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Sunday, Nov. 17, 2024

11:31am EST

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Turkey DrumStik

a day ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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.

1 Like

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').

1 Like

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

April 2, 2024 at 10:56am

#1067394

Sandpeople

Lots going on today, so I'll try to keep this short. It's from SciAm, and I'd find it amusing if it weren't, well, not amusing.

Inside the Crime Rings Trafficking Sand
Organized crime is mining sand from rivers and coasts to feed demand worldwide, ruining ecosystems and communities. Can it be stopped?

Good thing this didn't come up yesterday. People might think it was a prank.

He had been in Mali doing fieldwork on the drug trade when a source noted that most cannabis in Mali came from Morocco and that sand trafficking was also a major market in that country, with drug traffickers involved. “I think that when you talk about sand trafficking, most people would not believe it,” Abderrahmane says. “Me included. Now I do.”

I'm still not entirely sure I believe it.

Very few people are looking closely at the illegal sand system or calling for changes, however, because sand is a mundane resource. Yet sand mining is the world's largest extraction industry because sand is a main ingredient in concrete, and the global construction industry has been soaring for decades. Every year the world uses up to 50 billion metric tons of sand, according to a United Nations Environment Program report.

For comparison, a metric ton is slightly more than an Imperial ton, less than a shit-ton, and a lot less than a metric shit-ton.

A 2022 study by researchers at the University of Amsterdam concluded that we are dredging river sand at rates that far outstrip nature's ability to replace it, so much so that the world could run out of construction-grade sand by 2050.

Of all the things I'd expect us to run out of, sand is way down there on the list, right above human stupidity.

Sand is any hard, granular material—stones, shells, whatever—between 0.0625 and two millimeters in diameter. Fine-quality sand is used in glass, and still-finer grades appear in solar panels and silicon chips for electronics. Desert sand typically consists of grains rounded like tiny marbles from constant weathering. The best sand for construction, however, has angular grains, which helps concrete mixtures bind.

I've gone on about categorization problems in here before. This is one that has apparently been well-defined.

In case you're wondering, since "angular grains" and "diameter" appear to be mutually exclusive, sand is, in my experience, graded by successive sieves. Any grains that fit through a 2mm sieve but not a 0.0625mm sieve can qualify as "sand." Those limits are arbitrary, but useful in concrete design, with which I've had some familiarity.

There's a lot more to the article, but, as I said, I'm trying to be brief, finishing before too much sand passes through the hourglass. I don't have an answer for the problem, but it's one I'd never considered before, so I thought it was worth sharing.

7 Like • Comment • View 2 comments

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