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 11 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

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?

July 25, 2024 at 12:13pm

#1074374

Let Me Give You a Hand

Wrapping up this round of "Journalistic Intentions"

[18+] with the final randomly-selected prompt:

Antwerp

And seriously, I used a random number generator. It's pure chance that it landed on a city in Belgium at a time when I'm planning a trip to France and Belgium.

And I'd love to visit, really, I would. The city hosts the Bollekesfeest every year, which features local beer; if you need to know anything about me at all, it's that Belgian beer is my Kryptonite. Sadly, the event takes place in August, and the timing doesn't work out for me. And it's home to the Antwerpse Brouw Compagnie, which I've heard brews a beer called Seef or Seefbier whose origin dates back to the 16th century. For context, that was before European colonization of the Americas. Hopefully the beer itself isn't that old, but damn, it sounds like something I need to try. (Their more modern brews look awesome, too.)

The problem is I don't have unlimited time. Tant de bières, si peu de temps. Yes, that's French, though Antwerp is in the Dutch-speaking part of the country. Hell, it's practically Nederland, another country I'd love to visit. My primary interest in Belgium this trip is their famous abbey breweries. Monk beer. So, Brussels and maybe Bruges. I think. Yes, I'm aware that Brussels and Antwerp are quite close together, maybe 40km; I'm perfectly capable of reading a map. There's still not enough time to do everything.

Besides, I gotta save some cool stuff for the next trip.

And no, I still haven't picked up much of the language. I tried. It's harder than French, because it's simultaneously very similar to English and very, very different. All of the Dutch-speaking people I know have perfect English, and it's embarrassing that I can't seem to return the courtesy. Well, maybe not "perfect" English (because such a thing does not exist), but still better than that of most Americans.

As you know if you've been following along, one of my other interests is etymology, and the origin of the name "Antwerp" is debatable. But, like Rome, in addition to the actual etymology, it has a mythological etymology, and it's glorious.

I'll just let Eupedia

relate it:

The legend says that the name 'Antwerp' comes from the Dutch "hand werpen" meaning "hand thrown".

The story has it that a mean and nasty giant controlled the river traffic, demanding exorbitant tolls. Those who refused to pay had one of their hand cut off.

But one day, a young and brave Roman soldier named Silvius Brabo managed to kill the giant. He cut the giant's hand and threw it in the Scheldt River, giving the city its name.

I need details, dammit. How did Silvius Brabo defeat the giant? More importantly, how did he keep the giant from landing on him when he killed it? I did find that the giant's name was Antigoon, which somehow doesn't mean "he who fights against goons." And is this the first example of classic rent-seeking

behavior and its inevitable consequences?

Sadly, the likely actual origin of the name is far less interesting: "The name comes more probably from the word ‘aanwerp’ (alluvial mound’), which describes the first settlement's geographical feature." (Same Eupedia source as above.) Yeah, apparently "werp" served double duty, much as its English descendant "warp" does. Though one could make the argument that the alluvial soil was "thrown" there by the river, I suppose.

And now I want to visit Antwerp more than ever. Dammit. I don't have infinite funds, either. Maybe I can start charging tolls.

5 Like • Comment

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