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:

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

March 6, 2019 at 1:00am

#953793

You Get A Shiver in the Dark

The Story Behind The Song: Sultans Of Swing by Dire Straits

https://www.loudersound.com/features/the-story-behind-the-song-dire-straits-sult...

Singer/guitarist Mark Knopfler had written it in 1977, after ducking into a deserted pub one rainy night and witnessing a lousy jazz band. Undeterred by the lack of both talent and punters, their lead singer finished the set with a mildly enthusiastic, “Goodnight and thank you. We are the Sultans Of Swing.”

Much of writing, be it fiction or non-fiction such as articles about music, depends on point of view. We have to choose it carefully - and I don't mean just first person or the various third person perspectives (or even the less common second person narrative), so I suppose another word for it would be focus.

In Avengers:Infinity War, for example - yes, I know it's a movie, but writers were involved, and yes, there is a spoiler ahead but if you haven't seen it by now, that's on you - the story is told from the focus of the various heroes we've come to know from previous movies. And the ending sucks for them; they lose. This evokes sadness in the audience. But if you tell the story from Thanos' point of view, it's a tremendous victory. Sure, maybe it would be harder to identify with the guy who wants to eliminate half the sentient beings in the universe - for some people. Other people kind of saw his point.

But I digress. Point is, the linked article is far from the first rags-to-riches story in the music world. It's compelling; I mean, no one wants to hear about bands that got their start by, say, inheriting ten million dollars. No, we want to hear how the underdog became victorious. It's kind of an American thing, though not exclusively; after all, the band in question here is purely British. The details are always different, but the story's often the same: talented and hard-working musician(s) start out unknown, begging for scraps, using secondhand instruments, sleeping rough; then, boom, big break, hit single, concerts in packed stadiums.

I mean, really, it's the archetypal story of rock and roll - but again, not limited to that one genre. I was fortunate enough to personally witness it once, as the Dave Matthews Band went from playing drunk, sweaty fraternity parties to international stardom.

But you never hear the story of the Sultans of Swing.

I don't mean the song; I mean the desultory jazz band that provided Mark Knopfler's inspiration in the linked article. There you are, just trying to earn a living, maybe getting a few quid from the landlord, more likely just hoping for exposure. Along comes a desperate guitarist, putting up with your music just to stay out of the rain for a couple of hours. You don't recognize him because he's not famous yet. He's just sitting there having a pint or three. The set ends, you say "Goodnight and thank you," and you maybe go home and smoke a bowl and try to reevaluate your life.

And then a year or two later, you hear it on the radio. Sultans of Swing.

I wish some music journalist could have tracked them down and asked them what they thought about it. How they felt about their sole purpose in life being to make Mark Knopfler and the other band members millionaires by way of one brief moment of inspiration. Where are they now? Dead from some overdose? Doddering about in a nursing home, still cursing the name of Dire Straits for all eternity?

Did they even remember that one fateful night, or could it have been, to them, any one of a hundred lousy pub gigs?

"Then, of course, it’s a story. And let’s face it, all good songs have a story."

Everything has a story. Most of them don't have happy endings. Most of us are not destined for greatness. Most of us are the Sultans of Swing.

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