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:

tj-Merry Mischief Maker

about 20 hours ago
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🎁

a day ago
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

a day ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

a day ago
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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 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.

6 Like • Comment

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