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:

Wordsmitty ✍️

about 6 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 9 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 11 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

about 23 hours ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

Lilli Munster 🧿 ☕ 🎃

Sep 24, 2024 at 9:10am
In response to "All Out of Louvre"

Have a wonderful time in France. Have you run into Annette in France yet? She's there too!

2 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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