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 ✍️

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

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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

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"

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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 8, 2023 at 9:40am

#1059144

Harvey Wallbanger Day

Prompted by "Invalid Item"

, another alcohol-positive observance... sort of.

National Harvey Wallbanger Day

Believe it or not, there are drinks I haven't tried. This is one of them.

According to the above link, a Harvey Wallbanger is made with "vodka, Galliano liqueur, and orange juice." Proportions aren't given, but I'm confident that a quick internet search, which I can't be arsed to do, would provide numerous recipes.

Now, don't get me wrong: I appreciate cocktails or whatever else you want to call mixed drinks. One of the great joys of life is creating new ones, such as the Star Trek themed libations I've collected here: "Ten Forward"

[18+] And I'd drink a wallbanger if it were available; it's just that as long as there are other options, some things just aren't on my radar, and this is one of them.

To understand why that is, consider the most popular liquors, all of which are often used as the basis for cocktails: vodka, tequila, rum, gin, and whisk(e)y.

Of those Big Five, vodka is, by far, the least interesting. Unless you get flavored vodka, it's generally neutral in character. The advantage of this is, clearly (pun intended), that it mixes well with other beverages. I don't hate it. I don't love it. It's just there to put alcohol into something that doesn't naturally contain it, such as, for example, orange juice.

Sure, there's some variation in quality between different brands of vodka, but in general, it doesn't have much flavor of its own. This is good if you don't like the other available flavors, or if you're just trying to get drunk.

I have a higher purpose in mind, which is to enjoy what I'm drinking while I'm drinking it.

So, as I'm sure we all know, if you mix vodka and orange juice, you get a drink called a screwdriver in the US (In the UK, I think it's less creatively called "vodka and orange.") A quick search turned up the fun fact that this very simple cocktail wasn't really invented until WWII, which surprised me until I remembered I did an entry long ago about the origins of commercially produced orange juice, which I seem to recall weren't much earlier than that. I even bothered searching for that entry, and here it is, from last year: "Orange You Special"

As with the subject of today's entry, the origin of the name is disputed.

A Harvey Wallbanger, then, is basically a screwdriver with the addition of, specifically, Galliano, an Italian liqueur. And quoting from the headline link again: "The drink did not gain in popularity until the 1970's, when George Bednar of McKesson Imports Company promoted it, as a way to sell Galliano, which his company handled."

Thus, it's just another marketing gimmick. So was orange juice, if you read my earlier entry.

That's a good enough reason for me not to order one at a bar, and I have little desire to clutter my liquor cabinet with yet another flavored liqueur.

Besides, why get one of those when there are so many cocktail options involving gin, rum, tequila, or whisk(e)y?

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