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Sunday, May. 5, 2024

11:34pm EDT

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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:

🌕 HuntersMoon

about 15 hours ago
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

a day ago
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

May 3, 2024 at 11:58am
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

May 2, 2024 at 11:37am
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

May 16, 2023 at 9:47am

#1049662

A Frank Discussion

I never sausage a thing.

Which Hot Dog Brand Is Best? A Blind Taste Test of Oscar Mayer, Hebrew National, and More
Because your summer BBQs deserve the best. (Or at the very least, not the worst.)

Dammit! It was sitting right there. They could have said "not the wurst," but no, they had to play it straight.

Do you know the difference between sausages, wieners, frankfurters, and hot dogs? If, like me, you hadn’t ever really thought about it and assumed they were all pretty much the same, I’m thrilled to tell you that you’re wrong.

Because of course we are.

It's been many years since I've actually eaten a hot dog, frankfurter, or weiner; anything requiring a hot dog bun. At home, I got really tired of the mismatch between number of franks and number of buns in their respective packages, and while out, there are other foods that appeal to me more. Not to mention I know what they're made of, but that doesn't stop me from eating breakfast sausages.

So, unlike those two other staples of American haute cuisine, hamburgers and pizza, I don't have a dog in this fight. Pun intended, as usual. I just relished the article and found it amusing.

Regardless of which type of sausage is your favorite, there’s one that screams summer louder than the others: hot dogs. Our staff tasters were sure childhood standbys like Oscar Mayer and Hebrew National would sweep the competition but, as always, the results of our taste tests are full of surprises.

Naturally, I enjoyed hot dogs when I was a kid. Our cylindrical delicacy of choice was, unsurprisingly, the Hebrew National brand. When those were unavailable for whatever reason, the replacement still had to be made of cow, because my mom tried to keep a kosher house as best she could out in the boonies.

To cut down on variables, we boiled and tasted only all-beef hot dogs.

So this is why I picked this article to go into my queue.

And though we offered up buns, ketchup, and mustard, most testers boldly chose to taste their dogs plain.

This makes sense from a pure taste-testing perspective, but out in the wild, you're looking for a whole experience, including bun and condiments. I believe that the right choice of bun influences that experience. Would you taste-test pizza without the crust?

As for condiments, in a taste-test, you at least want them to be consistent across all the samples.

And finally, I know they didn't do this in Chicago, because in Chicago, they track down anyone putting ketchup on a hot dog and run them out of town on a rail.

In the end we blind tasted seven of the most popular brands and judged them on flavor, casing snap, and the satisfying firmness of the meat each bite.

Phrasing!

Of course, for full effect, you'll need to go to the article for details. I'm just highlighting things here.

The Biggest Loser: Oscar Mayer

Quelle surprise. Their dogs are terrible.

Unflinchingly Flaccid: Ball Park

I'm starting to think this author has issues.

Not because they don't like Ball Park. That's normal. It's just, again, phrasing.

Happily Herby: Hebrew National

Here's the thing: it's hard to be objective about food (or drinks) during a taste test. Taste is, well, a matter of taste. Beer, for example, is highly personal; some love *shudder* IPAs, while I prefer darker, less hoppy brews. For colas, Coke will, for me, always be far superior to Pepsi. And Hebrew National is always going to be my Platonic ideal of hot dogs, even if I don't eat them anymore, because it was our go-to brand when I was a kid.

The two that beat it on the list, Sabrett and Nathan's, were only available to Kid Me on trips to New York City. Either they hadn't yet expanded distribution to the rest of the country, or we just didn't get them in our off-the-beaten-path area. In both cases, I always wished it was HN.

So, in the end, you'll have to make your own taste test if you care to determine which is best. Or be like most people and just eat whatever's cheapest; this is why many Americans have no sense of taste.

But you like what you like and it's not my decision.

One final note: there is perennial debate over whether a hot dog, nestled as nature intended in its bun, is a sandwich. I've heard even a Supreme Court justice once weighed in on the matter (RBG, if it matters), though in an unofficial capacity.

This is, ultimately, a categorization problem, like whether the Blue Ridge are actually mountains, or Pluto's planetary status. So there's no official answer. Categories are, in the end, a social construct. However, when you consider that the hot dog is generally served with the split side of the bun facing upwards so that the toppings don't fall out—something you never do, and can never do, with a sandwich—and that the bun itself is always solid at the bottom, barring accidents, and also given its origins as handheld street food, there is only one True Conclusion to which someone can arrive:

A hot dog isn't a sandwich.

It's a taco.

4 Like • Comment • View 2 comments

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