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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 10 hours ago
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

a day ago
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

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 12, 2024 at 12:49pm

#1073868

Arsenic and Old Rice

Apologies for the Torygraph link today. If it helps, I'm going to rag on the article. Today in "You're doing it wrong!":

How we are all cooking rice incorrectly - and could be endangering our health

Millions of cooks are endangering their health by cooking their rice incorrectly, scientists believe.

Which scientists? The link provided with that lede goes to another Torygraph article scaremongering about lung cancer risk from high glycemic index carbohydrates, which violates at least two principles: 1) make your link relevant to the argument; 2) don't freakin' cite your own prior article as a source.

Putting more water in the pan or even steeping it overnight is the best way to flush out traces of the poison arsenic, they found.

Arsenic is well-known as a poison, but it's part of the natural environment (yet another reason I distrust "natural" claims).

I can't be arse(nic)d to go into much detail here, but it's an element on the periodic table, not some factory-produced compound.

The chemical contaminates rice as a result of industrial toxins and pesticides which can remain in the soil for decades.

It is true, however, that certain processes can concentrate and redistribute the element.

Experts have long debated what level of arsenic is safe, with new limits set by the EU in 2016.

"Safe" is misleading. The only thing I found on limits has to do with American public drinking water, where the limit is 10 parts per billion.

Usually. But that's for inorganic arsenic; organic arsenic (which, as that link points out, has nothing to do with the marketing concept known as "organic") is less toxic. Remember, it's an element, so it shows up in different kinds of molecules.

But experiments suggest that the way rice is cooked is key to reducing exposure to the toxic but naturally occurring chemical.

Or we could, I don't know... stop polluting everything?

Prof Andy Meharg, from Queens University Belfast, tested three ways of cooking rice for the BBC programme Trust Me, I’m a Doctor.

I'm way more inclined to trust the BBC than the Torygraph. Which doesn't mean they're perfect.

In the first, he used a ratio of two parts water to one part rice, where the water is “steamed out” during cooking.

In the second, with five parts water to one part rice, with the excess water washed off, levels of arsenic were almost halved.

And in the third method, where the rice was soaked overnight, levels of the toxin were reduced by 80 per cent.

That first one is the usual method, at least here in the US.

There's just one problem with the alternative methods. Well, perhaps more than one, but the one that jumped out at me was: that's a lot of water wasted. Which I guess isn't much of a problem if you're in the UK, where, I'm assured, it rains all the goddamned time. But in many parts of the world, water is precious (and becoming more so as time goes on).

This isn't usually much of an issue in what we've called "first-world" countries, but they get a different problem: insecure and sometimes questionable water supply. Which, as I've noted above, has a maximum arsenic level greater than 0.

Wells and other sources aren't formally tested at all.

To summarize, rice is a staple for more than half the world; rice has been found to have an unspecified but presumably risky level of arsenic; and using 2-3 times as much water, and possibly even more for rinsing after, might reduce arsenic levels... at the cost of time, water, and effort.

And once again, fixing the problem of industrial pollution falls squarely on our shoulders, rather than those of the companies who did the pollution in the first place.

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