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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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.

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1 Like

Turkey DrumStik

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

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Yeah, that's a little bleak."

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1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
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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!"

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September 21, 2019 at 12:27am

#966518

Take This Waltz

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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