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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

a day ago
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

a day ago
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

a day ago
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

a day ago
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?

June 17, 2021 at 12:04am

#1011997

Souperman

Entry #5 of 8 for "Journalistic Intentions"

[18+]

Don't cry because it's over, smile because the onion soup will taste great. -Max Kwoa

I suppose I could launch into a monologue that blasts the social expectation that men aren't supposed to cry. The pressure to conform to that stricture is still there, because we have yet to fully move beyond the whole patriarchal gender roles thing.

But you know, I'm not in the mood for social criticism right now. So instead I'll talk about how I finally learned to make French onion soup.

Cooking is another thing that men were traditionally not supposed to do. Well. Cooking at home, anyway, because that's not a paid job. It's always been quite alright for men to be restaurant cooks. This has to do with the perceived difference in value between "women's work" and "men's work," and...

Dammit. There I go again.

Anyway, fortunately, that particular taboo has been greatly eroded. Which is a good thing because if I were not allowed to cook, I'd be broke or dead from either eating out all the time or not eating at all. This is why single men don't live as long, on average.

Yeah, most of what I make is of the quick and easy variety. This is because I don't see much point in slaving over a hot stove for hours to make an elaborate meal for just me. Sometimes I'll make something and share it with my housemate, but mostly we do our own cooking.

But late last year, I got it in my head to make French onion soup because it was winter and I wanted some. Besides, everyone needed a pandemic hobby, right? It's a dish that takes a good bit of time, but there aren't that many ingredients and you can make a large batch to have leftovers.

It might surprise you that the principal ingredient in French onion soup is onions. Not French. And it takes beaucoup d'oignons. Er, I mean, a lot of onions. Oodles and oodles of them. And you gotta halve each one, and then slice each half into thin half-rings. While it's not as hazardous to the tear glands as completely chopping the annoying buggers, it still has the potential to aerosolize onion juice, which can lead to eye irritation.

There are a few ways around this hazard.

The first is to use white or red onions, not yellow onions. This, however, results in a less flavorful soup. Or, well, I think it does; I haven't tried it with anything but yellow, but in other recipes the other varieties just aren't as tangy.

The second is to have someone else cut the onions for you. This is out of the question for me, but it's always good to delegate. Just don't make your kids do it, or guaranteed you'll end up with bloody onions and them in the ER.

The third is to cut them under running water. This is a pain in the ass, and has the potential to make things more slippery -- see above about blood.

The fourth is cheating: wear goggles. That way the onion spray can't get to your eyeballs. Just remember not to touch your eyelids after you're done and have removed the goggles. Also remember to clean the goggles afterward.

Honestly? I don't do any of these things. Sometimes my eyes get a little irritated, yeah, but then I just wet a paper towel and wipe my eyelids with it.

Anyway, chopping all those damn onions is the most labor-intensive part of making French onion soup. The rest is mostly just caramelizing the onions, simmering them in broth and a few other secret ingredients, and, after topping each bowl with bread and cheese, making sure you don't broil the cheese so long it becomes charcoal. (I've always managed to get it right because I'm awesome).

I'm not going to provide the whole recipe. I found it online and so can you, and you might even find one that suits your taste better than the one I found. But I will note a few tips:

Experiment with the bread. I have had excellent results with challah, but that may be too chewy for some people. Traditionally, one uses baguette slices.

The optimum cheese combination is (in my own opinion of course) Gruyere mixed with a little bit of grated Parmesan. Sure, Parmesan isn't French, but who cares? Other than the French. Don't tell them, but all of their cuisine came from somewhere else too, just like almost everyone's.

Get a set of those purpose-made ceramic French onion soup bowls. It just doesn't work as well in a standard bowl. They're not all that expensive and you might even find some used.

In any case, the bread and cheese are not optional.

Wow. Normally after doing an entry like this I get hungry, but I don't have the three hours or so that it takes to do the recipe. Besides, I'm out of onions. So sad it makes me want to cry. But I won't, because I'm a guy.

3 Like • Comment • View 2 comments

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