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:

Wordsmitty ✍️

about 4 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 8 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 10 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 22 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, 2020 at 12:08am

#994223

O'zapft is!

Hey, this one's pretty easy for me.

PROMPT September 26th

It’s apple-picking season where I live and I’m so excited! Write about your favorite seasonal activities. What is something that only comes once a year, but you always look forward to?

I'm not a fan of Christmas trappings, and New Year's just brings fresh horrors. Thanksgiving does little for me, because my family is mostly gone and I won't cook a turkey for just myself. Halloween is kinda fun with all the spooky stuff, but it's not really a big deal for me, either. My birthday would be kinda cool, but it's in February, and I hate February. Easter isn't something I celebrate, and Pesach means schlepping to my few remaining family in New York City (not this year, though) and trying to find a parking spot in Manhattan on a holiday weekend. You'd think that being a would-be comedy writer, April Fools' Day would be my holiday, but I tend to spend it in my room under the covers, lest someone mistakenly assume that it would be funny to play a prank on me (it would not). Memorial Day, Independence Day, Labor day -- all just another day for me because I'm retired.

But there's one thing I look forward to every year, and -- you guessed it -- it involves beer.

I speak, of course, of Oktoberfest (which is really celebrated in late September because Germans are weird). (I can say that because my dad's side of the family was German.)

So it's going on right now, and the one thing I look forward to every year is getting to drink Oktoberfest beer, aka festbier, aka Märzen lager.

The story behind why Oktoberfest beer is named after the month of March is an interesting one, but I can't be arsed to look it up right now. As I recall, there was a period when Germany, or rather the nation-states that would later become Germany, forbade the brewing of beer between, like, April and August, or thereabouts. This had to do with refrigerators having not been invented yet. So they'd brew some beer in March and let it sit in barrels in caves or some shit until it was time for Oktoberfest (which itself started out as a wedding celebration for royalty).

Like I said, not looking it up, so I might have some of the details wrong, but that's the gist of it.

Anyway, point is, these days Märzen lager starts hitting shelves around late August, and I drink the hell out of that shit. My favorite is a local one, an Imperial version called Hansel and Kettle that's high-octane and delicious. The "Imperial" style name is borrowed from Russian Imperial Stout, which isn't actually Russian but British, and that's a whole 'nother story that I won't get into right now. It mostly just indicates that a beer has a higher ABV than usual, and that other ingredients have been balanced out to compensate.

But I'll try any festbier, if I'm at a taphouse or whatever, and they usually have a consistent quality to them because the recipe doesn't allow for too many variations.

Another thing that lights up what's otherwise a crappy season for me -- I know a lot of people like fall, but I despise cooler weather and prefer the sweltering heat of summer and, especially, its mild nights here -- is pumpkin beer.

But pumpkin beer varies wildly in quality. Some don't even use much pumpkin, just the ever-present fall staple of pumpkin spice. Some are way to sweet for my taste. Others are entirely too much pumpkin, not enough spice. But when I get a good one (Pumking from Southern Tier, out of New York State, another Imperial variant, is my favorite), I drink it. Still, it could go away and it wouldn't affect me much, as long as I get my Oktoberfest.

I have a vague desire to go to actual Germany for the celebration sometime, but it's usually really crowded there, and I don't do well with crowds, beer or no beer. Still, for the experience, it might be worth it.

So that's the big thing I look forward to. As the weather continues to turn against me, and Märzen becomes but a memory, I'll probably be drinking more Russian Imperial Stout. But that's not seasonal. Festbier is.

I'm told that the title of this entry translates to "It is tapped!" which traditionally opens the festivities in Munich. But I don't know much German, so hopefully it's not actually something rude.

10 Like • Comment • View 7 comments

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