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!

about 2 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 4 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.

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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.

August 2, 2022 at 12:14am

#1035995

Hallelujah

If you're somehow tired of the song "Hallelujah," don't bother with this entry today. Also, what is wrong with you?

‘More than a song’: the enduring power of Leonard Cohen’s Hallelujah
In a new documentary, fans and experts explore the legacy of a song originally shunned before becoming a timeless classic

This Guardian link is basically a plug for a film called Hallelujah: Leonard Cohen, A Journey, A Song, but as long as they talk about Leonard Cohen, I don't much care. I know I've gone on about him in here before, but it's been a while, so deal.

Directed and produced by Dayna Goldfine and Dan Geller, the film takes both a micro and macro view of the song and Cohen, along with their respective and deeply intertwined places in culture.

I've probably said this before, but my introduction to Cohen was through his poetry, not his music. Although the music is also poetry, of course.

“Leonard Cohen, in short, was a prophet,” said Goldfine, who, along with Geller, has assembled a stacked career directing expansive documentaries focusing on music including 2005’s Ballets Russes, about the early 20th-century Russian ballet company. “Leonard [was known for] timeless writing and timeless poetry that floats outside of any particular epoch,” Goldfine said. “It addressed the deepest of our human concerns about longing for connection and longing for some sort of hope, transcendence and acknowledgment of the difficulties of life.”

See, now, that's not what I generally get from his work. But okay. That just makes it greater: that different people can have different interpretations, mostly valid.

Why Hallelujah reached the heights it did is due to a unique mix of inventive cover versions, cultural happenstance and a magic-in-a-bottle quality the song no doubt possesses. Search the track on Spotify today and it’s Jeff Buckley’s version, not Cohen’s, which is the top result; the pairing of the song’s seemingly haunting subject matter and Buckley’s raw 1994 recording, coupled with the singer-songwriter’s drowning death aged just 30, adds another layer of weight.

I mean, the Buckley arrangement is clearly superior, musically. And that's okay. Hell, the two times I saw Cohen live (which I feel absolutely lucky to have done before he died), he drew more from Jeff Buckley for the music. The important part, though, is the lyrics, and those are entirely Leonard Cohen's poetry.

But oddly enough, the song’s modern ubiquity can be traced to its prominent placement in Shrek, the second-highest grossing movie of 2001, which effectively launched Hallelujah into the upper echelons of popular culture.

Also oddly enough, it was almost exactly one year ago when I did an entry here wrestling with this same outlet's brutal takedown of Shrek. Here: "Whipshrek"

. In that entry, I quoted the idiot reviewer as saying this: "(Nothing screams “unearned gravitas” like slipping in a cover of Leonard Cohen’s Hallelujah.)"

Perhaps it’s fitting that an animated comedy would propel Cohen’s legend, considering that according to Geller the biggest misconception about Cohen himself is that he was originally considered the singer of gloom and doom.

I'm not going to argue. Cohen's work tends to the philosophical depths. That's why I like it, though. To paraphrase a line from Doctor Who, "What's wrong with sad? It's happy for deep people."

In recent years, Hallelujah has had dozens of placements in TV and movies (from Scrubs to Zack Snyder’s Justice League) and has been sung by everyone from Bob Dylan to Bono, BrandiCarlile to Il Divo.

Okay, I might have inserted an editorial emoticon there. A few points to unpack from that brief quote:

I have enjoyed every movie Zack Snyder has made. Yes, that one. Yes, also that one. And oh yes, absolutely that other one that you hated. Around the third Snyder movie, though, I started to notice that he puts Cohen in most of his movies. It's led me to believe he's a fan, which would help explain why I like Snyder movies when so many people don't.

I can't imagine Dylan singing it. (Hell, I can't imagine Dylan "singing" anything, and I've seen him on stage twice.) But it has got to be better than Bono's ridiculous interpretation. Also, fun fact: I saw Cohen and Dylan perform on the same stage, separated by several years: the Beacon Theatre in Manhattan (I also once saw David Gray there).

The Brandi

Carlile version is on her Live at Benaroyal Hall album, and it's gorgeous. I wrote about it before. Here: "Hallelujah"

Who the fuck is Il Divo?

“It’s a Rorschach test,” says Light of the various interpretations of its lyrics, including the idea that it’s meant to be a Christian song. In reality, as the film chronicles, Cohen was Jewish.

With a name like that? Say it ain't so! ...Okay, but it's not that simple. Saying he was Jewish is like saying I am: true from a cultural perspective, but his relationship with religion was quite complicated. The complications are explored in much of his music.

Adds Geller of Cohen’s musical output: “He has these lyrics and very beautiful musical arrangements that step out of time and can last and be relevant for audiences of all ages.”

And if you only know that one song, please do yourself the favor of discovering more of his work. You can thank me later.

As usual, I can't let an entry on music go by without including a video. Since by now everyone who's still reading this has heard at least one version of Hallelujah, allow me to provide a different Leonard Cohen song. With any luck, it's one I haven't linked in here before.

Ah, you loved me as a loser, but now you're worried that I just might win
You know the way to stop me, but you don't have the discipline
How many nights I prayed for this, to let my work begin
First we take Manhattan, then we take Berlin

4 Like • Comment

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