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

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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.

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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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October 30, 2023 at 9:32am

#1058293

Martian Law

So you don't want to live on this planet anymore?

Everything About Mars Is The Worst
But this jerk planet is still humanity’s best hope for another home in the cosmos.

The idea of life on Mars (whether alien or transplanted from Earth) is obviously one of the oldest themes in science fiction and fantasy. It's so pervasive that some people take for granted that 1) we will colonize Mars and 2) at some point after we do, the colonists will rebel and declare independence, most likely resulting in a war.

If 1 then 2, because Mars is, after all, a god of war, and because all colonies eventually revolt; but 1 is still questionable.

At first glance, Mars seems pretty nice.

Compared to the other planets we know of that aren't Earth, sure.

No other world in the solar system offers us this chance. Mercury is way too close to the sun. Nearby Venus has far too much atmosphere, whose pressure and noxious gases would crush and choke visitors from Earth.

I vaguely recall a thing I linked here a while back that ran the numbers and concluded that the closest planets to Earth, on average, are 1) Mercury 2) Venus 3) Mars. I think it depends on how you calculate it. If this doesn't make sense, consider all the time that these planets spend on the other side of the solar system from us.

And yet, thanks to quirks of orbital dynamics, it's easier to get to Mars. Which doesn't mean it's easy.

At night, temperatures drop to -100 degrees Fahrenheit. Dust devils and shifting sands cover up solar panels and will test even the most tightly sealed spacesuits and habitats. During dust storm season, Martian winds can stir up haboobs that cover the entire globe in clouds of sun-blotting microscopic particles.

That's with an atmosphere that's less than 1% the density of our own troposphere. Also, "haboobs" still makes me chuckle because I'm actually 12 years old.

Humans have been slinging spacecraft Marsward for 57 years, and we’re still not even batting .500.

It has been pointed out that Mars is the only world believed to be populated entirely by robots.

So far, the U.S. is the only country to land anything on Mars, and we’ve stuck the landing on eight of nine attempts.

If I recall correctly (and I might not), the reason the one mission failed is that someone forgot to convert metric to imperial units or vice-versa, which is actually kinda hilarious.

Anyway, the article continues with details of some of the technical challenges, which I won't go into. Numerous SF authors have, naturally, imagined possible ways around each of them, up to and including terraforming. I've done a bit of fictional speculation along those lines.

But I've read enough science fiction to know that it's not going to end well, in any case.

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