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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10 Most Recent Blog Comments:

Wordsmitty ✍️

about an hour ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 4 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

June 21, 2024 at 11:10am

#1072978

Cool Rockin' Daddy

When I found today's article, it was like my cat finding the stash of catnip I'd left for her.

Red, White, and Misused: How “Born in the U.S.A.” Became an Anthem for Everything That It Wasn’t
On the deep, rich history and many misinterpretations (accidental or otherwise) of the Bruce Springsteen classic, which turns 40 on Tuesday

Except for the unfortunate subhead, that is. "On Tuesday," forsooth. For context, the article is dated June 4 of this year.

Well, that, and the article ventures into territory I try to avoid in here, so it took me a while to decide to put it on my list.

As this is a fairly long article, though, I'll just highlight a few key (to me) parts.

The “Born in the U.S.A.” video is a lament and a tribute. Mixed in with... 16-millimeter concert footage of a denim-draped Springsteen belting out the song, there are handheld shots of factories, construction workers, an amusement park ride, a check-cashing store with a long line, soldiers, a small Asian American child, and a military cemetery.

I'm not sure I ever saw that video. Not then, when I was in college and didn't have cable TV; not later, before MTV turned to shit and I lived in an apartment and still didn't have cable TV; not after the internet and YouTube somehow made music videos widely available on demand.

I know, seems strange, right? You'd think I'd have searched out and viewed every single Springsteen video, but that turns out not to be the case. I did see the famous "Dancing in the Dark" video (the song is from that same album), several times, though never because I sought it out.

It's not like I have anything against music videos, as should be obvious; it's just that, for me, the music stands on its own terms, sans visual interpretation.

And of course I like the song. That summer of '84, I was doing hard physical labor, outdoors, with a land surveyor. That summer job gave me sunburn, poison ivy, yellow jacket stings, mosquito bites, ticks, muscles, an appreciation for desk jobs, exhaustion, and not a lot of money, but I'd get up before dawn as required by the boss (not the Boss; the surveyor boss) and limp back home after dark. Never called in sick, never shirked. Except for the day Born in the USA came out; that day, I took off so I could go buy the LP at the music store, as was the custom at the time.

The misuse of “Born in the U.S.A.” has been so blatant that it’s distracted us from its message. As Nietzsche put it, “The text has disappeared under the interpretation.” The misinterpretation is so glaring and has gone on for so long that it’s still a punch line.

I honestly didn't even expect there could be misinterpretation. Some of his songs, sure; but that one? The message was, to me at least, pretty damn blatant. We're not talking about mumbled lyrics, something Bruce has been known to do, or mondegreens, like "cut loose like a douche" from his "Blinded by the Light." No, it's straight-up "my country has failed me." Well, more nuanced than that, but the point is, it's no "God Bless America." (barf)

But, again, this was 1984. War is peace. Freedom is slavery. Etc.

The article veers into the political sphere, albeit from a 1984 perspective. The year, not the book. It was, of course, an election year, and Reagan would go on to win a second term in what turned out to be the biggest electoral landslide (I'd even call it an avalanche) in history. As the article notes, at one point in his campaign, he famously misinterpreted the song we're talking about today. But politics is something I try not to harp on in here, especially shit that happened 40 years ago. Nevertheless, this paragraph in the article stood out for me:

Springsteen spoke out again the next night, this time without mentioning Reagan by name. “There’s something really dangerous happening to us out there,” he told the audience. “We’re slowly getting split up into two Americas. Things are getting taken away from the people that need them and given to people that don’t need them.”

As big a fan as I am of the guy's music, I don't think he has special mystical powers (beyond an abundance of creativity and talent). But that quote was so spot-on, so prophetic, that I started to think he might have been channeling Madame Marie.

Madame Marie was a fortune-teller in Asbury Park. Bruce ended up immortalizing her in his song "4th of July, Asbury Park (Sandy)" from his second album around, oh, 1973 or so; don't make me look it up. She, of course, wasn't actually immortal, but lived into the 21st century. The story goes that, at some point before he released albums, she told Bruce he would become a famous musician. At the time, you couldn't swing a guitar in Asbury Park without hitting a musician; reportedly, Bruce shot back with something like "I bet you say that to all the guys."

She ended up being right, of course, in that particular prophecy, perhaps by coincidence, or perhaps by the shotgun effect. Or, I don't know, maybe she actually heard him perform and knew talent when she saw it. That Bruce quote, though, seems more like genuine insight. Enough like genuine insight to verge on the prophetic.

But you can read the article for the political stuff if you want. I have more musical commentary.

Born in the U.S.A. was released on June 4, 1984. It was the poppiest record Springsteen had ever made. The first single, “Dancing in the Dark,” reached no. 2 on the Billboard Hot 100 chart and is still the biggest hit of Springsteen’s career.

I gotta disagree with the "poppiest" description. I mean, okay, sure, compared to his previous six albums, maybe. But again... this was 1984, in the throes of a worldwide epileptic seizure of synthpop and electronica. The biggest hit of the year was Prince's "When Doves Cry," which I won't say anything bad about from a creative perspective, but it, like most of the "hits" that year, suffered from overproduction. The only song from BitUSA that I'd describe as "pop" in the context of 1984 was that one, "Dancing in the Dark."

At one point, I can't remember when, I heard Bruce talking about the making of BitUSA the album, and he said something like "most of the songs were done in two or three takes. Not a lot of editing. Except for maybe one." Well, I knew right away what that "one" was. It stands out like a broken middle finger on an album that's otherwise just two steps away from a live performance.

I'm not saying it's a bad song, mind you. I'd never say that about any Springsteen song. Just that when I first heard it in the context of the whole album, and yes, I remember this clearly from 40 fucking years ago, my thought was, "Someone told him he needed a pop hit."

“‘Dancing in the Dark’ was grabbed on to immediately,” says Carter Alan, who back then was a DJ at Boston’s WBCN, one of the country’s most influential rock stations. “A lot of [Born in the U.S.A.] was danceable. And that’s a very danceable song. I mean, they made a 12-inch dance mix out of it.”

Of all the qualities that make me like a song, "danceable" isn't even in the ballpark. I care primarily about lyrics (which shouldn't be too surprising in a writer). Sure, the music's important, too, but as much as I've tried, I just have no musical talent whatsoever, so I can't always articulate why I like a song musically. Sometimes, though, I get the impression that I can write words and explain why I appreciate what others write.

No surprise, then, that one of the things that does make me enjoy "Dancing in the Dark" is the line: "I'm sick of sittin' 'round here tryin' to write this book."

To Alan, that message was obvious. “Some singers mumble, but Springsteen’s all about the words,” he says. “And he makes damn sure you can understand them. He doesn’t bury himself in the mix. There’s a lot of sound and noise going on in that song. I mean, those drums are like cannon shots. But you can hear everything he says.”

Um... have you heard Springsteen? Hell, the only way I learned what he was actually singing sometimes was via the liner notes. And sometimes, even those were wrong.

But yes, I think he tried a lot harder to make himself understood on BitUSA.

So, I usually follow up a music entry with an appropriate video (with a few exceptions, like my Total Eclipse of the Heart entry a couple of days ago). As I said up there, I don't recall ever seeing the BitUSA video, though I'm sure I've listened to the song over a thousand times, including live performances. Consequently, what could be more appropriate than to post it right now so that I can see it for the first time?

There are approximately 45,000 versions on YouTube, though. I hope I picked the right one.

Down in the shadow of the penitentiary
Out by the gas fires of the refinery
I'm ten years burnin' down the road
Nowhere to run, ain't got nowhere to go

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