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:

tj-Merry Mischief Maker

about 9 hours 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

a day 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

September 30, 2024 at 3:01am

#1077469

Owl Miss This Place

For my final full day in Dijon, I didn't have anything planned out—no wine tours, no mustard discoveries, not even a "let's meet for lunch" (because unlike Paris and, by lucky chance, Tours, I don't know anyone here). So, as the day was fine and sunny and neither hot nor cold, I took a walk.

Dijon, like most European cities, is really quite incredibly old (from a US perspective). And, as with other cities, there's only so much one can do in a day. Turns out that Dijon has a self-guided walking tour embedded into it. Literally; there are brass plaques with engraved owls

on the sidewalks, pointing the way.

Why an owl? Well, on the side of the cathedral here (also named Notre Dame, because Catholics weren't very original) perches a small stone owl, barely recognizable as such from the accumulated wear of people touching it. Supposedly, touching it (but only with one's left hand) will grant a wish. Apparently, no one has yet wished for "restore the Notre Dame owl to its original carved glory." Or maybe they have, and it's just another silly superstition.

Following the trail eventually gets you to the owl, and then it moseys on along through other historic sites. It's like following the quest markers in a video game, which made me want to play a video game, but that can wait until I get home again.

I didn't touch the owl. For one thing, it's a silly (if mostly harmless to everything but the owl itself) superstition. For another, even if there were something to it, making a wish is tantamount to acknowledging that there's something missing in my life, and there isn't. For yet another, over half a century of reading and writing stories, and playing fantasy role-playing games, has instilled in me a deep, abiding distrust of wishes. And, for a final thing, some beggars were approaching with their cups out, so I hoofed it out of there, continuing to follow the brass owl arrows.

These led me next to a mustard shop, on the same street (Rue de la Chouette, naturally), where I finally got to see the thing Dijon is most famous for and holy condiments, Batman, there's a lot of mustard there.

At some point, though, around the Ducal Palace, I lost the trail and never did pick it up again, despite looking and even trying the internet. All I could discover was that it was not, in fact, the end of the trail. Oh, well.

Much later, in the evening, I returned to the palace area (Palais des Ducs) and its accompanying semicircular courtyard, Place de la Libération, which is surrounded by cafés, bistrots, restaurants, bars, and brasseries, the differences between which are subtle and very French. (Brasserie also means "brewery," but not all brasseries are actually breweries, much to my disappointment.)

Point is, I went back there for dinner because it's pretty, and I sat outside with a beer and a burger as the sun set and the Palais des Ducs lit up, and the shifting, multicolored lights in the Place's fountains illuminated their waters.

This sort of thing is one of the reasons I wanted to go to France.

9 Like • Comment • View 2 comments

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