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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Elisa: Middle Aged Stik

a day ago
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WakeUpAndLive️~Happiness

a day ago
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a day ago
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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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November 6, 2023 at 10:21am

#1059020

Worlds Enough

Today's article is long. There's a link to the podcast version on the site, if you're into that sort of thing; I'm not. The text is presumably a transcript of the podcast, or the script. It touches on topics I've covered here in the past, related to astronomy, so in contrast to the article, I'm keeping my commentary short.

Journey to the Invisible Planet
Long-Form/Podcast: The tangled history of humanity’s search for the solar system’s uncharted planets.

I wrote about the potential Planet Nine back in August, and I've taken to calling it "Planet Ix"

.

What we have here is a comparatively brief (compared to the actual time frame involved) history of our discoveries about gravity and the solar system, starting with Newton. There's a discussion of Neptune—the first planet to be predicted by calculation prior to being telescopically observed.

But then they tried the same trick on the orbital perturbations of Mercury, thinking that with the success of finding Neptune due to anomalies in (the seventh planet)'s orbit, they could discover a world orbiting between the sun and Mercury, which would throw off our numbering system. Mercury would become the second planet from the sun. Earth, fourth. The seventh planet whose name I will not type in an attempt to avoid juvenile puns, the eighth.

Apart from its apparent influence on Mercury, the only evidence for another planet was a long history of spotty, unreplicated glimpses and blurry solar photos. It was the sea serpent of the solar system. Nevertheless, some manufacturers of solar system charts began including the intra-Mercurial body as a presumptive planet, and astrologers began to include Vulcan’s movements in their horoscopes.

Said horoscopes would have been just as accurate as those without it, to be fair.

In short, this proposed and popularly accepted planet, named Vulcan for its presumed temperature, didn't exist. Instead of modifying the other planets' numbers, we had to modify the theory of gravitation that otherwise worked so brilliantly. By "we," I mean "Einstein."

The International Astronomical Union, a body responsible for naming celestial objects, continues to reserve the name “Vulcan,” just in case we do one day detect an invisible planet lurking in the heart of our solar system.

You want to tell them, or should I?

How this relates to the maybe-Planet Ix is that we're still finding anomalies, even with relativity taken into account, that point to the possibility of another large-ish planet out beyond Neptune. Besides Pluto. Besides Eris, which after all is probably about the size of Pluto and largely the impetus for demotion of that world to dwarf-planet status.

If it does exist, it might have an orbit measured in thousands of years. Add that to your charts, astrologers.

The article/podcast concludes with a broader discussion of the limitations of the revised gravitational theory.

It is tempting to draw parallels between dark matter, dark energy, and planet Vulcan⁠—all of them elusive, invisible influences plugging holes in accepted theories.

I have, indeed, drawn such parallels in the past.

After all, the universe isn’t obligated to agree with human intuition.

Nor does it.

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