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 1, 2007 at 12:50pm

#484943

More On Philosophy

panthera: Would you agree that nothing is truly 'discovered', that it is just a question of a new awareness of this or that, which may have existed all along?... we live in a bigger world than what we are 'aware' of, proven by many 'discoveries' starting with the planet not being flat... being newly 'aware' of its real shape does not take away the fact that it always was round.

(for full text see comments to "The Father of Invention"

)

Thanks for the thoughtful comment, Chatty.

I like to think that my philosophy is pragmatic and realistic; that is, I don't concern myself much with the question of whether all y'all exist or not. Solipsism is an easy trap to get into, especially on the Internet, but it's also easy to get out of: I'm real, you're real; my computer is real. I'm also not "a butterfly dreaming I'm a man;" that observation strikes me as some straw-grasping from a guy who doesn't want to deal with reality. I'm willing to accept variant definitions of reality, but they all have to embrace the fact of me sitting here on my lunch break and typing in my blog; the idea that this is all illusion is simply preposterous, not worthy of consideration outside the walls of University philosophy departments and Buddhism.

Now, that doesn't mean that we don't all run around with illusions, or that these illusions don't affect our reality. That's the symbolism of all the math discussion in my blog header, if you haven't figured it out by now: the idea that there's a real part and an imaginary part, and that most of what we see isn't purely one or the other but some mixture of both.

So, yes, to me the definition of "discovery" is finding something that existed before you knew about it (as a species or an individual) AND incorporating that into your worldview. Like, for instance, I have a "Discover" card, so named because you have to run around and discover places that actually take it. They took it before I found them; I merely found what was already there, and others already knew about. Once I find them, they're on my mental list of "places that take Discover cards."

It's the "incorporating that into your worldview" that's tough for a lot of people. There are still people who think the world is flat, despite all evidence to the contrary, and despite this being known since at least as far back as Aristotle. Others deny that anyone ever walked on the Moon, or that the Holocaust happened. You can deny all you want; it doesn't change the facts - but it does change your worldview, your perception of the meaning of events (or even if events have meaning, which I'd dispute).

So I'd turn around what Chatty said: Being newly aware of the Earth's real shape doesn't take away from the fact that a lot of people have thought it was flat. And for some, even all the evidence to the contrary won't convince them.

We're just stubborn that way, we (mostly) hairless apes.

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