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 9 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day 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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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

September 26, 2024 at 3:45am

#1077301

Two Castles and a Cave

My schedule yesterday was pretty much: go on a tour.

Now, I've been trying really, really hard not to make a joke about Tours tours. You have no idea how difficult that is for me. It's sitting right there; how can I not pick it up?

I've been unable to find a reliable etymology for the city's name, for which, incidentally, the s is silent. Wiki says there's a "popular folk etymology" that it's named after a Roman named Turonus, nephew of Brutus (but not that Brutus). I've long thought that it comes from the French word that translates as "tower" in English, as in La Tour Eiffel. Far as I can tell, though, there aren't any significant towers here. Perhaps there were, once. Hell if I know.

The point is, I'm trying really, really hard not to make a pun about it.

What I do know is that it's wine country. Pretty much everyone has heard of Bordeaux to the west and Burgundy to the east, as well as Champagne, which is located between Paris and Belgium. I think the Loire valley, or Val de Loire, is not quite as well-known internationally, though it absolutely should be.

Getting back to what I originally set out to write, the travel agent had arranged a small group tour to run outside the city. By "small group," I mean there ended up being three of us. Continuing my run of coincidences, the other two were a couple from Fairfax, in northern Virginia. The guide was this small French woman who'd prepared an entire booklet of illustrations, some quite amusing.

First stop: Château de Chenonceau. I'd never heard of it before this trip, but apparently it's the second most visited castle in France, after Versailles. The really interesting thing about it is that it straddles a river.

Like, the brief version is: someone built a castle on the bank of the river Cher. Later, wanting an easy way to get to the other bank, they built a bridge from the castle to the other side of the river. And still later, a massive three-story structure got built atop the bridge. The day was rainy and nasty, and I didn't get any really good pics of the outside, but you can see the bridge-castle here.

I was starting to gain an understanding of what might have led the French peasants to revolt in the 18th century.

Then we went to a cave.

Not just any cave, of course, but a wine cave. And yes, of course, a wine tasting was involved, complete with a brief tasting class led by a sommelier who supposedly passed on some of the "secrets" of his profession. I mean, they're not secrets anymore, now that I know them. What are they? I'm not telling.

After eating lunch (with wine) just outside the cave, we went on to the next castle: Chambord.

Sadly, it had nothing to do with the berry liqueur called Chambord. Or, really, with any booze. It was another example of the excesses of the French nobility, further deepening my understanding of why the guillotine was invented.

Still raining (which the sommelier had assured us was a Really Bad Thing for this year's grape harvest, because apparently it hardly ever rains in the Val de Loire in early autumn, and I didn't point out that this was probably my fault because I've been known to end droughts in California just by visiting the place), so again, none of my photos are very good and I'll defer once again to Wikipedia.

So, with all the crappy weather, what pic from this tour can I share? Oh, I know! This dude was hanging out in a stable at Château de Chambord. You might need to enlarge it, but I promise the white streak on the right side belongs to a horse.

6 Like • Comment • View 3 comments

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