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 ✍️

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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

a day ago
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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"

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"

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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 27, 2023 at 10:14am

#1058145

Carmine

Carmine
an entry for "Journalistic Intentions"

[18+]

I always wondered about "carmine." Is it a bomb you put on a car to make it blow up? A hole in the ground where you dig for cars? Or is it pidgin for "my car?" Oooh, I know: it's a corruption of "karma mine," which is a place where you take out only what you put in.

Well, no. Sadly, it's none of those things, although they would absolutely make the world a more interesting place.

It is a not-unpopular name, especially for those of Italian origin, and one with several cognates, including Carmen (as in Sandiego) and Carmelita. But it turns out that the name has two origins, like a river with a fork: one from Latin, meaning song, and one from Hebrew, meaning garden. Not that famous "first" garden, though; that was something like paradise. (No, really, the original Hebrew word for it is also translated as paradise.) Just your regular old garden, though presumably one with flowers and bushes rather than cabbages. If one were poetically inclined, one could say that the name Carmine might mean "garden of song," which would presumably be attached to Leonard Cohen's "Tower of Song."

The way Hebrew might have gotten mixed up with Latin to create that name, I leave as an exercise for the reader.

I say all this because one might be forgiven for thinking that the name and the word for a certain deep red color share an origin. They do not. Though one Carmine I heard of, Carmine Infantino, was a comic book artist well-known in some circles (comic readers) for a great run on the Deadman series, and Deadman wore a carmine-colored onesie. Coincidence? Well, actually, yeah. Infantino was drawing an established character. Also, only the reader (and maybe other dead people in the comic) could see Deadman, as he was a ghost. His superpower was possession, incidentally.

What's not a coincidence is that carmine and crimson, the color words, share a linguistic root. In the analogy above, that would be two rivers sharing a single source, which is kind of uncommon for rivers when compared to the other way around, but not so rare in language.

No, carmine-the-color came from bugs.

Apparently, there was this one species of insect which, when tortured, killed, and ground up, produced the carmine dye. I have no idea if this is still the practice, but at least no insects were harmed in making that color in your photo editing software. After all, something's gotta live in your garden.

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