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 7 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 10 hours ago
In response to "Landmarks"

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 21 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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December 1, 2023 at 12:43pm

#1060387

Travel Advice, Part 3

When traveling, it's important to be aware of local customs. For example,

Pronunciation

When traveling around the US, you need to be aware that, often, place names aren't pronounced the way you'd think based on the usual rules of English. For example, there's a town near where I live called Staunton. You'd think, looking at it, it's pronounced stawn-ton, right? No, it's pronounced che-vee. Okay, no, just kidding; it's stan-ton, with an a like in hat.

Worse, though, are places that are pronounced like you'd expect in English... but they're Spanish or French. Like, if you pronounce Versailles, Indiana, as if you'd just stepped off the plane from Par-ee, the locals will kick your derrière all the way to Illinois (a state whose name is pronounced like the French would).

But you just never know, so unless and until you're absolutely, 100% certain of something's pronunciation, usually by hearing a local speak the word, it's best to not mention the name at all. Some folks don't take kindly to strangers, and their best way of identifying a stranger is by their mispronunciation of the local town, street, or dive bar.

Like, in my hometown, we have a Rio Road. I can always identify the tourists because they pronounce it like Rio de Janeiro or Rio Grande. But no. It's pronounced like rye-oh. Don't ask me why; I just know we use it as a shibboleth so we know who to send to the more expensive restaurants.

As I said, though, you never know. You could have a place in the US named some common, easy-to-pronounce English words like, I don't know, New Haven, and it'll turn out that Haven is pronounced like havin'. As in, don't bother me, I'm havin' lunch. As far as I know, that's not the case with the one in Connecticut, but you know it could happen.

Or, like, I can see myself visiting a small town with a common name, and someone asks "How do you like it here so far?"

I'll be like "Oh, it's nice. I really like Riverside."

And their face will immediately close up and they'll look sideways at me with great suspicion. "It's 'reverse-a-dee.'"

Look, I'm just saying I wouldn't be surprised, okay?

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