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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Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

September 26, 2019 at 10:37am

#966808

Pappy? Happy!

PROMPT September 26th

A prompt from Jaelynn

!

Write about a special place where you might go to meditate, unwind, de-stress, relax, or just be.

It's generally known a bar.

Some people go to bars to meet other people. That's not for me - dealing with people is the opposite of relaxing. Which is not to say that I don't like to do it; I'm pretty social for an introvert. But I like bars that are mostly empty and I don't have to deal with the noise and crowds. That's why it's best to go at like 8 am.

No, I use bars for their gods-intended purpose: drinking.

Let me give you a recent example.

Just the other night, I was, as I mentioned here a few days ago, in Washington, DC. On the last night of the conference, Tuesday, the organizers had arranged for the opportunity to go see a Washington Nationals game. Turns out that's something called "baseball." Okay, no, I'll just do what I always do when I travel, which is find the local craft breweries and sample their wares.

So I did, with a little help from Uber.

First place I went was a nice little wood-decor pub, not too crowded. There's a mural on the wall featuring a fox with laser eyes, two pandas fighting, and a squirrel with a flamethrower, all kaiju-sized and rampaging around DC. Sat down at the bar, ordered samples of their various beers... and became disappointed. They weren't bad beers, exactly, just not to my taste. That's okay - plenty of styles for everyone, and the world doesn't exist to cater to me.

So I Ubered the hell over to the next place on my list, which turned out to be a) not a brewpub but a taphouse (fuck you, Google) and b) crowded as the Beltway on a weekday morning. Pass. (I should clarify here that I enjoy taphouses as well; I was just looking for local beer.)

The next place was only a short walk from the crowded taphouse. It looked promising. Just a few blocks from the Capitol, so I was afraid I'd run into gods-be-damned politicians, but the crowd looked more like politicians' flunkies trying to deal with the stress of having to work with politicians. That's fine. And I say "crowd," but it wasn't full enough to keep me from finding a nice comfortable barstool. But the place turned out to have only two of their own beers, and it was unclear where said beers were actually brewed because there's more than one location for this particular business. Not bad beer, mind you, but still not the experience I was looking for.

So I walked back to my conference hotel. Yes, walked, not staggered; the reason I took Uber in the first place was not because I planned on becoming too drunk to drive, but because it was motherhumping DC on a motherhumping Tuesday evening, and where does one park?. The good news is that I did a mile and a half at a brisk pace without getting in the least bit winded; I guess all the exercise is paying off.

I sat at the bar in the hotel, which also wasn't crowded because the bulk of the conference attendees were either at the sportsball game or had already left. Told the bartender, "I'm disappointed by the breweries around here. I want a positive drinking experience before I leave."

"What can I get you?"

I pointed out a fine 18-year-old scotch I saw standing lonely and dusty behind the bar. "That. Neat."

A lady came in and sat next to me, and I immediately felt myself tense up. What am I supposed to do? I don't go to bars to pick up women; I go to enjoy fine libations. I hear that in such situations, it's customary to buy the lady a drink, but I also hear that's a social signal that shows interest in such activity. So I basically just concentrated on my scotch, which, I will have to note, was indeed a very good scotch.

We ended up talking, anyway, and within two sentences she'd made it clear she was there with her husband, who had gone to the game. Whew. No pressure, then; I don't mess with married women. But there's no rule I'm aware of about not carrying on an ordinary conversation with one while in a public place. Maybe there is such a rule, and I just don't know it. Anyway, she said I looked like The Dude, and if you're going to compliment me like that, I'm happy to talk to you.

I had another scotch during the conversation, and then I noted to the bartender that I was surprised that a fancy place like this doesn't have Pappy van Winkle.

Allow me to digress for a brief time to explain that for non-whiskey folks. There are dozens of bourbon distilleries in Kentucky; it's kind of what they're known for, along with horses, bluegrass, and intolerant politicians. Many of them make fine bourbon. (The distilleries, that is, not the horses. Maybe some of the politicians; I don't know.) But the Holy Grail of bourbon in the last few years has been a label called Pappy van Winkle, after the old white guy who developed it.

Bourbon has particular requirements in order to be called "bourbon;" one of these is that it has to be made from at least 51% corn (that's maize for my British friends, not barleycorn). Now, to save money, most places use cheaper grains like barley or rye to make up the remainder of what's called the "grain bill." There is absolutely nothing wrong with this, but it does impart to bourbon its characteristic "bite," and I tend to prefer other whiskeys (or whiskys or whiskies, whatever).

Pappy, I'm told, makes up the rest of its grain bill with wheat. My understanding is that because wheat is in high demand due to its use in this thing called "bread," it tends to cost more per unit weight. And it's aged for various times in magic barrels that give the flavors time to mellow. Pappy - which I've only encountered once before, in California of all places - is, as a result, smooth as silk. No burn, no bite. It is also not cheap. As with all whiskey, as a general rule, the longer it's aged, the more mellow it becomes, and the more expensive it tends to be.

"Oh," says the bartender, "We have some in the safe."

"You have a safe?!"

"Indeed. But the only Pappy van Winkle we have are the 20 year and 23 year."

"Okay. Sounds great. But believe it or not, I'm going to need to know the price first."

So he told me, and I'm not going to relate it here; suffice it to say that it's not cheap, but it also wasn't going to send me into bankruptcy. I ordered the 20 year, and it was glorious.

After, he took me to the safe and opened it and holy mother of suck, there were some impressive distilled spirits in there. Also rare and expensive ones, even pricier than Pappy. I was not tempted - well, not too much - but it sure was pretty to look at.

So there it was, a positive drinking experience. Fully de-stressed, relaxed, and serene. A fine ending to a crowded conference.

I guess what I'm trying to say with all these words is: I like bars.

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