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Monday, May. 6, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

🌕 HuntersMoon

about 21 hours ago
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

a day ago
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

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.

4 Like • Comment • View 3 comments

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