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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 3 hours 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

about 5 hours 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').

1 Like

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.

April 23, 2021 at 12:01am

#1008989

As If It Happened Only Yesterday

Not doing my usual random article thing today. I found out after I posted yesterday that one of my favorite songwriters died on the 19th, so this one's about Jim Steinman.

Jim Steinman obituary
Composer and songwriter who masterminded Meat Loaf’s Bat Out of Hell, one of the biggest-selling albums of all time

I'll let the obituary do most of the work, here. I'll be getting together with a friend of mine on Tuesday, someone who's also a fan, and pour one out for the Master of Melodrama.

Jim Steinman, who has died of kidney failure aged 73, made a spectacular career of being bigger and more bombastic than the rest, and his achievement in masterminding Meat Loaf’s album Bat Out of Hell will guarantee his immortality.

Meat Loaf is, of course, a great singer and performer, and one of those whose career survived having been in Rocky Horror. But he wouldn't have enjoyed nearly the success that he did without Steinman's songwriting.

Bat’s producer, Todd Rundgren, thought the album was supposed to be a parody of Bruce Springsteen – members of Springsteen’s E Street Band played on it – and Steinman did not entirely reject parallels with the Boss. His own songs, Steinman said, “are dream operatic, his are street operatic. He’s more West Side Story and I’m more Clockwork Orange.”

I am, of course, also a fan of Springsteen, and I noticed long ago that they shared a lot of backup musicians. Roy Bittan's piano in particular lent itself just as well to Steinman's operatic compositions as it did to Springsteen's more mainstream rock and roll.

Steinman’s career as a solo artist only managed to encompass the album Bad for Good (1981), which reached the UK Top 10 and spun off the modest hit single Rock and Roll Dreams Come Through, but he was in demand from many directions.

I had that album. Still do, actually. Most of its songs have been redone by Meat Loaf and other artists, because Steinman's voice just wasn't up to signing the checks that his fingers wrote. Still, I have a fondness for the album.

In the 1990s Steinman ventured into musical theatre by collaborating with Andrew Lloyd Webber on Whistle Down the Wind (based on the 1961 film).

Because really, there was only one composer who could out-angst Steinman, and that's Webber.

He reunited with Meat Loaf for the last time for the singer’s album Braver Than We Are (2016).

I hope they finally got over their turbulent past for that. Never got that album, though; it was after albums stopped being a thing, really. But now I'm going to have to track it down. (That's a pun, yes. "Track?" No? No? Damn. Just not in the groove today.)

The obit I linked includes a video of I Would Do Anything for Love (concerning which comments about what "that" is will be received with great disdain), but I thought I'd include a video of the song I rule at, when I have a suitable partner, on the karaoke stage.

It never felt so good, it never felt so right
And we're glowing like the metal on the edge of a knife...

3 Like • Comment

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