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Saturday, Apr. 27, 2024

2:37pm 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:

Elisa the Bunny Stik

about 4 hours ago
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

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

October 26, 2021 at 12:04am

#1020145

What Do You Mean, "Leftover Pizza?"

It's not just Cracked that tackles today's hottest topics. Sometimes it's also Popular Science.

The best way to reheat pizza (and some things you should never do)
We ruined some pizza so you won’t have to.

And you're damn right the pun up there was intentional.

While ruining pizza is normally sacrilege, I can somewhat forgive it for SCIENCE! because, hopefully, ruining a few now is better than ruining more in the future. It's kind of like a trolley problem, only for pizza. Pizza cart problem.

Order pizza, and there’s a good chance it’s gone within hours.

At least the first sentence addresses my incredulity: I mean, what even is "leftover" pizza?

Something about that round wheel of dough, melted cheese, warm tomato sauce, and seemingly countless topping possibilities is simply irresistible.

Seemingly? Let's assume there are one hundred possible toppings. I believe the actual number must be higher than that, because more than a hundred items of food exist. But just make it 100. The total possible combinations of potential toppings for pizza is therefore 100 factorial, which is written 100! because holy shit! that's a big number -- you get it by multiplying every integer from one to 100. Try it. It starts out slow -- 1x2 is 2, times 3 is 6, etc. but long before you get to 100 you will break your calculator.

Hence, "countless." Which is not the same thing as infinite, but if you'll never get to try every possible pizza topping combination in your lifetime, what's the practical difference?

Still, it’s hard to resist the temptation of a leftover slice as you rummage through the fridge for food the next day.

Big assumption there. But okay, yes, I admit that sometimes I have leftover pizza. And before you say anything, eating it cold is fucking disgusting. I'm not a cockroach. Hell, I'm pretty sure I've seen cockroaches sitting around a campfire reheating pizza; even they know it's an abomination.

Maybe you like cold pizza—there’s no shame in that...

Yes. Yes there is. There is ALL the shame. Ugh. Foul.

...but if you’re looking learn how to reheat pizza in a way that restores some of that fresh-pizza magic, you’ll need to know what you’re doing.

It always helps to know what you're doing.

The very moment pizza is born out of the oven, it’s too hot to eat and might not even be done cooking. But right around 140 degrees Fahrenheit—the temperature experts recommend you dig in so you don’t burn your mouth—it begins its inevitable march toward complete decay, just like everything else on Earth.

Existential nihilism from Popular Science? Okay. Incidentally, if you're hungry enough, you know you're not going to wait; you're going to shove that gooey mess right into your piehole right away. It will burn the roof of your mouth and particularly inflame the spot right behind your upper incisors. We even call that spot the "pizza bump" for that reason.

You see, cheese only likes to be melted once, because when it does, it loses its integrity.

So, it's like a politician, then.

You should never leave pizza out on the counter or in the oven overnight (because of bacteria), but putting it in the fridge doesn’t do it any favors. Low temperatures congeal everything the dough has absorbed and accelerate the staling process, or retrogradation. In short: The starch in the crust recrystallizes, and all that fresh-pizza chewiness goes out the window.

I will say that the absolute ideal pizza isn't fresh out of the oven. No, it's removed, sliced, and then left under a heat lamp behind glass somewhere in New York City for some indeterminate amount of time. Then, when you order it, it's returned to the oven for a couple of minutes. This twice-baking technique makes the absolute best pizza in the world. I have spoken.

PopSci is based in New York City, so we tested each method with the thin-crust style of pizza the Big Apple is famous for.

Good, at least they're starting with the Platonic ideal of pizza.

The official reheating method of the /r/pizza subreddit, this calls for placing your cold pizza on an oiled, preheated non-stick pan and cooking it for two minutes over medium-low heat (or until the bottom of the slice is crispy). Then, pour two drops of water (less than a teaspoon) into the pan as far from the pizza as you can get. Cover the pan with a lid and turn the heat to low. Cook it for another minute.

Way, way, way too much work for reheating pizza. Wash a pan, oil it, preheat it, pizza it, water it, cover it, uncover it, retrieve pizza, wash pan again.

Hot tray in a hot oven

Put a baking tray in your oven and heat it to 500 degrees Fahrenheit. If you don’t want to clean the tray later, you can line it with foil. Once the oven has reached the proper temperature, use an oven mitt to take the hot tray out and put your slices on it.

Do you know how long it takes for an oven to preheat to 500F? This method is absolutely incompatible with my core philosophy of Instant Gratification.

Air fryer

Specialized equipment.

Straight-up rebaking

Seriously, this is what I usually do.

The microwave + oven combo

Bread, including pizza crust, should never, ever, ever go in a microwave. Not even for a moment. Not even if you don't turn the microwave on. Just pretend the microwave is a bread repellent. Yeah, I know it's faster, but in this case Instant Gratification is trumped by Taste. (The one exception to this rule is things like frozen breakfast sandwiches if you follow the instructions to the letter, because they were designed to be nuked, but even there, if there's a baking option, I use that instead.)

Low and slow in the oven

"25 to 30 minutes?" I can get a whole new pizza delivered by then.

The other methods listed are even worse, mostly also involving microwaves. I mean, yeah, the microwave is one of the greatest inventions of all time, but it's important to know its limitations.

One-Sentence Movie Review: Dune:

Watching Dune is like viewing a painting at a museum... for three freakin' hours; they were too busy making it look visually stunning to make me care about any of the characters, and too busy setting up a sequel to actually resolve anything.

Rating: 3.5/5

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