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 17, 2024 at 10:24am

#1068972

Down Under

Something interesting from the BBC last year:

The Australian town where people live underground

I thought all Australians lived underground. Otherwise, they'd fall up into space.

I mostly saved this article so I could use the entry title I did. I'm inordinately proud of that pun. But it's also a fascinating concept, at least to me.

These are the first signs of Coober Pedy, an opal mining town with a population of around 2,500 people. Many of its little peaks are the waste soil from decades of mining, but they are also evidence of another local specialty – underground living.

That name sounds as Australian as kangaroos and Crocodile Dundee. I wonder if the article will state its origin.

In the winter, this troglodyte lifestyle may seem merely eccentric. But on a summer's day, Coober Pedy – loosely translated from an indigenous Australian term that means "white man in a hole"...

Of course it did, and it's hilarious.

...needs no explanation: it regularly hits 52C (126F), so hot that birds have been known to fall from the sky and electronics must be stored in fridges.

Yeah, I prefer heat to cold, but that's just going too far. I thought Phoenix was bad in the summer.

As the blistering three-month heatwave continues in the US – with temperatures even cacti can't handle – and wildfires incinerate swathes of southern Europe, what could we learn from Coober Pedy's residents?

Again, this article is from last year. August, specifically. At least it was winter then in Australia.

People have been retreating underground to cope with challenging climates for thousands of years, from the human ancestors who dropped their tools in a South African cave two million years ago, to the Neanderthals who created inexplicable piles of stalagmites in a French grotto during an ice age 176,000 years ago.

Well, duh. It's not exactly news that humans and related apes lived in caves.

Caves served another very important purpose, as well. As the article notes, most underground locations that are deep enough—provided one isn't near a volcano or whatever—stay around 13C, or 55F. Before refrigeration, this was an common way to keep beer and wine relatively chilly, and thus preserve it longer. Even now, a lot of fermented beverages are best served at "cellar temperature."

Apart from comfort, one major advantage of underground living is money.

That, and your HOA can't give you shit if you happen to paint your house the wrong shade of pink.

On the other hand, many underground homes in Coober Pedy are relatively affordable. During a recent auction, the average three-bedroom house sold for around AU$40,000 (£21,000 or US$26,000).

I'd imagine that at least part of that is its remote location.

The question is, could underground homes help people to cope with the effects of climate change elsewhere? And why aren't they more common?

Guessing that at least part of the answer to that last question is "rock" and "water tables." And also "building codes" and "zoning laws." It's my understanding that, to be considered a bedroom in the US, a room must have a window. Windows aren't exactly a common feature of underground living.

Besides, if your neighbor lives in a cave, how are you supposed to judge their lifestyle by how they keep their yard?

There are several reasons why making dugouts in Coober Pedy is uniquely practical. The first is the rock – "It's very soft, you can scratch it with a pocket knife or your fingernail," says Barry Lewis, who works at the tourist information centre.

Okay, so I might have cheated a bit with my "rock" answer, because further up in the article, they mentioned it was sandstone and whatnot.

However, the feats at Cooper Pedy would not be possible everywhere. One major challenge with any underground structure is damp.

I did say "water tables." Civil engineering education is good for something.

But in Coober Pedy, which sits on 50m (164ft) of porous sandstone, conditions are arid even underground. "It's very, very dry here," says Wright. Ventilation shafts are added to ensure an adequate supply of oxygen and to allow moisture from indoor activities to escape, though these are often just simple pipes sticking up through the ceiling.

The downside of building, even underground, in a desert is the other side of that equation: where do you get your water? I missed it if the article addresses this. I imagine there's an aquifer further down, but I don't know.

The article also doesn't note another important thing about learning how to live underground. If we end up with people living on the Moon and/or Mars, they're pretty much going to have to do so in underground habitats. Not only does the rock above you provide a barrier to space radiation, but it all but eliminates potential problems with small meteorites punching a hole in your bio-dome or whatever.

Also, nearly unlimited beer storage.

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