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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Wordsmitty ✍️

about 20 hours ago
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

a day ago
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

a day ago
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

Lilli Munster 🧿 ☕ 🎃

a day ago
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

Wordsmitty ✍️

a day ago
In response to "Bière Snob"

Great to see you got a good shot. And the picture is nice too.

1 Like

Writer_Mike

Sep 22, 2024 at 10:52pm
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

Sep 22, 2024 at 6:44am
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

3 Like

Wordsmitty ✍️

Sep 22, 2024 at 3:41am
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

Sep 21, 2024 at 4:44pm
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Go Forth📖

Sep 21, 2024 at 12:51pm
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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December 6, 2020 at 12:02am

#999730

Tradition!

I enjoy making up my own traditions and then following them... or not.

"30-Day Blogging Challenge ON HIATUS" [13+]:
6. Marooned Without a Compass Day!
So where are you and what are you going to do?

or

Bartender Appreciate Day! (Europe)
Write about an encounter with a bartender,
make one up if you have to.

"JAFBG" [XGC]:
Some traditions are a lot of fun. Others are a pain in the ass. What are your least favorite holiday traditions?

I try not to be too negative around this time of year. Not only am I prone to seasonal depression, but I think some people take my grumpiness as a personal affront. But, dammit, it's hard sometimes. Especially this year, when the one inviolable holiday tradition -- that of traveling to reconnect with friends and family -- will only result in more illness and death. Just know that, apart from dangerous behaviors, I'm not trying to criticize others' traditions just because they don't work for me. I really do wish everyone, in the spirit of the season and for the sake of humanity in general, peace and joy; just understand that, in general, things like blinky lights, yard decorations, and most especially, the despicable practice of dressing your car up to look like a reindeer, annoy the living shit out of me.

Probably, it would be more succinct to mention those holiday traditions that don't cheese me off. Most of these involve drinking, which should surprise no one. It's an excuse to drink eggnog or spiced wine, for example, which is kind of frowned upon at other times of the year.

I actually kind of like the traveling, to a certain extent, although with millions of other people doing it, things get crowded, and I dislike crowds. When I first got my driver's license, lo these many years ago, I started my own tradition: go out on Christmas Day (first making sure I had a full tank) and drive around, enjoying the solitude as, on that one day, most people were staying home or at their holiday destinations, leaving the roads wonderfully empty. Back then, fewer things were open on the 25th, and for the most part, I had the road to myself. It was strange, especially after weeks of traffic, but also very calming -- provided, of course, there wasn't an ice storm.

Since then, the world's population has doubled and more things are actually open on that holiday, so it's not as solitary; and besides, driving interferes with my drinking time. So last year, I started a new tradition for myself: walk over to the drafthouse cinema, about a mile away, and see how many of their 32 beers I can drink before I would be unable to stumble home (and finding an Uber on Christmas Day was not something I wanted to attempt).

The primary purpose of the cinema, of course, is to show movies, but I didn't feel like seeing a film that day, so I parked myself at the bar in the mostly-empty restaurant part of the cinema complex. Because it was mostly empty, I got to talk to the bartender, whose name was Hunter. He seemed a decent guy and not too grumpy about having to work on Christmas Day; and, most importantly, the guy knew his beer.

Turned out from talking to him that he was the one who got to choose the kegs for the taps, and let me tell you, Hunter has good taste in beer. I'm used to going to places that are heavy on IPAs and, more recently, sours, which are not my favorite styles of beer, but they're popular so a lot of beer vendors devote the bulk of their taps to them, leaving me to choose from maybe one or two preferable styles, such as English browns or Belgian tripels.

Not this guy, though. Oh, there were a couple of IPAs and sours, and I think two of the taps were also devoted to ciders. We got to talking about that. One of my former go-to local beer places has only twelve taps, and the last time I went there, ten of them spouted pale ales, one of them was devoted to Pabst Blue Ribbon because hipsters, and the other was Allagash White, which is nice, but I need variety. Like I said, it was the last time I went there.

Hunter agreed with me -- at least out loud; I'm sometimes incompetent at figuring out when a bartender is humoring me for the sake of a tip and when they actually share my feelings about beer, but in this case I'm pretty sure it was the latter. Turns out he'd worked at the IPA-heavy place for a while, and also preferred more variety.

It's customary, at least in states that allow it, to let customers sample small quantities of a beer to help them determine whether to order a whole pint or not. It's best not to abuse the privilege, say by asking for samples of every one of the 32 beers/ciders and then not buying anything. Don't do that. It's a dick move. But since Hunter quickly pegged me as a Beer Guy (and, most importantly, one who was willing to spend money on pints and a pizza), I got to sample a lot of what was their selection at the time. There are some beers that are standards in each taphouse, ones they'll order over and over to keep them on the menu. Others generally rotate; when one keg kicks, it gets replaced by an entirely different magical concoction.

That's what keeps me coming back, of course. While some people stick to one style or brand of beer, others, like me, enjoy trying new ones, and there are always a lot of new ones. Oh, sure, sometimes I like to go back to old favorites, too. And neither way is wrong; it's a matter of personal taste. It's just that some taphouses are heavy on the "standards" while others focus on the "rotation," but this place strikes a good balance between the two.

You can do that with 32 taps.

And so it was that when I was done -- I think I had four full beers in addition to a dozen or so samples, for a total between the beer and pizza of maybe 40 bucks, I handed him a $20 bill.

"No, no, that's too much."

"Dude, you're working on Christmas Day. Call it a present."

I resolved then to make that my new Christmas Day tradition, since pretty much nothing else was open. But then the... well, you know what happened. The place closed entirely for several months, and I have no idea if they'll be open this Christmas or not. Like I said before: so much for tradition.

Since I drafted this, I actually checked the theater's calendar. Turns out they're indeed open on Christmas Day, and they're showing a bunch of screens of Wonder Woman 1984. This is relevant to my interests. Now I need to decide if it's worth the risk.

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