Complex Numbers (Book)

Blog Calendar

◄ October ► 2021
S	M	T	W	T	F	S
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
31
Archive				RSS

About This Author

Contact Me

Come closer.

Complex Numbers

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:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 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

8 Like • Comment • View 5 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.