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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QPdoll

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

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Wordsmitty ✍️

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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.

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Annette in France

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WakeUpAndLive️~Happiness

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

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WakeUpAndLive️~Happiness

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

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Elisa: Middle Aged Stik

a day ago
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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

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WakeUpAndLive️~Happiness

a day ago
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/

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WakeUpAndLive️~Happiness

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

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

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Aiva Raine

a day ago
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.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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November 2, 2023 at 7:36am

#1058615

Stout Day

Something different today, inspired by "Invalid Item"

International Stout Day

I really don't know who comes up with these promotional or commemorative "days," but they amuse me. Sure, some of them are blatantly commercial, but then, most official holidays are, too. Others are icky heartstring-tuggers, but we have a few of those on the official list, too.

The important thing is that, sometimes, there's a day related to fine fermented and/or distilled beverages, such as this one.

First, to be clear, the theme of the day is the beer style. The word "stout" itself, as the above link indicates, initially meant proud or brave, but acquired the meaning of strong... though, even later, it became a less offensive way to call someone fat.

But that was after we started calling certain beers stout for their strength.

Now, sure, you can read the text at the link, which, while I can't vouch for its accuracy, tracks with what I know. And you can also view one of the many entries I've done here talking about stout, such as this one from five years ago: "Stout"

But I have one thing to add: as noted at the title link, "stout" used to refer to any stronger beer. Gradually, though, the style came to refer only to stronger porters. So, to recap: Ale refers to almost any beer that's not a lager (it's mostly a matter of what kind of yeast is used, but that's not important right now). Porter is ale made with dark roasted grain. Stout used to be more intense anything, but now is reserved for more intense porter. But that wasn't enough, so they had to invent the imperial stout. A quick rundown of how the first imperial stout came to be can be found here, in another of my earlier entries: "The Yeast You Can Do"

And now, "imperial" is used to modify any beer style that has a higher than normal alcohol content. Imperial stout, such as one of my all-time favorites, Old Rasputin. Imperial lager. Imperial pumpkin ale. That sort of thing. Amusingly, imperial porter.

Such is the evolution of language, style, and beer.

Sure, these definitions can get a bit fuzzy. Where does porter end and stout begin? What's the ABV cutoff between metric and imperial? (No, regular beer isn't called metric; it's just one of my stupid puns.) Is it possible to have a beer that's both ale and lager? (Yes. Cream ale.) These things are at least as much art and marketing as science.

It just so happens that I have some Old Rasputin waiting for me right now. Perhaps it's time for breakfast.

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