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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. ![Merit Badge in Quill Award
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PROMPT September 21st Today’s prompt is straight from Fivesixer  ’s notebook!  It’s come to my attention that Leonard Cohen has a posthumous album coming out, and the first single dropped today. It's spare and beautiful and sad and lovely. Listen to the song below and write anything you want about how it makes you feel. https://youtu.be/mszJwXsZwKM Some of these prompts are, admittedly, a struggle. This one is not one of those. Still - what can I say about Leonard Cohen that I haven't already said? I'll tell you a story. It's a true story, though the details may be worn away by time and the elements. Once, long ago, there was a boy and a girl. He'd spent the vast majority of his short life in Virginia; she, in Ontario. They were young, but not too young. Sixteen, maybe, or seventeen. They met at a camp in upstate New York. I'd like to say they fell in love, but he knew, even then, it wasn't love. Did she? Maybe. Maybe not. That wasn't the foremost thing on his mind. I think you can guess what was the foremost thing on his mind. He leaned against her, and she opened up a book of poems, and she read some of them aloud. The boy became entranced: not so much by her, though he would forever remember her fondly, but by the poems. He'd never heard of their author: a man named Leonard Cohen who, like the girl, was proudly Canadian. Thus began a lifelong journey into beautiful darkness. I found out later that he was not just a poet, but also a singer/songwriter, one who hobnobbed with the likes of Bob Dylan and Janis Joplin. He even wrote a song about Joplin. It remains one of my favorite songs. You remember when I said (probably several times now) that sad songs make me happy? This was the beginning of that. Oh, yeah, in case it wasn't painfully obvious, the boy in the story was me. I don't know much about his personal life, only that he struggled with deep depression, a struggle that was reflected in his work. Like I've said, I'm not one for celebrity-worship, and he barely qualifies as a celebrity. In fact, a lot of people still don't know about him. Chances are they've heard his music, though, and not realized it. Zack Snyder, the movie director, seems to be a fan - Cohen's music turns up in his soundtracks, which is one reason I will always appreciate Snyder's movies, even the ones that people like to rag on. And there was a period where just about every performer covered "Hallelujah." Yeah - that song was Cohen's. He wrote that dark-ass shit. You're probably more familiar with the Jeff Buckley version of it, but my all-time favorite rendition was by Brandi Carlile: https://www.youtube.com/watch?v=N1sFYdwlXtI I could go into exactly why that song is the greatest piece of poetic music ever penned, but this is already dragging on. By the way, Cohen is enshrined in the Rock and Roll Hall of Fame. Yeah. I wouldn't call what he did rock and roll, but here: https://www.youtube.com/watch?v=t9IZfiHEgd8 I count myself lucky that I saw him on stage - not once, but twice; the first time was on one of my birthdays, a cold and dreary February in New York City. The second time was in a half-empty auditorium at Caesar's Palace in Las Vegas. But I'm supposed to talk about how that video embedded above makes me feel. Honestly, I'm not sure I can. Cohen was one of the victims of the unholy year 2016, having died that November at a ripe old age. Others may mourn other victims of that year, but for me, the greatest loss was that of Leonard Cohen. Hearing the posthumous release - well, it brings back the whole flood of memory associated with his poetry and music. So I'll do what I always do when I can't really express my feelings: I'll let Leonard say it for me, because he always has been, and always will be, able to describe my emotions better than I can. Now in Vienna there are ten pretty women There's a shoulder where Death comes to cry There's a lobby with nine hundred windows There's a tree where the doves go to die There's a piece that was torn from the morning And it hangs in the Gallery of Frost Aey, aey, aey, aey Take this waltz, take this waltz Take this waltz with the clamp on its jaws Oh, I want you, I want you, I want you On a chair with a dead magazine In the cave at the tip of the lilly In some hallway where love's never been On a bed where the moon has been sweating In a cry filled with footsteps and sand Aey, aey, aey, aey Take this waltz, take this waltz Take its broken waist in your hand This waltz, this waltz, this waltz, this waltz With its very own breath of brandy and Death Dragging its tail in the sea There's a concert hall in Vienna Where your mouth had a thousand reviews There's a bar where the boys have stopped talking They've been sentenced to death by the blues Ah, but who is it climbs to your picture With a garland of freshly cut tears? Aey, aey, aey, aey Take this waltz, take this waltz Take this waltz, it's been dying for years There's an attic where children are playing Where I've got to lie down with you soon In a dream of Hungarian lanterns In the mist of some sweet afternoon And I'll see what you've chained to your sorrow All your sheep and your lillies of snow Aey, aey, aey, aey Take this waltz, take this waltz With its "I'll never forget you, you know" This waltz, this waltz, this waltz, this waltz With its very own breath of brandy and Death Dragging its tail in the sea And I'll dance with you in Vienna I'll be wearing a river's disguise The hyacinth wild on my shoulder My mouth on the dew of your thighs And I'll bury my soul in a scrapbook With the photographs there, and the moss And I'll yield to the flood of your beauty My cheap violin and my cross And you'll carry me down on your dancing To the pools that you lift on your wrist Oh my love, oh my love Take this waltz, take this waltz It's yours now, it's all that there is Source: LyricFind Songwriters: Garcia Lorca / Leonard Cohen Take This Waltz lyrics © Sony/ATV Music Publishing LLC |
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