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

June 27, 2008 at 3:33pm

#593415

Born in the USA

But currently touring in Europe, which I believe is a mythical land of castles and fairies that lies beyond the Great Eastern Ocean.

http://www.brucespringsteen.net/news/index.html

Springsteen and the E Street Band are currently on tour in Europe and are earning accolades for their stellar performances. In a five-star review (out of five), London's Sunday Telegraph raved, "Bruce proves music can save your soul." The Observer called a London show "a performance of blood, sweat and tears." In another five-star review (out of five), Financial Times of London dubbed the concert "masterly, memorable and utterly exhilarating." The Daily Mail testified, "as good a show as I've ever seen."

Okay, so the Daily Fail's review notwithstanding, those are some serious praises. But hey, Observer? Blood, Sweat and Tears hasn't performed in the past quarter-century. Stick to Bruce.

I guess they like him in England. How about other places in the mythical lands outside of the Real World, AKA the US?

'Magic' has been certified triple Platinum in Norway; double Platinum in Ireland and Sweden; Platinum in the United States, Czech Republic, Denmark, Italy, Spain, and Canada; and Gold in the UK, Austria, Belgium, Germany, The Netherlands, Hong Kong, Singapore, Australia and New Zealand.

Singapore? Hope he doesn't play there; that's a caning.

Now, I think there's two things going on here. First, but least, is that the Magic album is Bruce's most political album ever. By that I mean that most of the songs on it are in some way related to the mess in Iraq (funny how Iraq isn't on that list up there. Hmmm.) And I daresay this war is unpopular pretty much everywhere. But most of all - and I may be biased, but I've seen enough Bruce concerts to get a feel for the stage dynamics - the Magic tour has been one incredible musical experience. The band has never been better. Oh, sure, they were probably more energetic in the 70s, when they were just getting their feet wet. And they were a tight ensemble even in the 80s, on the Born in the USA tour. Then they went their separate ways for a while, reappearing in the late 90s (a tour I, sadly, missed) and then again for The Rising in the early 00's, which is when they really started their apotheosis.

But this tour confirms their godhood. Even without Danny Federici - and possibly even because of his untimely loss - they have a way of working together that I, as a pitiful non-musician, can only grok a small part of. All I remember is seeing Nils Lofgren do whirlies on the stage during an extended guitar solo on "Prove it All Night" and thinking, "This is better than anything I've ever seen before, ever."

And now I find that they're going to wrap up the tour with a few more US shows - including one in Richmond, where I saw them for The Rising.

This morning, I got tickets.

The Coliseum is going to rock.

I hear the guitars ringin' out
Ringin' out down Union Street
I hear the lead singer shoutin' out, girl
I wanna be a slave to the beat
Yeah, tonight I wanna break my chains
Somebody break my heart
Somebody shake my brains
Downtown there's something that I wanna hear
There's a sound, little girl, keeps ringing in my ear

I wanna be where the bands are
I wanna be where the bands are
Where the bands are
I wanna be where the bands are...

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