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 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.

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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.

April 9, 2021 at 12:02am

#1008074

There's a Blaze of Light in Every Word

This one's been languishing in my queue for quite a while now.

The Many Lives of Leonard Cohen's "Hallelujah"

Let me just start by saying this: If you're "tired" of the song, if you're sick of hearing about it, if you're weary of its many versions... I don't want to hear it. There are few perfect songs out there, and this is one of them. I will never get tired of hearing it, and I will never get tired of hearing about it. You can just stop reading here if that's the case; chances are tomorrow I'll have something else to talk about (though I should note that I have at least one other Cohen article in the waiting room).

That said, Bono's version of it straight-up sucks.

Anyway, if you're still with me:

In the late 1970s, Leonard Cohen sat down to write a song about god, sex, love, and other mysteries of human existence that bring us to our knees for one reason or another. The legendary singer-songwriter, who was in his early forties at the time, knew how to write a hit: He had penned "Suzanne," "Bird on the Wire," "Lover, Lover, Lover," and dozens of other songs for both himself and other popular artists of the time.

I was first introduced, in my teens, to Leonard Cohen as a poet. It was only much later I discovered he did songs, too. The lyrics reflect the poetry. Hell, at least one of them started out as a poem.

I consider myself extremely fortunate to have seen him in concert twice before he died, once in New York City and once in Las Vegas. And he was one of the many musical casualties of 2016, overshadowed in that (as usual) by more well-known artists such as David Bowie and Prince.

I say all this because I want to emphasize that it was not "Hallelujah" that brought me to his music, but rather vice-versa.

In the 35 years since it was originally released, "Hallelujah" has been covered by more than 300 other artists in virtually every genre. Willie Nelson, k.d. lang, Justin Timberlake, Bono, Brandi Carlile, Bon Jovi, Susan Boyle, Pentatonix, and Alexandra Burke...

Brandi

Carlile does a completely amazing version of it. It's on her live album if you're interested. But none of them that I've ever heard actually suck -- except Bono's.

With its synth-heavy ’80s production, Cohen’s version of “Hallelujah” doesn’t announce itself as the chill-inducing secular hymn it’s now understood to be.

And I will admit that other artists did a much better rendition than the original. Such is the case with many of his songs. His voice is... an acquired taste, like Tom Waits', or gin.

“I wanted to push the Hallelujah deep into the secular world, into the ordinary world,” Cohen once said. “The Hallelujah, the David’s Hallelujah, was still a religious song. So I wanted to indicate that Hallelujah can come out of things that have nothing to do with religion.”

Which is probably one of the reasons the song has always appealed to me.

Cale’s pared down piano-and-vocals arrangement inspired Jeff Buckley to record what is arguably the definitive “Hallelujah,” a haunting, seductive performance found on the late singer-songwriter’s one and only studio album, 1994’s Grace. Buckley’s death in 1997 only heightened the power of his recording, and within a few years, “Hallelujah” was everywhere.

It's really the Cale / Buckley version that has spawned most other covers, including Brandi

Carlile's. I'm pretty sure a lot of people still aren't aware that Cohen was the songwriter.

And yet, I've never fully been able to put my finger on exactly what the lyrics mean, metaphorically. I mean, obviously they have meaning; they're evocative, and they come from a truly amazing poet. This article ended up helping me with that, which is one reason I'm linking it here:

Putting aside all the biblical allusions and poetic language, “Hallelujah” is a pretty simple song about loving life despite—or because of—its harshness and disappointments.

And I don't think I can sum it all up better than the article does, in Cohen's own words.

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