego-free bliss.

Give me that apple. It won't make me happy, but it will set me free.

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January 24, 2007 at 12:16am

#483266

Okay, okay...

I know... enough with the political rants, right?

Someone suggested I talk about the weather. But that would likely lead to a discussion about global warming, which at present is a political hot potato.

Damn Al Gore for taking away the Last Safe Topic!

Oops, I talked about politics again. Damn.

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