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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.
September 8, 2021 at 12:02am September 8, 2021 at 12:02am
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This one might not make a lot of sense unless you're a Springsteen fan. It's also from The New Yorker, generating cognitive dissonance within me. In this case, though, the content supersedes the medium.
A Springsteen Mystery Solved 
Jon Landau, the Boss’s longtime close collaborator in matters musical and financial, offers a definitive answer about what Mary’s dress is doing in “Thunder Road.”
Thunder Road happens to be one of my all-time favorite songs, one that cemented my lifelong appreciation for Springsteen. It's almost fifty years old now. Bruce himself has expressed amusement at his twentysomething self writing the line, "maybe we ain't that young anymore."
The Internet is an uneven contribution to the human prospect...
Thor's balls, TNY, just once get to the point at the beginning and stop meandering around like a rabbit in Central Park.
All the way in the middle of paragraph 3:
Early this month, on a day too grim for dogs or snakes, it was best to stay inside, scroll numbly through Twitter, and wait for a virtual brushfire. Maggie Haberman, the tireless chronicler of the Trump Administration for the Times, unintentionally provided one, tweeting a photograph of a half-empty theatre and stage along with the lyrics “A screen door slams, Mary’s dress sways.” The lyrics are the opening to “Thunder Road,” arguably the best song on Bruce Springsteen’s breakthrough album, “Born to Run.”
There is no "best song" on Born to Run. The entire album is a masterpiece.
Haberman was blasted for getting Springsteen’s lyric all wrong, and, in the days since, people have continued offering confident opinions. It’s not “Mary’s dress sways”! It’s “Mary’s dress waves.”
My original LP of BTR was lost in a flood many, many years ago, but I remember the lyric sheet. And I'm pretty sure it said "waves." But it's not like you can tell from the actual song; Springsteen is an epic songwriter and musician, but he's not the world's greatest enunciator.
The article goes on to re-create a Twatter spat and mention several other sources, none of which is very enlightening, in typical TNY fashion.
Both on Springsteen’s official Web site and in his songbook, the word is “waves.” And yet Springsteen uses “sways” on page 220 of his memoir, which is also called “Born to Run,” and in his handwritten lyrics, which were auctioned off a couple of years ago by Sotheby’s.
Yes, I have a copy of the book Born to Run. No, I don't remember the lyric being different. You think you can't understand Bruce's voice sometimes? Try reading his handwriting. Dude should have been a doctor. I'm not trying to be mean, here. My handwriting sucks ass, too. In Bruce's case, you don't have to have good handwriting to write powerful lyrics. Or, for that matter, write a damn good book.
I e-mailed Jon Landau, who, as a critic for The Real Paper, in 1974, declared Springsteen to be the future of rock and roll, and then became his close collaborator in matters musical and financial. Short of Springsteen himself, no one could answer the question more definitively than Landau.
That's from the penultimate paragraph. It's like they have to drive from midtown through Delaware to get to the Bronx.
“The word is ‘sways,’ ” Landau wrote back. “That’s the way he wrote it in his original notebooks, that’s the way he sang it on ‘Born to Run,’ in 1975, that’s the way he has always sung it at thousands of shows, and that’s the way he sings it right now on Broadway. Any typos in official Bruce material will be corrected. And, by the way, ‘dresses’ do not know how to ‘wave.’ ”
Well. I'd rather get confirmation from Bruce himself, but I guess Landau's the next best source.
And you know, now I think of it, I don't see how it could be any other way. Early Springsteen was absolutely obsessed with rhymes. The first song (Blinded by the Light) on his first album was, by his own admission, the result of him sitting down with a guitar and a rhyming dictionary. And the last word of the second line of Thunder Road? The one right after the Mary's dress one? It's "plays." Of course the first line had to end with "sways." Why I never saw this before, I don't know -- but now I'll never unsee it.
I remember the shock and dismay I felt when I found out some people don't care about lyrics. I mean, what the hell? Especially with people like Springsteen, they're, like, at least 70% of the power of the music. Regardless of the source, this was another surprise for me -- I'll try to remember it next time I'm belting out that song at karaoke.
Thunder Road breaks all the rules of songwriting. There's no real chorus. There's no melodic hook. It just jumps out (it was the first song on the album) like a bull in a rodeo, bucking and bellowing, never letting up... just, eventually, fading out like you're bored with the show and looking for something else to do, but you know that somewhere, the bull's still raging, and it will not quit until the last recording of it succumbs to the entropy that will eventually take us all.
And of course, I can't let this go without at least including the song in question.
The screen door slams, Mary's dress wavesways
Like a vision, she dances across the porch as the radio plays
Roy Orbison singing for the lonely
Hey, that's me, and I want you only
Don't turn me home again
I just can't face myself alone again...
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