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 5 hours 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.

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Adherennium Plotting something

about 7 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.

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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.

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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