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 11 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 14 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

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

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
In response to "Paris, I Guess"

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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October 20, 2023 at 11:11am

#1057739

Navy

Navy
an entry for "Journalistic Intentions"

[18+]

Of all the weird, strange, or just plain incomprehensible names for colors, "navy" stands out as one that almost makes actual sense... from a certain point of view. Under certain ideal conditions, the ocean appears that deep, dark blue color, for reasons I can't be arsed to go into right now but you're right; it does involve physics.

But our word, navy, comes from a Latin word that referred more to ships than the ocean. Still, you know, you can't have a ship without something to float it in. Which reminds me that another word from the Latin navis is navigate, whose meaning should be limited to finding one's way around on a body of water. But, naturally, it's not, because we're pretty good at making new meanings for old words, and new words for old meanings.

Take, for example, space. Thanks to years and years of science fiction, we know what a vessel that carries things around in space is: a spaceship. And again, that makes sense from a certain point of view. But the harsh reality is that if we do get to the point where we have spacecraft transporting people or things around out there, the vessels will have far more in common with a submarine than they do with a ship. And, by naval convention, submarines are always boats, not ships.

"Spaceboat" just doesn't have the same ring to it, though, does it?

Don't ask me to define the difference between boats and ships further. My dad was a sailor, and I never fully understood the distinction he made, when he bothered to make one. Near as I can tell, a boat goes out from port or ship and returns to the same port or ship, while a ship carries cargo and/or passengers from one port to another (hence the verb "to ship," which also refers to sending parcels by road or rail). But by that definition, a ferryboat should be a ship, but it's not (I sidestep this by calling it a "ferry"). A ship can carry a boat, but a boat can't carry a ship, though a tugboat can push a ship, despite "tug" having the connotation of "pull."

Anyway. The other thing we get from science fiction is the use, in spaceships, of a "navigator," like on the original Enterprise. Properly, this should be "astrogator," but that would just lead to alligator puns (though some SF does use this term), so probably best to repurpose the word. And don't get me started on the etymology of "bridge," as in a spaceship's control room, which takes its origin from Mississippi River steamboats... dammit, I said don't get me started.

Still, sometimes it bugs me that landlubber GPS uses "navigation." I mean, we want fewer people not paying attention and driving their cars into lakes, right? But again, there's not much to choose from in terms of better words. "Orienteering" is the process of finding one's way around on land, but that doesn't really work in cars or trucks, does it? So we're stuck with navigation.

But even that makes more sense than calling a web browser a "navigator." Which you don't see much these days in English, but one early web browser was Netscape Navigator. And the French word for browser is "navigateur." This wouldn't bother me so much if the defining metaphor of the Internet weren't a spiderweb, rather than an ocean.

You could say all the contradictions give me the blues.

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