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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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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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

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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
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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
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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
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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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September 23, 2023 at 10:13am

#1056105

Crepuscular Musings

Rather appropriate for the first day of fall...

Why Are There Three Twilights?
And we’re not talking about vampire books.

Yes, I said fall. I get really, really tired of saying "or autumn if you're in an Anglophone country that's not the US, or spring if you're in the southern hemisphere, but some cultures considered this to be the midpoint of the season, so it gets complicated." As mine is the only perspective that matters, it's fall, and today sucks because we're getting spitting rain thrown off from a minor hurricane.

Anyway. Why is it appropriate? Because while "equinox" literally means "equal night," implying that day and night are of equal length, that's not technically the case on the equinox, and part of the reason is twilight.

For example, this is the (badly formatted) almanac for my area today (courtesy of Weather Underground):

Sunrise 7:03 AM Sunset 7:11 PM
Civil Twilight 6:37 AM 7:38 PM
Nautical Twilight 6:06 AM 8:08 PM
Astronomical Twilight 5:35 AM 8:40 PM
Length of Visible Light 13 h 0 m
Length of Day 12 h 8 m

"Length of visible light" takes twilight into account. "Length of day" is, you'll note, 8 minutes longer than it would be if day were actually equal to night. That's because the atmosphere refracts sunlight, so the sun appears to rise earlier and set later than it would if we didn't have an atmosphere, which, if that were the case, there would be no one around to be picky about these things.

The three twilights there... well, that gets us back to the Atlas Obscura article.

When the sun slides out of sight—or, more accurately, when Earth rotates so that your particular place on the planet is no longer exposed to our day star—but before darkness consumes the landscape, there is the magical, muted time of twilight.

One of my all-time favorite words is "crepuscular." Animal activity can be crepuscular, or there might be crepuscular rays visible in the sky due to distant clouds. It's an adjective meaning "of or pertaining to twilight." I'm unaware if it has a noun form; "crepuscule" just sounds wrong.

Most of us think of twilight as a single period of transition from bright day to deep night, but there are actually three twilights: civil, nautical, and astronomical. Each is defined by how far below the horizon the sun is, and hint at our lives before artificial light and GPS.

Or, for those of us who care, our lives now.

Civil twilight has nothing to do with being polite; it’s simply the evening’s first twilight, which starts at sunset: the moment when the sun’s center is exactly at the horizon or, to get technical, 0 degrees below it.

Again, this is not precisely true because of that refraction thing. It also happens, in reverse, near sunrise.

Once the sun is 6 degrees below the horizon, nautical twilight arrives. If you’re not a degrees kind of person, hold your arm out to the horizon, with your hand turned so your palm is facing the horizon and your fingers, parallel to the horizon, are together but extended. Close one eye. The width of your first three fingers is roughly 5 to 6 degrees, depending on the size of your hand.

That's nice to know and all, but at that time, you can't bloody see the bloody sun, so how do you know how far below the horizon it would be without math and a clock?

Nautical twilight is the period after sunset when, though no longer able to read a chart without a lantern or torch, sailors could still take accurate readings.

Reverse that for morning. Also, this doesn't take into account cloud cover or moon phase, which can shorten or lengthen the time when one could see the horizon. Also also, things are a bit different on land because you sometimes have hills and mountains and stuff. Your own height of observation plays into it as well.

Regardless of all these variables, it's still cool stuff to know.

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