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"

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 16, 2023 at 9:51am

#1059568

Beaujolais Nouveau Day

Once again prompted by "Invalid Item"

, I present yet another wine-related celebration, one that I look forward to every year.

Beaujolais Nouveau Day

Unlike some of the other observances I've noted, which are international or US in nature, this one's French. Or, at least, it's French the way Oktoberfest is Bavarian: no one's stopping us ugly Americans from celebrating.

There's a bit about Beaujolais Nouveau at the link above, but, naturally, I'm going to say my piece (actually pieces) about this utterly delightful wine.

The "Nouveau" part of the name means, it doesn't take four years of Duolingo French lessons to know, "new." But it's not like it's a new product; no, I think it's "new" in the sense of "fresh." The "Beaujolais" part is a place name, like Bordeaux or Champagne. Apparently, according to that link, the capital of the region is Beaujeu, which might translate to "beautiful game," which honestly would be really cool if so.

But I digress. This is a "new" wine, meaning the time from harvest, through fermentation, to the important part (drinking it) is very short. This results in a clean, crisp, fruity wine, without many of the more complex flavors or profound textures of the slower reds.

It is also, unusually for a red wine, best served chilled, like a white.

Another digression: it's well-known, even outside of wine circles, that most reds aren't served chilled. But this does not mean they should be served at room temperature (around 20C). Same goes for many dark beers. No, these heavier fermented beverages are at their best at cellar temperature, which is around 13C, though there's some leeway there. Owing to its status as a young wine, Beaujolais Nouveau is, instead, served cold like a white, or that better-known French product, champagne.

As delicious as the wine is on its own, there's a special, serendipitous connection for those of us in the US: Beaujolais Nouveau Day is, as is also noted at the link, celebrated on the third Thursday of November.

If this sounds vaguely familiar, it's because Thanksgiving is celebrated over here on the fourth Thursday. And yes, Beaujolais Nouveau is, normally, available in the US. Today is the day I usually make a pilgrimage to the grocery store (as opposed to ordering a delivery therefrom) to pick out a bottle. Might have to be tomorrow, though; I have a beer dinner to attend tonight and I need to make sure I'm well-rested for that.

And I can tell you this from experience: there is, quite simply, no better wine to accompany a traditional Thanksgiving feast.

Like I said, it's not a complicated wine. But usually, one of the distinctive fruity flavors it fronts is that of cranberry. And what better accompaniment for turkey than cranberry?

So there's that linking the two countries, as if the Statue of Liberty, the design of our capital city, and, you know, the entire concept of representative democracy weren't enough.

"But, Waltz, the Pilgrims didn't have French wine to drink." Okay, fair. And Beaujolais Nouveau hadn't even been invented yet, not to mention the refrigeration that would keep it at its ideal temperature. But if you think your Thanksgiving feast is a perfect reconstruction of what the Pilgrims ate, or, more appropriately, what the settlers at Jamestown managed to scrounge up, think again. I don't know; maybe you try to do that, keeping it all to truly native American harvest foods. If so, great. You do your thing.

As for me, while I have no intention or desire to be part of a crowd at Thanksgiving, I may give the holiday a nod by warming up a frozen turkey pot pie.

It will be accompanied by a nice, cold bottle of Beaujolais Nouveau.

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