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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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.

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September 28, 2024 at 2:17am

#1077370

To the East

Most of yesterday was taken up by traveling. To get from Tours to Dijon, apparently there's no direct way, or at least not one that gave kickbacks to my travel agent. So it started with me taking an Uber from my hotel to the Tours train station, because now I know better than to roll luggage for that mile. Then the TGV back to Paris, during which, for some reason I absolutely cannot comprehend, my assigned seat was right next to someone else... while there were maybe 2 other people in the entire car.

But the real fun was when I got back to Paris and met the chauffeur for the ride from Gare Montparnasse across the Seine to Gare de l'Est, which is the station with the train to Dijon. The chauffeur didn't speak very good English or French, but he spoke excellent Ukrainian. At least I assume he did; I know almost nothing about that language. But he was very good at pointing out the Paris sights along the way, in some version of English with a Ukrainian accent.

"There is Tour Eiffel. And this building on left?" He indicated one with scaffolding. "Ukrainian mob has construction contract." We drove a bit further. "There on right is Notre Dame. And over there, building by Ukrainian mob." I wanted to ask him if the mob also did the Notre Dame reconstruction, but I knew better. He pointed out other important Ukrainian construction jobs along the way, too.

Now, look. It's entirely possible that "mob" was a bad translation of something else in Ukrainian. Something innocuous like "Fine Upstanding Ukrainian Construction And Renovation Company And Not Mob At All We Promise And Pinky Swear." But I choose to believe that my chauffeur, who was about my age, was an actual member of the actual Ukrainian mob and did the chauffeur thing as a day gig.

Just in case, I gave him an enormous gratuity.

As Skinny Pete said in The Italian Job, "If there's one thing I know, it's never to mess with mother nature, mother-in-laws, and mother-freaking Ukrainians."

After that, the train (the slow kind) to Dijon was a letdown. Hell, the rest of the trip might well be a letdown.

Things picked up again, though, once in Mustard City. The hotel, this time, is like 100 meters from the station. It has a lovely little bar. I had dinner elsewhere, but came back for the bar, where I got acquainted, for the first time, with a regional (the east of France) custom: Picon bière.

Mostly, they drink Belgians here, because, as I've said numerous times, traditional French beer is ass (also, it's wine country). But apparently, it's not flavorful enough for them (QUOI?!) so they add a shot of a kind of orange bitters called, you guessed it, Picon.

And whaddayaknow, it's actually a pretty good combination. Sadly, Picon isn't generally available in the US, so I'll have to enjoy it while I'm here and then smugly brag about it every time I drink a Belgian beer back home.

8 Like • Comment • View 1 comment

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