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Friday, Nov. 22, 2024

2:47pm EST

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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:

Robert Waltz

a day ago
In response to "Powers of Three"

🌕 HuntersMoon

- that's pretty much what quantum computing is, from what I've read.

2 Like

Brandiwyn🎶

Nov 20, 2024 at 2:32pm
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

Nov 20, 2024 at 1:20pm
In response to "Powers of Three"

With binary, it's either on or off. Wouldn't ternary introduce Schodenger's Cat? On, off, or both?

2 Like

Turkey DrumStik

Nov 19, 2024 at 7:43pm
In response to "Paramount Purpose"

I suspect old web filter attempts

would have combusted when confronted with this article.

1 Like

tj-turkey-jobble-jobble-hard-J

Nov 19, 2024 at 4:04pm
In response to "Losing Weight Is Easy"

Some of them break out in hives if you ask them to add 10% to something.

Yesterday I purchased an item that cost $6.42 after tax was affixed by the register. I handed the cashier a ten dollar bill and watched as she placed it in the cash register, which like most now days, showed her how much change to return. She had to try three times to count out the proper amount, and as I was leaving commented to the bagger, "Why do people have to pay with cash, don't they know how hard it makes my job?"

I didn't say anything, but was trying desperately not to break out in hysterical laughter.

1 Like

🌕 HuntersMoon

Nov 19, 2024 at 9:28am
In response to "Paramount Purpose"

At your age, Bob, take heart in the fact that it can be restored...

2 Like

Wordsmitty ✍️

Nov 18, 2024 at 12:34pm
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

3 Like

Brandiwyn🎶

Nov 18, 2024 at 11:27am
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

Brandiwyn🎶

Nov 18, 2024 at 11:19am
In response to "Losing Weight Is Easy"

Has anyone ever told you how annoyingly pedantic you are?

Damn your pedantic ways. I was hoping to justify a vacation to Venus (which I still argue would be a much better holiday destination than fr-fr-freezing Mars, Elon. Ugh.) Now I guess I'll have to settle for the Cancun plans we already made, since my mass (and waist diameter, and thigh diameter) will be exactly the same regardless.

3 Like

tj-turkey-jobble-jobble-hard-J

Nov 17, 2024 at 4:04pm
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

November 22, 2024 at 8:03am

#1080330

Quoth the Singer, "Evermore."

Mostly, I'm just pleased that there exists a magazine called Far Out.

Which war was Led Zeppelin’s ‘The Battle of Evermore’ about?

Really, I didn't think this was much of a mystery.

Led Zeppelin’s fourth studio album might be most famous for the all-out hard rock of opening tracks ‘Black Dog’ and ‘Rock and Roll’, as well as the slow-burning firepower of ‘When the Levee Breaks’, and, of course, the ethereal majesty of ‘Stairway to Heaven’.

They apparently used up all their creative energy on the songs, because they never bothered to give the album a name. I guess, technically, the album name is Led Zeppelin, but most of us called it Zeppelin 4.

But ‘The Battle of Evermore’ is an underrated favourite for many fans, as well as the band’s singer Robert Plant.

It's absolutely near the top of my list.

It sounds as though ‘Evermore’ could suit a fictionalised or dramatised narrative of Agincourt or Bosworth Field, but that’s not what’s going on here. Zeppelin’s lyricist has taken his inspiration from the realm of pure fantasy, as compelling as that realm appears in this context.

That fantasy drew from reality, so it's not unreasonable to think that a real-life historical battle may have been the ultimate inspiration.

Plant didn’t just suck this legendary battle out of his thumb, however.

I'm just including that line here because I wish I'd come up with that particular turn of cliché.

He draws heavily on Tolkien’s high-fantasy novel The Lord of the Rings, including specific details in certain lines of the lyrics that indicate exactly which battle he’s referring to.

And somehow, I realized that even before I read Tolkien. Perhaps someone told me of the connection.

Lord of the Rings fans pinpoint Plant’s descriptions as a lyrical account of the Battle of the Pelennor Fields, which serves as the climactic battle in the War of the Ring during the third volume of the novel.

And that's a level of detail I'd never even considered exploring. Plant's lyrics on Zeppelin songs were always good examples of "This feels like it could be a metaphor, but for what?" It's probably more true for Stairway, which, incidentally, was the only song on the album with printed lyrics included, so I could only assume that there was something important there that he was trying to convey. I finally figured out what it was, decades later.

But Evermore was always pretty obvious: he was channeling Tolkien.

The legendary island of Avalon, the place where the sword Excalibur was forged in Arthurian legend, which also appears in the works of Geoffrey Chaucer, gets a mention, too. The frontman seems to have played around with mythical imagery without much care for its relation to any pre-existing narrative structure.

Okay, maybe not that obvious. Still, I thought the reference to Avalon in the song was mostly because it scanned well.

Normally, when I do an entry about a song, I'll include the video. But there's little need, as there's one at the linked article.

5 Like • Comment

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