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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Writer_Mike

about 8 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

1 Like

QPdoll

a day ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:46am
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

Sep 21, 2024 at 6:16am
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

March 10, 2023 at 12:12pm

#1046182

Musk I?

Another one from "JAFBG"

[XGC] today.

Choose one famous person and rant or rave - your choice.

Funny thing about "rave." It's not precisely a contronym, like "cleave" or "sanction." But it can mean "talk crazy talk" and "heap praise upon." Unlike with "cleave," it's possible to do both. In fact, any raving I do about someone famous would probably be both.

And praising someone presents risks. I seem to remember, several years ago, praising Joss Whedon as a director. Well. That didn't age well, did it? Once it was revealed he was behaving like a douchecanoe, no amount of good that he did could ever make up for that. No matter how great someone is, at some point, they'll turn out to be human after all, and you'll look like an idiot for ever supporting them, because people are unforgiving. So praising someone has its dangers.

On the contrary, once someone reveals themself to be a cockmuffin, no amount of good stuff that they do can ever make up for that. They are always, forever, known as a cockmuffin, because, as I noted, people are unforgiving. That's why it's easier to rant about someone than to praise them: by ranting, you will always be right; if you rave, you'll eventually be laughed at for being so hopelessly naïve.

The risk, there, even with public figures, is that there are laws about talking too much shit about someone. In the US, from what I understand, it has to be both shitty and untrue. But I don't understand much, so I try not to shit-talk a lot.

But Muskmelon is too rich a target not to.

Oh, I admit, I was, like many people, taken in at first by his apparent business sense. I loved the stuff SpaceX was doing, and cheered the first time I saw a rocket booster land itself. I thought Tesla was doing great work: introducing electric cars that didn't look or handle like you were doing penance. Good stuff, all of it. Top of the world.

And then the raving began.

I don't remember all the details. Something about a bunch of people trapped in a cave, and false accusations of pedophilia? Making promises he couldn't keep, like with tunneling machines or high-speed rail? Some half-baked (actually, fully baked and burnt) scheme for founding a colony on Mars? Managing to turn public opinion against autonomous vehicles by leaping them into production before they were ready? I don't know. The details don't matter.

Taking over Birder and sinking it into the lava pits of Hell was just the cherry on top of the parfait of doom. I never liked that site anyway, so it barely affected me, but I still felt like it made Rachel Carson's prophecy of "Silent Spring" come true.

I get the feeling that if he hadn't had those initial successes, Muskmelon would be just another cracked pot. But he did, which makes the crash-landing even worse.

Maybe he should have paid more attention to his engineers. Not just the ones at the bird site, but the ones at SpaceX, who became really good at soft, controlled landings. Instead, he'll now be forever known as a lithobraking expert.

6 Like • Comment • View 5 comments

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