Complex Numbers (Book)

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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/

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Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

April 22, 2020 at 12:01am

#981725

Sit Back

No.

https://www.outsideonline.com/2391763/stop-reclining-your-seat-airplanes

Stop Reclining Your Seat on Airplanes
You heard me. On cramped flights, the person who reclines their seat in front of me really chafes.

You know what really chafes? Complainers.

I love airline travel.

You're a monster.

The predictable flow of airports soothes me, and I get giddy during takeoff because I still don’t quite understand the science (read: magic) that keeps planes aloft.

And you're willfully ignorant.

Let me be clear: reclining is perfectly acceptable on flights longer than, say, four or five hours, especially if it’s an overnighter.

And who appointed you Arbiter?

So when we all pile into our clown car in the sky, we do so knowing that for whatever time we’re at cruising altitude, we’re going to sacrifice some personal space and our legs are going to fall asleep. Which is why my fellow economy-class members who recline their seats are the worst.

So don't fly.

Let’s get real. Those extra three degrees of slope on your reclined seat back aren’t going to make you more comfortable.

Yes. Yes, they are. And you're engaging in hyperbole.

Did they have a drink on their tray table? You didn’t think to check before you reclined onto their warm Diet Coke, did you? Were they working on their laptop? Now their screen is hunched forward at an angle that makes productivity impossible.

How about: Be aware that the person in front of you will recline their seat, and act accordingly. And for fuck's sake, if you need to spend every waking minute being "productive," you're the one with the problem.

The thing is, you know you suck when you hit that recline button, but just in case you truly are ignorant, hear this: your actions have direct consequences for the people around you—any space you take is taken from someone else.

Damn right. I have one magic button on my shitty airplane seat that makes me marginally less uncomfortable, and I am going to use it.

So if you want to lounge back and make life a little bit worse in an already untenable situation, I suggest you get rich, leave us common folk behind, and live it up in business class, which United is expanding while, you guessed it, shrinking economy.

Oh, I've done that. Not just business class, but first class. And believe me, nothing -- nothing -- gives me more pleasure than sitting in a wide, comfortable first-class seat, sipping my complimentary Bloody Mary, and watching people like you struggle past with disease-vector offspring and more stuff than will ever fit into an overhead compartment, let alone under the seat in front of you.

But when I do fly cattle-class, I always make sure to take off my shoes, fart silently, eat fragrant food, and, yes, recline my goddamned seat.

Want to make air travel more comfortable? Take it up with the airlines, and be willing to pay more for the privilege. You know, if there are still airlines after all this.

2 Like • Comment • View 2 comments

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