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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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').

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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October 6, 2024 at 4:59am

#1077831

Like a Bosteels

One thing I insisted upon during trip planning was to visit Brouwerij Bosteels.

This is an ancient brewery which creates two of my favorite Belgian beers: Tripel Karmeliet and Pauwel Kwak. (The link there goes to the English website.)

The difficulty: the brewery is located in a small village called Buggenhout, kind of between Brussels and Antwerp. And I'm still in Bruges, to the northwest. Yes, Belgium is a relatively small country, but it's not that small (about the size of Maryland, though more compact), so trains were involved.

This meant that I had to walk the mile (roughly 1600 meters) from the hotel back to the train station and, yes, my nemeses the cobblestones were heavily involved, though at least I wasn't rolling luggage. From there, I caught a train directly to Ghent (or Gent), then switched trains at a heavily-under-construction station to a stoptrein (which I gather is the equivalent of a "local" in the US) going through Buggenhout. All while being mostly unable to read signs or understand announcements.

While this is part of the adventure of traveling, I was on a schedule; I had to be at the brewery by 1:00pm (or 13:00 in the superior Euro style).

I managed. But then I had to walk again, this time about half a kilometer, to the gigantic brewery, and here, Google failed me in a way that signs in Dutch did not, giving me the wrong location of the entrance. As the brewery is, as I noted, gigantic, this meant walking around an enormous block to find the tour location and, yes, there was a preponderance of cobblestones.

I found it, albeit a few minutes late, but they let me in anyway. After a quick taste of a beer, the guide (who resembled Mirror Universe Jean-Luc Picard, in that he was bald but had a goatee) took us around the brewery operations, which, it being Saturday, weren't active at the time. This is a good thing, because we didn't get in anyone's way and I could actually hear the guide because the machines weren't operating. Point is, you know the Wonka movie? Yeah, that was me gawking at this enormous brewing operation.

Now, my appreciation for Bosteels wavered a bit a few years ago, when AB/InBev bought them. (AB/InBev is an enormous Belgian company, but it's best known for its first initials, which stand for Anheuser-Busch.) But they didn't change the beer in any noticeable way. I'd hoped, because of the big corporation's greater reach and distribution network, that this meant we would be getting more Kwak in the US, but I still can only rarely find it in Virginia. I see Tripel Karmeliet quite often, though, so that's okay.

Interesting thing about Kwak. Each Belgian beer has its own dedicated glassware for serving; this is one of the delightful aspects of Belgian beer. Most of them are variations on a theme, though; something resembling a goblet or Holy Grail, or perhaps a pilsner glass, or, you know, another kind of glass you'd expect to find beer in. Kwak glassware is unique, resembling instead a test tube on a wooden stand. Today's photograph, below, is of a serving of Kwak, so you can see what I'm talking about. I possess two such glasses back home, because on those rare occasions that I do find the beer near me, it's imperative that I use the proper receptacle, and there are two of them because, sometimes, I share with friends.

The funny thing is that a bartender here told me that the glassware, being so different, tends to get stolen from bars. So the bars here, he said, require that you put up one of your shoes as collateral to ensure they get the glass and stand back when it's empty.

This would definitely be a deterrent for me because, well... cobblestones.

I didn't have to get one of my shoes taken hostage at this brewery, but they did let me have a Kwak, and yes, it is much better to drink it at the source.

Between the opening beer, a serving of Tripel Karmeliet, and the Kwak, I was pretty far gone by the time the tour/tasting ended (I'm no lightweight, but Belgian beer tends to be high-octane), and I had to retrace my steps, and the trains, back to the hotel. Easier said than done, because I was still kind of out of it when I got off at Bruges, and had to walk the mile back to the hotel. Along the way, I gave up and had dinner (and a beer) at one of the many sidewalk bars in the old town.

All of which is to say: Mission Accomplished. I only have a few days left here, and rumor has it there's a hurricane (yes, a hurricane, though "only" of tropical storm intensity) coming, but I've now done what I most wanted to do.

As promised, a picture of how Pauwel Kwak is served (bonus: Mirror Universe Picard):

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