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🎁

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 11, 2024 at 7:33am

#1078100

Beersday

As expected, my sleep schedule is all screwed up now, and it might take me a few days to get back to my normally-screwed-up Eastern Time schedule.

Fortunately, I don't have a lot of requirements for being at certain places at certain times, so I'll take as long as I need to recover. I do have a role-playing game scheduled for tomorrow evening, as is the case for most Saturdays when I'm not traveling, but that's about it.

One of my cats remembered me right away. The other remembered me after I gave her some food. "Oh, yeah, you're that human."

The true miracle of the trip, though:

When I did the Bosteels tour, as part of it, they gave me a Tripel Karmeliet boxed set—a bottle of that delicious beer, packed with two glasses (remember, most Belgian beers have dedicated glassware). There's nothing extra-special about the TK glasses; they're fairly normal tulip chalices, but with the beer logo printed on them.

The problem is, they're somewhat fragile.

My suitcases were already stuffed, so I spent the rest of the trip wondering how I was gong to get this boxed set home without everything breaking. Maybe I could mail back some clothes, so I'd have room? Mailing the beer to myself was right out, because of stupid laws. US Customs allows up to one liter of alcoholic beverage in checked luggage, and this bottle is under that limit, though of course you can't bring that quantity of any liquid in a carrion bag.

Well, I spent some time over there creatively rearranging stuff in my bags, and trying to ensure that the box containing the fragile, precious cargo was well-padded with dirty clothes on all sides (look, I wasn't about to waste time in Belgium looking for a laundromat). Still, I've done similar things before, a bottle of beer from England several years ago, and it came back all shattered. The beer soaked into the clothes, but I'm used to that and I was going to have to wash them upon returning, anyway.

So I wasn't holding out much hope. Between the rough handling of the bags, and the pressure and temperature differentials at flight altitude, I mentally prepared myself for returning home with a box of shattered glass and beer-soaked clothing.

But, somehow, the beer, and even the glasses, survived intact.

I should make a holiday of this, like how Hanukkah celebrates the miracle of the lamp oil lasting longer than it should have. October 10 will be Beersday for me from now on. It's even easy to remember, since it's 10/10 in both US and Euro formats, and it's also the birthday of one of my best friends.

Really, it's not even that big a deal, except that this was my only physical souvenir of the trip (like I said, suitcases were already stuffed, so no room for plastic Eiffel Towers or Manneken Pisses). Tripel Karmeliet is generally available in my area, albeit at exorbitant import pricing, and the glassware can be ordered online. I'm just glad my luck held out.

7 Like • Comment • View 6 comments

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