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 13 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 16 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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September 29, 2023 at 11:19am

#1056392

Ring World

It's been a few weeks, I think, since the last time one of the solar system articles came up from my queue. This one's about Saturn.

Saturn Could Lose Its Rings in Less Than 100 Million Years
Recent discoveries suggest that the planet’s distinctive feature may be gone in the cosmic blink of an eye

I guess this is considered noteworthy because, with a few notable exceptions such as comets or sunspots, we tend to think of astronomical objects as relatively unchanging. And, from a human perspective, 100 million years is a very long time indeed. 100 million years ago, our ancestors were mouse-sized quadrupeds, hiding from dinosaurs. I did an entry a while back on Jupiter, that other gas giant in the solar system, and we've witnessed the shrinkage of its iconic Great Red Spot over the past couple hundred years—in comparison, lightning-fast. Now it might be more properly called the Okay Red Spot.

Anyway, back to Saturn...

If someone asked you to draw a planet other than ours, you would likely draw Saturn, and that is because of its rings.

Rings are very common in SF art, to indicate "hey, a different world." And yet, of all the planets we're most familiar with, only the one has easily identifiable rings.

It was Galileo Galilei who first spotted something there. His primitive telescope gave him only a slightly better view of the heavens than did the naked eye, and in 1610 he thought he saw two undiscovered bodies flanking Saturn, one on each side.

I think he called them "ears." Some of his, and others', early drawings are reproduced in the article. As the article notes, when they're edge-on, they basically disappear from Earth-based views.

The particles in the inner rings move faster than those in the outer rings, because they are fighting against a stronger gravitational pull.

That's... well, that's mostly correct. Except they're not "fighting against" anything.

The average thickness of the main rings is believed to be no more than 30 feet. A recent study showed that parts of the B-ring—the brightest ring of all—are only three to ten feet thick.

I've known this for a while, but it still amazes me. Considering the overall breadth of the rings, this makes them far thinner, proportionately, than a sheet of paper.

Most of Saturn’s rings lie within what’s known as the Roche limit—the distance a satellite can orbit a large object without the planet’s tidal force overpowering the object’s own gravity and tearing it apart.

Which might well be how they formed in the first place.

The article goes on to describe how one researcher figured out (probably) how the rings were disappearing. In 2012. Just goes to show that there's no apparent end to discovery.

There's also lots of cool pictures of the rings, of course. From close-up. Because we've sent robots there. We're also treated to what I consider an excessively long biography of the researcher in question, which, admittedly, I skimmed.

“Take a Good Look at Saturn Before It’s Too Late,” Time magazine cheekily warned, “Because It’s Losing Its Rings.”

And this is why I have issues with most science reporting. There's another article in my queue, which I'll get to eventually, about the recent discovery of potential life-indicating chemicals on an exoplanet. Popular media immediately jumped to techno-aliens in flying saucers.

But. This article shies away from such sensationalism.

Because the reality of it is sensational enough. I mean... just look at those fucking rings!

4 Like • Comment

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