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

November 8, 2007 at 8:33pm

#547834

B minus 3 Days

After The Rising came out in 2002, I missed out on tickets again. Tickets for Bruce shows, as I might have mentioned, can be difficult to get. Before the Internet, they tended to sell out in a matter of minutes; having Ticketmaster online only sped up the sellout.

And then the local rock station DJ said he'd give away two tickets to the Richmond show to the first caller who could identify the five songs whose opening bars he would play.

It's as if the contest had my name written all over it.

Funny thing, though, what excitement will do. He didn't play openings to songs like Hungry Heart or Born to Run or Pink Cadillac. It's been five years now, so I don't remember exactly which ones they were, but they were songs that rarely if ever got airplay, even on the album rock stations. Things like The Promised Land and Two Hearts. Things that only a true Bruce fan would know. And I missed one.

I was completely devastated. What could that one be? I replayed the opening bars in my head, tried to continue and recreate the lyrics, but it kept morphing into something it wasn't. Something with a similar beat and guitar riff.

Meanwhile, I knew other people were calling in. Other people who would, if I didn't calm down really soon and let the music come to me, would end up with my damn tickets.

And then I got it: I remembered the one song.

I called back - and there still wasn't a winner! I gushed out all five songs, in order, in fast forward speed, unable to contain my excitement, forgetting that people would be listening. People I knew.

The DJ paused on the phone, trying to decipher my high-speed babbling, then, "Yes! You're a winner!"

I think you could hear my "WHOOHOO!" all the way to Richmond.

The actual concert wasn't until March of 2003. They turned out to be seats in the upper section of the Coliseum, in front near the stage, with an awesome view. I took Kirstin, of course.

Now, what a lot of people don't know (which, of course, I do) is that while Bruce is most associated with the Jersey Shore, in the early days he had a massive cult following in another place: Richmond, Virginia. So this show was not only the first time I'd seen Bruce in almost 20 years, but was something of a homecoming for him. A lot of the older folks in the crowd had seen him 'way back then, playing in clubs with Steel Mill, his first band. And towards the end of the set, when Bruce called out former Steel Mill partner Robbin Thompson, and noted Virginian Bruce Hornsby, the crowd went fucking nuts.

As a side note, I was marginally involved in the start of Bruce Hornsby's career. His brother, John Hornsby, was a law student at UVA and a lyricist on the first album. I was trying to do some album photography for them, but they ended up using a shot by Robert Llewellyn instead. John asked me once whether I thought it should be "Bruce Hornsby," "The Range," or "Bruce Hornsby and the Range." I told him that "Bruce Hornsby and the Range" was a lot like "Bruce Springsteen and the E Street Band," and that worked, so they should go with that. I was also the first person to write a review of their first album when it came out. Unfortunately for my budding career in the music industry, I wrote an honest review, and they never talked to me again.

But back to the Richmond show. Aside from the guy next to us, who was not only The Drunk Guy (there's always one near you at a concert) but also The Guy With The Cell Phone (which I wanted to grab and throw off the balcony, except that would have gotten me ejected), the concert was so full of awesome that I vowed, there and then, that I would never again miss a nearby Springsteen concert if I could find tickets at all.

I've kept that vow, but details will have to wait.

Oh yeah - it was around the time I won the tickets that my wife and I got married. It was the perfect way to start out a marriage. I think my life was on hold for a long time, and when this concert came along, well, I felt I was finally on my way.

Now a life of leisure and a pirate's treasure
Don't make much for tragedy
But it's a sad man my friend who's livin' in his own skin
And can't stand the company
Every fool's got a reason to feelin' sorry for himself
And turn his heart to stone
Tonight this fool's halfway to heaven and just a mile outta hell
And I feel like I'm comin' home

These are better days baby
There's better days shining through
These are better days
Better days with a girl like you

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November 8, 2007 at 10:02am

#547702

1984

A few entries back, in "B minus 8 Days"

I talked about the 1984 release of Born in the USA. Shortly thereafter, I mentioned my friend Pat in "Blast from the Past"

- he's in town now, and we spent yesterday evening smoking cigars, drinking beer and catching up.

So my wife tracked down the picture I talked about in "Blast from the Past"

, from 1984. Ladies, try not to faint. Dudes, try not to be jealous.

(Pat's on the left; I'm on the right. You know, kind of opposite to our political leanings

)

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