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 12, 2024 at 8:39am

#1078145

A Few Observations

Tomorrow, I intend to get back to my usual annoying self here by posting a Revisited entry. As for today, well, I'm still in recovery, so I'll just summarize some of my observations on the trip that I may or may not have already blogged about (because I can't remember and can't be arsed to re-read what I've already posted).

I gained a greater appreciation of the intricacies of French wine. Contrary to French beliefs, it's not necessarily superior to other countries' wine, but it does take a different approach. Not that I know a whole lot of details; that takes years of study and practice. I don't mind practicing.

It's popular in some circles to tout the general superiority of Europe over the US. In other circles, the opposite is the case. As it turns out, in my opinion at least, that they're just different. Europe does some things better, like public transportation and healthcare delivery. The US does other things better, like... um... well, I can't think of anything offhand, but I'm pretty sure I would if I weren't so tired. I guess we invented the Internet and GPS, so there's that.

There are whole sections of Brussels where there are more chocolatiers on each block than all other businesses combined.

There are others that are pretty much entirely restaurants.

I gather that it is easier and takes less time to become a medical doctor anywhere than it does to become a chocolatier in Belgium.

Real estate prices (both rent and buy) are amazingly cheap there, compared to the US. At least in the country and small towns. And that's with a denser population.

Unlike my above comment about French wine, Belgian chocolate really is the best. Sorry, Switzerland. At least you won't send assassins after me for that observation. ...Right?

The old joke about how Brits think 100 miles is a long way, while Yanks think 100 years is a long time? It applies to mainland Europe as well. You just have to convert to SI units.

Best quote of the trip, by a Belgian tour guide, uttered between bites of a Belgian waffle topped with whipped cream and chocolate: "You Americans eat like you have free health care."

So that's all I'm going to post about the trip... unless, of course, something else reminds me of a thing I did there. Like I said, tomorrow, we're back to our regularly scheduled programming.

2 Like • Comment • View 4 comments

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