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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 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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