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 9, 2023 at 10:07am

#1059195

International Tempranillo Day

Hot on the heels of "International Merlot Day"

and "Harvey Wallbanger Day"

, it seems that today is...

International Tempranillo Day

Don't look at me; I don't make these things up. Apparently, it's set for the second Thursday in November every year, which, sadly, means it can never fall on the same day as Merlot Day. I was looking forward to blending them.

I wanted to highlight this observance for several reasons:

1) I like Tempranillo. It's usually a blending variety, but I'm pretty sure I've had pure Tempranillo and liked it;

2) In my discussion of wine regions the other day, I shamefully left out Spain, Portugal, and Italy, where this grape tends to thrive (I also left out Germany, which, with a few exceptions, really should stick to beer);

3) The promoter of International Tempranillo Day is named "Tempranillo Advocates Producers and Amigos Society (TAPAS)," and that made me laugh.

Clearly, TAPAS

is an American organization, because only English speakers would dare to create such a boldly horrible forced acronym, and the UK wouldn't want another war against Spain; Canada doesn't produce much Tempranillo, if any; and Australia and New Zealand are too classy to torture language like that.

As for the grape itself, obviously, it's associated with Spain in the way that Merlot and others are associated with France, but it's a very popular variety, so it's been transplanted to other places, like California. I suspect, though, that more people have heard of Merlot. That's partly because, as I noted up there, it's often used in blends; and partly because France likes to think that good wine can only come from their side of the Pyrenees (and their side of the Alps), and they have a better marketing cartel.

The name, incidentally, refers to it ripening earlier than other grapes (same Latin root as other "time" words like tempo, tempus, temporary, temporal, and tempura). I imagine this is a desirable quality for winemakers, as it means their harvest and production periods (and thus, output) can be extended, if they have other vines planted as well.

As for the wine itself, it's been a while since I've had any, so I couldn't describe the taste even if I could bring myself to use oenophilic language like I did in the Merlot entry. I don't keep wine-tasting notes the way I do for beer, but I do remember that it's quite tasty, though probably not as bold as the common French varietals.

It's also, as I understand it, a common grape used in port.

Port can only be labeled "port" if it comes from Portugal, which, as I'm sure you can guess, has roughly the same wine-growing conditions as Spain, being on the same peninsula and all. I mean, I don't know what penalties they could impose if someone in California called their product "port;" war, I suppose. Trade, shooting, nuclear, or otherwise. This is analogous to how champagne has to come from Champagne, or tequila can only be called tequila if it comes from a small region of Mexico, outside of which it must be labeled mezcal or agave spirits or whatever.

But the point here is that port can be made from many different grape varietals, and blends thereof, but the ubiquity of Tempranillo in the region means it's a popular one for the famous fortified wine.

So there you have it. If you like red wine, give it a try next time you see it (doesn't have to be today); it may surprise you, and that surprise might even be pleasant.

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