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

October 4, 2024 at 1:59am

#1077735

Coats Do What Now?

For my final full day in France, I took a small-group tour to the wine region of the Côtes du Rhône.

In contrast with the previous day's trips, I wasn't very familiar with the Côtes du Rhône vintages. This, of course, is why I went. I find that, when traveling, some mixture of the familiar and the new is ideal, though I expect the proportions would differ for everyone.

While I don't think I'd ever had a wine from this region before, some of the grapes they're based on are fairly popular to grow in the US.

The tour group itself was pretty interesting. Some of us ugly Americans, a couple from Quebec, and three people from Australia, who, while initially distrustful of my Americanity, quickly warmed up when I assured them (out of earshot of the tour guide) that I very much enjoy Australian Shiraz. While the guide insisted that Syrah, one of the grapes in Côtes du Rhône, is not the same thing as Shiraz, it totally is; it's just grown on very nearly the exact opposite side of the world.

The tour guide himself, who looked a bit like Kiefer Sutherland, was otherwise very knowledgeable, and spoke good enough English that all of us varied Anglophones mostly understood him.

The problem comes in when some unknown place name, or grape name, pops up in spoken French.

Like, if someone comes up to me and says "Bonjour, peux-je vous aider?" I can work out what they're saying. But if I'm at a wine tasting, even if it's mostly in English, it might go something like this:

"This wine comes from the pl***du*** region, near B***on and the S**** river valley. If you visit there, be sure to go to D****de*****, a fine restaurant that serves bl*****, v*******de la *******tion, and many very good wines, including Petit ********tre, Vo******** Blanc, and sparkling Dr*******"

It's tiring, to be honest. I started asking for spellings. To make things worse, my hearing isn't that great thanks to 50 years of rock concerts (no regrets), and I'm not asking you to repeat yourself because I don't understand French; I'm asking you to repeat yourself because I literally can't hear you over the ambient mumbles.

Ah, well, soon I get to complain about the same thing in Dutch (or at least the version of it known as Flemish; I don't know the details there), as I'm leaving for Bruges today. It's a tight schedule: catch a train to Brussels, transfer to a train to Bruges, get to the hotel from there somehow (Uber seems to be wonky in Belgium; apparently, only taxi drivers can do Uber, thus defeating the whole purpose but providing job security), then find a certain statue in a city saturated with statues to meet a beer guide who will take me... well, I don't care, as long as there's Belgian beer.

I'll wrap up with another landscape picture, this one of the Rhône Valley.

6 Like • Comment • View 2 comments

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