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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Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

about 2 hours 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 5 hours ago
In response to "Gray Matters"

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

1 Like

tj-Merry Mischief Maker

about 24 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 19, 2024 at 11:40am

#1072895

Matters of the Heart

Way back before the solar eclipse, I saved this Atlas Obscura article to my queue. But the topic itself is timeless, so there wasn't any rush, and the Random Number Gods saw fit to wait until today to bring it forth.

‘Total Eclipse of the Heart’ Was Almost a Meat Loaf Song
Bonnie Tyler’s classic rock epic nearly got the ‘Bat Out of Hell’ treatment.

Hardly the usual AO fare; they usually concentrate on place oddities (hence the name) or food. They were doing special "eclipse" articles, and this was one of them. Everything here tracks with stuff I already knew, but I didn't fact-check it further.

When Bonnie Tyler’s husky, powerful “Total Eclipse of the Heart” hit the radio in 1983, rock ballad fans around the world fell in love.

Well, some did. Some hated it. Still do. I can always tell if someone is worth hanging out with based on their feelings about that song alone.

But for one musician, the history of the tune, which was written by composer Jim Steinman, is as torn as the dark love ballad’s lyrics. Meat Loaf was the one who was supposed to have sung that song.

Confession time: I possess, and have read, a copy of Meat Loaf's autobiography (unsigned, unfortunately). But I can't remember whether I knew the above from that or through other sources over the years. But the way I see it, ML was famous anyway, still one of the best-selling performers in history; and it boosted Bonnie Tyler, so no real losers here. As the article notes later, ML didn't seem to hold any of this against Tyler.

Meat Loaf had long worked with songwriter Jim Steinman, including on his hit 1977 album Bat Out of Hell.

What I did get from the book, I think, was that that album was supposed to be a collaboration between singer and songwriter, not purely Meat Loaf with the writing credit added as an afterthought. (Many of the session musicians on it were also members of the E Street Band, adding to its awesomeness.)

Meat Loaf’s next planned album, the never-made Renegade Angel was poised for recording, and was supposed to include the track “Total Eclipse of the Heart.”

There are at least two things in music history that almost happened that I wish had happened, and that's one of them. The other also involves Jim Steinman—he was meant to collaborate with Andrew Lloyd Webber on a stage production. Just as well that the latter didn't happen; Broadway would have been sucked into a black hole of melodrama.

Amid mysterious and abrupt problems with his voice and a psychologically damaging aversion to fame, Meat Loaf’s success and partnership with Steinman began to slowly unravel, forever changing the sound and trajectory of the ballad “Total Eclipse of the Heart.”

It changed a lot of trajectories, really. And that sentence glosses over a lot of drama. The article dips into this a bit, then:

Meat Loaf, under contractual obligation, went ahead with his next record, Midnight at the Lost and Found, a collection of songs written by various songwriters, which Meat Loaf later came to regret. (Steinman meanwhile released songs that he initially thought up for Meat Loaf on his own solo album, Bad for Good.)

I bought that Steinman album on LP when it came out. I also acquired the CD several years later. I like it. It's not for everyone. Steinman's voice just didn't work in the same way Meat Loaf's did. Most of the tracks on it later showed up on different Meat Loaf albums, after they reconciled, and the difference is plain if you hear both versions of a song.

But that's kind of a sidebar; the point is, Steinman was a prolific songwriter, but lacked the performance talents needed to turn his ideas into popular music. For that, he needed other talent. Preferably people who could pull off melodrama, like Meat Loaf... or Bonnie Tyler.

This Rolling Stone article

has more on that.

So back to the actual topic of the article (and this entry):

The song’s music video, which featured prep school boys with glowing eyes, gymnasts, dancing men in leather jackets, and lots of floating fabric, also spawned a Literal Video spoof in the early 2000s that highlights the super weirdness of the visuals.

I've posted that Literal Video in here before, so I don't want to do it again. It's still up on YouTube, attesting to not only the power of the original song and video, but to the wonder of parody. Yes, I can like an original and also appreciate the parody.

Despite the song's title, though, it wasn't really part of my eclipse soundtrack. No, that was mostly Pink Floyd.

1 Like • Comment • View 2 comments

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