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:

🌖 HuntersMoon

about 4 hours ago
In response to "Meat Me In the Lab"

Congratulations, Bob! You managed to get through the whole subject without once mentioning Soylent Green.

Mmmmmm. Real meat!

1 Like

tj is ready for spring!

a day ago
In response to "Walk the Prank"

If a practical joke backfires is it an impractical joke?

2 Like

tj is ready for spring!

Jan 14, 2025 at 12:41pm
In response to "Failure Is Victory"

Just think, back in the mid seventeen hundreds, Ole Ben had himself an electric BBQ; he stunned the turkey, shocked himself and cooked the bird on a gadget called an electric jack. Personally, I think he should have used a pellet smoker.

2 Like

🌖 HuntersMoon

Jan 14, 2025 at 9:52am
In response to "Failure Is Victory"

"Yes, even I make mistakes sometimes."

You may remember, that once words are placed on the internet, they last forever. They WILL come back to haunt you.

3 Like

tj is ready for spring!

Jan 13, 2025 at 12:55pm
In response to "Rebel Without a Clause"

It's just like this sometimes.

1 Like

🌖 HuntersMoon

Jan 12, 2025 at 9:58am
In response to "Revisited: "Squash Court""

tj is ready for spring!/Robert Waltz

Oh My God!

Now I understand the etymology of "Pun-ishment."

3 Like

Elisa: Snowman Stik

Jan 12, 2025 at 9:44am
In response to "Revisited: "Squash Court""

Ironically I'm eating scrambled eggs with zucchini as I read this entry.

As someone who has shopped for and cooked delicata squash, I can say that they're mostly seasonal. I occasionally see them in the summer, but they're far easier to locate in the cooler months. They haven't quite broken out the way zucchini has or even acorn squash. I can find acorn squash year round, although admittedly I don't eat it in the warmer months. Speaking of which, if I had to guess, I think pumpkins aren't sold year round because A. summer demand and applicability is low and B. size is a factor. Acorn squash is grown in similar areas/conditions as pumpkin, but it's a smaller size. Grocery stores can carry it year round in a smaller footprint compared to pumpkin even if fewer people are buying it in the warmer months.

4 Like

Robert Waltz

Jan 12, 2025 at 9:41am
In response to "Revisited: "Squash Court""

tj is ready for spring!

- if he has lesser relatives in the produce section, that must be why he's called Bigfood.

4 Like

tj is ready for spring!

Jan 12, 2025 at 9:38am
In response to "Revisited: "Squash Court""

Here in the north woods there are rumors of a giant squash that roams the woods; Sasquash! I've never seen him, but do, on occasion, come upon his lesser relatives in the produce section.

4 Like

tj is ready for spring!

Jan 11, 2025 at 12:48pm
In response to "Time Marses On"

I agree. When we finally get close to inhabiting Mars, I doubt anyone will be scouring through sci-fi books to find a design, asking, "How 'bout this, it looks pretty cool."

1 Like

January 16, 2025 at 9:13am

#1082410

Meat Me In the Lab

Today's article possesses the unfortunate combination of being long, four years old, and from a source (Town & Country) that I don't usually link.

Fake Steak, Well Done
Science is promising us steak that’s heart-healthy, eco-friendly, and still decadent. But will we eat filet mignon from a bioreactor?

I don't know about "we," but I absolutely would—assuming it's comparable in taste and texture to actual dead cow. When it comes to eating meat, I don't actually enjoy the idea of killing animals. It's just that the delicious taste of it is more than enough for me to overcome any moral objections.

See? I have principles. They're just selfish ones.

Now, like I said, this is a long article, so I'm only going to include a few highlights.

Aleph is one among an expanding field of companies racing to bring to market what they would rather not be called “lab-grown meat” (they prefer “cultivated” or “slaughter-free”).

What you call something matters. For instance, they could have marketed GMOs better as, I don't know, Power Plants or something. Instead, the freaks and astroturfers got a hold of the idea of calling it "frankenfood" and that was so catchy that it caused people to actually believe that there's something wrong with GMOs.

Though the technology did not exist even just a few years ago, today at least 33 startups in 12 countries are producing a variety of meats—from dog food to foie gras, pork to duck, chicken nuggets to beef patties. Some are promising cultivated meat in stores next year.

As far as I've heard, cultivated meat didn't, in fact, make it to stores that next year, which would have been 2022. Nowhere have I found anything that indicates that the technology has been scaled up for mass production yet.

Thanks to our palates, Americans don’t generally eat bats, the animals most widely suspected of harboring SARS-CoV-2’s precursor, but two other potentially fatal viruses, the influenza strains H1N1 and H5N1, have come from poultry and livestock in recent years—suggesting that more are on the way.

On the other hoof, this sentence turned out to be prescient.

And if pandemics aren’t enough to convince people, maybe antibiotic resistance is. Cattle producers discovered some time ago that giving their animals antibiotics to head off any possibility of bacterial infection also causes even healthy cattle to grow faster.

I will give them points for this: every other time I've seen cattle antibiotics mentioned anywhere, the phrase used is "pump them full of antibiotics." It's long past being a cliché, and I'm glad the article avoids this overworn, overused, tired phrase that nevertheless attempts to sensationalize the practice.

Overuse of antibiotics has accelerated the evolution of bacteria that can resist them, and now around 700,000 people all around the world die every year from what should be treatable infections.

Of all the things science should have seen coming, this is right there at the top of the list. Or maybe they did see it coming, but figured we'd just make better antibiotics. I don't know. But it stems from a basic principle of evolution: the organisms that have resistance to pressures can go on to pass that resistance onto their offspring.

Look at it this way: suppose that, every once in a while, a bulletproof deer is born. Normally, the bulletproof trait doesn't breed true. But after you send hunters into the woods with the mandate to reduce the deer population, the bulletproof deer won't be leaving with them. No, it'll stay in the woods with a few lucky stragglers. Eventually, another bulletproof deer is produced. Produce enough of them, and let them breed together, and within a few years, you have an entire population of bulletproof deer who proceed to take over the forest.

The only thing left to do, then, is nuke the site from orbit, which has the unfortunate side effects of a) destroying the entire ecosystem and b) accelerating the rate of genetic mutation in the nearby populations.

I was at the dentist yesterday, which meant I was subjected to television ads. They've only gotten worse since I last saw them. But I digress. One of them was, appropriately enough for the location, for mouthwash. "Kills 99% of germs that cause bad breath!" the commercial proclaimed.

Okay, yeah, sure. Even if that's true (which it's not, except maybe in a petri dish or something), that other 1%? Yeah, those will go on to reproduce into an entire population of Listerine-resistant bad-breath germs. Result: in 20-200 years or so, everyone will have bad breath and there won't be anything we can do about it except maybe nuke the site from orbit.

All of which is to say that bacteria evolving antibiotic resistance through selection pressures was entirely and completely predictable, even by non-scientists with some knowledge of science.

And that's a major digression, so I won't be quoting any more from the article. My only excuse is that this sort of thing won't apply to lab meats (or whatever marketing name they settle on).

As I said, I would absolutely try this sort of thing. Hopefully, it'll be better for the environment. I'm sure there are people, and some are mentioned in the article, who will oppose it on principle. Maybe because they're trying to promote their own meat alternatives. Whatever.

But I want to throw a hypothetical out there. If you can do this with pig and cow and chicken... what, besides the ew factor, would stop them from creating lab-grown human meat? There's already talk about 3-d printing human organs for replacement, and no one seems to have ethical issues with that.

I guarantee you, I'm not the only one who's had that thought.

2 Like • Comment • View 1 comment

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