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 21 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 3, 2024 at 12:09pm

#1077683

Beaujolais

The weather decided to give me a break, just in time for another wine tour out in the countryside.

Yesterday's excursion took me into the nearby wine region of Beaujolais, source of one of my personal traditions.

I've talked about it enough in here before, but, in brief: Beaujolais Nouveau is released on the third Thursday of November, just in time to ship it to the US for drinking with Thanksgiving meals, which it accompanies very well. I try to get a bottle every year.

This is not, of course, the only wine produced in the region, and their more traditional offerings are very good.

But the funny thing I found out on this trip is: the French consider it a trash wine. Which probably explains why the US gets so many bottles in November; they unload their stock on us so they don't have to drink it themselves. Every year when it's released on Beaujolais Nouveau Day,

though, they get together and party to drink the stuff and complain about how bad it is. And probably to laugh at the Americans who they can pass it off onto, but for some reason none of the tour guides I talked to would admit to that.

Thing about it, though, is the wine may be crap, but it's cheap crap, and to me, the price of the stuff figures into the taste. So yeah... I like it. Deal with it, France, because I have to deal with beer drinkers claiming to love Bud Light.

In other news, today is my last full day in France, but I'll talk more about today's events tomorrow, which is when I'm scheduled to get on a train to Belgium. I'm of the considered opinion that no trip to France is complete without at least one decadent, elegant, crazy expensive formal dinner. So I had mine last night, in the restaurant on the top floor of my hotel.

The only thing more boring than hearing about someone go on about their dream last night is hearing them talk about their decadent and crazy expensive French dinner, so I'll spare you most of the details, only to say it was everything I expected and not nearly as expensive as I thought it would be (and that neither snails nor frogs were involved). I accompanied it with a bottle of Pouilly-Fumé, which is not from this region, but what the hell, I like it and it was on the menu.

When I say bottle, I mean bottle. Damn right I polished it off.

Today's vacation photo was from an overlook where we stopped on the tour for a glass of wine. You will note that it's not raining, and that the view was incredible.

10 Like • Comment

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