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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:

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

March 11, 2024 at 8:26am

#1066054

Cheeses Crisis

Nooooooo! Someone needs to Do Something, Right Now!

Beware: A cheese crisis looms
Gird your curds! Say a prayer for Camembert! A collapse in microbe diversity puts these French cheeses at risk.

Just to get ahead of this: no, if this Vox article is correct, it's not because of climate change; yes, it's peoples' fault.

This looming cheese crisis, this Camembert calamity, stems from a much bigger problem: a collapse in microbial diversity.

On the plus side, Camembert Calamity would make a great name for a heavy metal Depeche Mode / Tears For Fears cover band.

Each hunk of Camembert or smear of brie is an ecosystem, an assortment of fungi and bacteria that turn milk fats and proteins into hundreds of different compounds. Those compounds produce the flavors, smells, and textures we love.

Hopefully, you already knew that the deliciousness of cheese is due to the action and presence of beneficial microorganisms. While we usually associate these things with spoiled food, they're all around us, and inside us, all the time; and as with multicellular plants or animals, the helpful ones outnumber the harmful ones.

Some are also responsible for beer and wine.

In recent decades, however, the genetic diversity of some of those microbes has caved.

The article will, of course, expand on this statement, but of course they gotta hook readers with the bad news.

To make cheese, producers typically take fresh milk and mix in bacteria and often fungi, including both yeasts and molds (fungi that tend to be fuzzy). Different microbe melanges produce different varieties of cheese.

It's obviously way more complicated than this, but the full process isn't really relevant to the story.

That's also a very simplistic description of mold... but an accurate enough one.

Historically, Camemberts and bries likely relied on mold strains from a species of fungi called Penicillium biforme, according to Jeanne Ropars, an evolutionary biologist who works at a lab affiliated with CNRS. Each strain was slightly different genetically, and so the resulting cheeses had slightly different colors, flavors, and smells.

I'm sure we all recognize at least part of that binomial.

Roughly a century ago, however, cheesemakers identified a particular strain of P. biforme that was fast-growing and albino; it produced a fluffy white mold that was, apparently, quite appetizing. This strain, known as Penicillium camemberti, was henceforth considered the gold standard for brie and Camembert (which differ from one another mainly in size). It quickly dominated the cheese industry, and the diverse group of other mold strains used to make Camembert and brie, and the colors they produced, vanished from disuse.

And, basically, that strain needs to be cloned, not bred, and cloning eventually results in DNA transcription errors.

The whole thing reminds me of the banana problem, which I've written about before, here: "Going Bananas"

Ironically, the problem there is one of the bad fungi (the kind that wear leather jackets, smoke cigarettes, and hang out in front of liquor stores). So it's analogous but not exactly the same thing. To its credit, the article mentions that.

This rapid caving of genetic diversity threatens other food industries, too, as the author Dan Saladino writes in his book Eating to Extinction: The World’s Rarest Foods and Why We Need to Save Them.

Hey, this was actually a book advertisement all along! That cheeses me off. At least be up front about it.

Ultimately, this doesn’t mean that we must bid farewell to brie, or that Camembert on toast is, let’s say, toast. There is a way to save these cheeses, though it requires some changes in our own taste and tolerance.

As for me, I look forward to greater cheese diversity. You want uniformity? Use that "pasteurized process cheese food" that America substitutes for culture (pun absolutely intended). As with beer and wine, I'd be happy to taste different cheeses produced with different friendly microorganisms.

As the Mandalorian might say, "This is the Whey."

5 Like • Comment • View 3 comments

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