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Saturday, Apr. 27, 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:

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 16, 2024 at 10:52am

#1066354

Substandard

Ironically, the tagline for The Takeout, which provides today's link, is: "Food is delicious."

Everything Tastes Worse Than It Used To
You've heard of shrinkflation, but what about skimpflation?

You know what else is inflating? My rage at portmanteaux. Every time I see a cutesy one, it's like I have an anger bar (like a health bar in a video game) and it just keeps getting fuller and fuller. I call it... rageflation.

It's okay when I do the portmanteau.

If you swear your go-to snacks and candies all seem to taste different—and worse—these days and you can’t quite put your finger on why, you’re not alone.

I mean... it is supposedly true that one's sense of taste dulls with age.

Business Insider has taken a deep dive into food manufacturers’ increasing adoption of what BI calls skimpflation or flavorflation, aka modifying recipes in order to (you guessed it!) maintain or increase profit margins.

Oh, good, now I know what entity to blame for the maddening portmanteau. Call it what it actually is: enshittification.

Ingredient costs, obviously, are a huge factor in the pricing of any consumer product.

So are employee wages, which is why we replace them with robots.

Business Insider cites an instance of Conagra reducing the fat content in its Wish-Bone House Italian Dressing by 10%, replacing it with additional salt and... water.

On the bright side, we're running out of fresh water, so the price of that is going to go up, too.

In 2013, Breyers, the ice cream of my childhood, had to legally change the labeling of its products from ice cream to “frozen dairy dessert.” Why? Because the company had reduced the amount of dairy fat in its product to the point that it didn’t legally qualify as ice cream anymore.

Good to know there are standards. I eat "frozen dairy dessert" so infrequently that I'd probably never notice. Part of the reason is that I have cold-sensitive teeth. But once a "quart" of ice cream became 3.75 cups or whatever, I quit buying it. Apologies for the shitty measurement system to anyone using a logical one; what you need to know is that one quart, which is a bit less than a liter, is equivalent to four standard cups.

Speaking of logical measurement systems, I remember when soda was sold in half-gallon bottles (and they were made of glass, prone to shattering, and heavy). At some point, they switched to 2-liter plastic bottles. As I noted, a liter is slightly more than a quart, and 2 liters is thus more than a half-gallon. Again, apologies to metric users: it's called a quart because it's a quarter of a gallon, okay?

I'm not really sure why. Perhaps they were thus able to increase the price beyond what it cost them to include those few extra drops in the bottle. If so, that would be one way to sneak a price increase past us: give us a bit more of the product, while charging disproportionately more.

I digress. This is about (ugh) skimpflation and not (blech) shrinkflation.

The most egregious example of so-called skimpflation we’ve seen recently was October 2022, when Conagra dropped the amount of fat in its dairy-free Smart Balance spread from 64% to 39%, which meant water became the most plentiful ingredient in the product.

I guess someone there found a smarter balance on their accounting spreadsheet. My solution? Use butter.

As an aside, I'm going to complain about English muffins (or, as I believe the English call them, muffins), which are one of my favorite foods. I even use them for hamburger buns. For a while, though, the only kind sold by my go-to grocery store was Thomas', so I didn't buy them (or I made a special trip to Whole Paycheck to get the good kind). Thomas' is to English muffins as Lender's is to bagels; that is, a piss-poor replica. Unsurprisingly, those are now both products of the same soulless corporation (aptly named Bimbo). Bread is food; everything else is a condiment.

Anyway, more recently, the nearby grocery store started selling its store-brand English muffins. One time, they were out, and the shopper (yes, I get groceries delivered, because I am remarkably lazy) subbed Thomas'. Not only are they inferior in taste, but I noticed that they seemed to be quite a lot thinner than they used to be, so thin it took hours of careful work with specialized tools to split them without destroying them. Now, that could have been my own perception, colored by comparison with the much heftier thickness of the store-brand muffins (which, I should note, are also cheaper), but it could also have been (gag) shrinkflation.

Either way, now I have to include a note with my delivery order: "DO NOT substitute Thomas'." I'd rather go without than deal with that bullshit.

In conclusion, however, the headline is wrong: Not everything tastes worse than it used to. Beer, for instance, has vastly improved in quality with the advent of craft breweries. More expensive? Sure. But worth every penny.

Well. This discussion didn't lower my rage bar. I'm going to go eat an English muffin with butter.

4 Like • Comment • View 2 comments

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