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Thursday, Nov. 21, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Robert Waltz

a day ago
In response to "Powers of Three"

🌕 HuntersMoon

- that's pretty much what quantum computing is, from what I've read.

2 Like

Brandiwyn🎶

a day ago
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

a day ago
In response to "Powers of Three"

With binary, it's either on or off. Wouldn't ternary introduce Schodenger's Cat? On, off, or both?

2 Like

Turkey DrumStik

Nov 19, 2024 at 7:43pm
In response to "Paramount Purpose"

I suspect old web filter attempts

would have combusted when confronted with this article.

1 Like

tj-turkey-jobble-jobble-hard-J

Nov 19, 2024 at 4:04pm
In response to "Losing Weight Is Easy"

Some of them break out in hives if you ask them to add 10% to something.

Yesterday I purchased an item that cost $6.42 after tax was affixed by the register. I handed the cashier a ten dollar bill and watched as she placed it in the cash register, which like most now days, showed her how much change to return. She had to try three times to count out the proper amount, and as I was leaving commented to the bagger, "Why do people have to pay with cash, don't they know how hard it makes my job?"

I didn't say anything, but was trying desperately not to break out in hysterical laughter.

1 Like

🌕 HuntersMoon

Nov 19, 2024 at 9:28am
In response to "Paramount Purpose"

At your age, Bob, take heart in the fact that it can be restored...

2 Like

Wordsmitty ✍️

Nov 18, 2024 at 12:34pm
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

3 Like

Brandiwyn🎶

Nov 18, 2024 at 11:27am
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

Brandiwyn🎶

Nov 18, 2024 at 11:19am
In response to "Losing Weight Is Easy"

Has anyone ever told you how annoyingly pedantic you are?

Damn your pedantic ways. I was hoping to justify a vacation to Venus (which I still argue would be a much better holiday destination than fr-fr-freezing Mars, Elon. Ugh.) Now I guess I'll have to settle for the Cancun plans we already made, since my mass (and waist diameter, and thigh diameter) will be exactly the same regardless.

3 Like

tj-turkey-jobble-jobble-hard-J

Nov 17, 2024 at 4:04pm
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

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):

5 Like • Comment • View 1 comment

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