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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.




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May 26, 2022 at 12:49am
May 26, 2022 at 12:49am
#1032870
More for "Invalid ItemOpen in new Window. today.

What would happen if Andre opened a bar in your town/area? (I did say maybe!)


Well, Andre, first thing you need to know, as I noted way back in "A Tour of the TownOpen in new Window., no bars are allowed in Virginia, so you'd be arrested, fined, and sent to toil away on the vast banana plantainations of the Piedmont.

I suppose if we're being technical about it, you could open a juice bar, or a vegan smoothie bar. Those are legal. But you'd fail. No one cares. Oh, there'd be a glut of people checking it out for the first week or so, taking selfies to show how virtuous they are, but then... crickets.

Now, if you opened a restaurant that also happened to have a place that serves various delicious fermented and distilled beverages, behind a long table with stools lined up in front of it, well, that would be okay. As I noted in the above entry, there are plenty of those around.

Which is, after all, the problem: there are plenty of those around. Market's saturated. You'd need an angle, and as far as angles go, bananas just ain't gonna cut it. Oh, sure, you could import (or grow in greenhouses, because I was just kidding about the banana "plantain"ations above) different varieties than the standard Cavendish, and introduce people to the wide range of genetically engineered berries that constitutes the musa family of plants. But that would only have limited a-peel. The advantage of bananas is that they're easy to eat -- probably the easiest fruit, because, unlike, say, apples or blueberries, you don't have to wash them first -- and they're easy to peel and shove into one's gaping maw. They're almost as easy as granola bars. This is important, because I'm lazy.

But it's that very ease that would make a banana bar, in reality, not a viable enterprise. We can eat them at home. Even the making of banana bread (which I think is actually bald cake; see also "carrot cake") is relatively simple. Not that I've ever done it. Every time bananas sit on the counter at my house too long, my housemate slaps them into the fridge to make banana bread with. She then proceeds to never make banana bread.

Still, there are two banana markets left untapped here in Charlottesville: wine and beer.

As I've noted before, there's plenty of wine and beer here. But not banana wine,  Open in new Window. which is actually a thing, or banana beer,  Open in new Window. if you can deal with the inevitable accusations of cultural appropriation.

Those articles I just linked are a bit light on details, I know, but you're a smart monkey. You'll figure it out.

I will, however, take this opportunity to do my wine vs. beer rant.

At their most basic level, beer is a fermented beverage made from grain, while wine is a fermented beverage made from fruit. As with many things, though, the definition from there gets as fuzzy as a monkey's back.

As I've noted repeatedly, I'm a fan (and a snob) of both, though given a choice, I'll usually lean in the direction of beer. But I tend to lean in all kinds of directions after I've had a few, so really, I'm not picky about wine vs. beer, only about the particular wine or beer. And I'm also technically-oriented, having achieved a degree in engineering, so this fuzziness bothers me. Allow me to explain.

Cider is also a fermented beverage made from fruit (usually apple or pear), though it's not marketed as wine, but usually alongside beer. For instance, it's found amongst the tap selections at the drafthouse cinema, and in the beer section at grocery stores around here. And yet it's not called wine or beer; it's called cider.

Additionally, you'll sometimes find a delicious and dangerous product called "barleywine." Barleywine is beer; it just has a higher ABV than is commonly found in beer, because, I suppose, wine generally has a higher alcohol content than beer. And yet many beers, usually Belgians or those called "Imperial," also have a higher ABV than ordinary beers, usually in the 8-12% range.

You could make the distinction that beer is fizzy, while wine isn't... but there exist exceptions to that as well, such as champagne. One could also make an argument involving sugar quantity, but there exist sweet beers and dry wines; the fermentation process works only on sugar, so the starches in grain (or potato or plantains) must first be converted to sugar, which is chemistry so basic that even your saliva does it (not that this is a recommended method of commercial production of beer). And sometimes you get residual sugars. Sometimes a lot, as with port wine, or sour beer.

As if the matter weren't complicated enough, there's also sake, which is made from rice. Rice is a grain, not a fruit. But when describing sake to an English-speaking person, it's generally called "rice wine." Sake is usually marketed like wine, not like beer. And it's not like rice beers don't exist. Most of them suck monkey ass, but they exist. The big, brightly-colored cases of macrobrewery product you find at convenience stores, for example, are often rice-based. In the case (other definition) of rice product, as with barleywine, it's largely a matter of ethanol content and the process they use.

All of which is to say that a monkey could, in theory, sell both banana wine and banana beer in a banana bar. And be functionally alliterate.


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