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 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🎁

about 24 hours 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 9, 2024 at 1:36am

#1077983

What a Pisser

Lots of cities have a recognizable symbol, an icon which represents the city's, and sometimes the country's, character. Paris has that Tower. Sydney has the Opera House. Washington has the Phallus- errr, I mean Monument. Hell, even my city has one, the Rotunda at UVA.

Brussels, ancient economic powerhouse, capital city of the European Union, world culinary treasure... Brussels has the Manneken Pis.

Like the Mona Lisa (reportedly), the statue/fountain's size can be shockingly disappointing to first-time visitors. As that link notes, it's a bit over half a meter in height, and it's set in a relatively large fountain so it seems even smaller.

Please feel free to make your own jokes and insinuations about that last paragraph. I'm too tired to bother.

The first connection one might make is to those pissing-Calvin stickers rednecks used to put on the back window of their trucks. Usually, if it was a Ford, it'd be pissing on the Chevy logo; or vice-versa. This is different, though I suppose that if it were mischievously pissing on anything, it'd be France. No, the Manneken Pis is more heroic than delinquent.

Anyway, you can read more about that that at the Wiki link above, so I'm not sure why I'm bothering with the background here.

According to my guide, they used to replace the water in the fountain with beer during their beer festivals, which, like the Manneken Pis itself, is a very Belgian kind of thing. I suppose the only more Belgian thing to do would be to replace it with liquid chocolate, but I'd imagine that'd be even harder to clean up, later.

When I was there, the statue was dressed up as a little graduate, because Brussels is having a graduation ceremony this week. Why Brussels schools graduate people in October, or exactly which schools are involved (high school or college or whatever), I never did find out and can't be arsed to look up. But this is the condition in which I found it.

6 Like • Comment • View 3 comments

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