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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Write_Mikey_Write!

about an hour ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 3 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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