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

about 9 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 12 hours ago
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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

about 23 hours ago
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Bienvenue en France.

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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

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WakeUpAndLive️~Happiness

a day ago
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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
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

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WakeUpAndLive️~Happiness

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

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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 15, 2023 at 10:05am

#1059521

Forgive Me, For I Have Zinned

Once more inspired by "Invalid Item"

, today marks another wine celebration.

National Zinfandel Day

When I discussed various reds last week, I failed to mention Zinfandel. There are probably hundreds, if not thousands, of grape varieties out there, and I don't even know all of them, but Zinfandel is kind of a biggie, not something relatively obscure like Mourvedre (one of my favorites).

Zinfandel is a bit of an outlier because it came to the US by way of Austria, though the wine is of Italian origin, and the name is... Croatian? Something like that. Actual zinfandel is bold, deep, unsubtle, and so red it's nearly black. The grapes themselves are extraordinarily dark-skinned.

But then they had to go and make White Zinfandel, which is emblematic of everything that's wrong in the world.

Now, to be somewhat fair, I've heard it's improved since the last time I had the misfortune of sipping it. But I'm still avoiding it on general principles.

"White" wine is, of course, not actually white, any more than my skin color is. It can range from pale, almost clear, to a brilliant off-yellow color. The first offense of white zinfandel is that it's actually a blush, or rosé. I'm not expert enough to know much about the processes involved, but from what I understand, the pale pink color in most rosé wine comes from only brief contact with the darker skins of a grape that's generally pale inside.

The second offense is that it's inoffensive. It's the wine equivalent of white bread, American cheese, and light beer: something seemingly crafted to appeal to the lowest common denominator, and I'm not low nor common nor a denominator.

And finally, the wine I tried when it was all the rage in the States was cloyingly sweet. (As I noted above, that may no longer be the case.) I'm not a hater of sweet wines in general; my first wine was, it should come as no surprise, Manischewitz, a fortified wine; I'm a fan of port and its non-Portuguese imitators; and I'll drink the hell out of icewine. Sweet wines have their place. This one just seemed tailor-made to be a wine for people who don't like wine.

"But, Waltz, doesn't that make it a gateway wine?" Maybe. But it's not like there aren't other fine fermented and/or distilled beverages if you're just looking for something alcoholic, and if you don't like wine, then you don't like it, and that's okay. I know people who simply don't like the taste. WZ seems like it's meant to help non-wine people fit in to the rarefied snobbery of wine culture. In that, it fails, because real wine snobs look down their well-trained noses at WZ quaffers, much as I look down my nose at Bud Light aficionados.

It's like they're trying to appeal to a broader market by making a product that's nothing like other instances of that product. Naked cash grab.

Finally, "white" zinfandel tastes completely unlike the red variety, such that when I finally got around to tasting actual zinfandel, it was a real epiphany. I might actually like it better than Shiraz.

There was a bumper sticker floating around some time ago: "Absolve yourself of white zin."

Indeed.

That said, if you like it, then you like it. Apparently lots of people do, or at least claim to; it still sells. I'm not actually ragging on your sense of taste, only how the winemakers addressed it.

5 Like • Comment • View 3 comments

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