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:

Write_Mikey_Write!

about 11 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.

1 Like

Adherennium Plotting something

about 12 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.

1 Like

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.

June 15, 2020 at 12:16am

#985674

Meat the Press

This article is a couple of years old, now, and there may have been some changes in the landscape of its subject matter since then. But what the hell, right?

https://slate.com/technology/2018/07/should-lab-grown-meat-be-called-meat.html

Is Lab-Grown Meat Really Meat?
A labeling war is brewing.

This sounds like more of a philosophical question than a scientific one. What we call something is largely a matter of perception. For instance, someone who eats dead people to survive is a "cannibal," but someone who takes organs from a dead person to replace their own in order to survive is called an "organ recipient." There's no philosophical difference (though there would be if it weren't a matter of survival), but in our culture, one is taboo and the other is a triumph of medical science.

After centuries of a veritable monopoly, meat might have finally met its match. The challenger arises not from veggie burgers or tofu or seitan, but instead from labs where animal cells are being cultured and grown up into slabs that mimic (or, depending on whom you ask, mirror) meat... But there’s another, more immediate battle heating up between the cattle industry and these new entrants into the meaty ring. So buckle up and put on your wonkiest hat, because the labeling war is about to begin.

So, it's worse than a philosophical question. It's a marketing one.

In February, the U.S. Cattlemen’s Association wrote a petition to the U.S. Department of Agriculture, asking the government to ban cultured-meat companies from using the terms meat and beef at all. In response, a rival cattlemen’s association, the National Cattlemen’s Beef Association, wrote a letter opposing the petition.

Splitters!

This is not be the first time that food products meant to imitate or replace more traditional fare have faced questions about their labeling. In 1869, margarine was invented by a French chemist. As the butter replacement spread to the United States, dairy farmers raised the alarm.

This is not the same thing at all. Health comparisons aside, margarine is made from completely different ingredients. In contrast, cultured meat is chemically precisely the same as dead cow, pig, chicken, whatever.

David Henderson, a representative from Iowa, compared margarine to the witches’ brew in Macbeth.

You're not supposed to say the name! No wonder margarine still exists.

The rise in vegetarian and vegan food options in supermarkets has given us a few more examples of mimics and their labels. Soymilk and almond milk have been a thorn in the side of the dairy lobby for more than 15 years. The Soyfoods Association of America petitioned the FDA back in 1997, asking for permission to call their products “soymilk,” starting a long battle between soy manufacturers and dairy farmers. Dairy farmers object to these beverages being called milk, but thus far the FDA hasn’t done anything to stop brands from using the word.

This, too, is a marketing thing and maybe one of classification. Standing in a garage doesn't make you a car, but if it quacks like a milk, maybe it's a milk? I don't know.

So the question of who is going to dictate the labeling of cultured meat is something of a riddle, because it really depends on whether you see cultured meat as meat. From a production standpoint, cultured meat is more in line with the way that drugs and supplements and additives are made in a lab, and that would make the FDA more qualified to oversee things. But from a final product standpoint, if the lab-grown meat is going to wind up on the shelf next to the traditionally slaughtered stuff, it seems like the USDA should take charge.

With bureaucracy involved, we should get a final decision on this sometime in the 2070s. Or, I don't know; maybe they've already made the decision since this article was published. Depends on who gives the government the biggest bribe.

But what do they want, exactly? It turns out that different cattle lobbying groups want different things. This brings us to our second question, which is less bureaucratic, and more philosophical: What is meat anyway? Is this cultured meat truly meat? Should it be called meat in the first place? The lab-grown meat companies I spoke with are clear on their answer to this question: yes. “Our products meet the statutory definition of meat,” Eric Schulze, the vice president of product and regulation at Memphis Meats, told me by email. “Does it comes from an animal? Does it have the same biochemical makeup as meat? If yes, then it’s meat,” says Josh Tetrick, the CEO of Just.

And what is the sound of one hand clapping?

Don't answer that. It's 99% likely that I've already come up with the same answer.

So, okay, questions of definition, philosophy, classification, legality and marketing aside, let's consider empathy.

I'm a carnivore, but I'm well aware of the issues surrounding traditional meat production in a developed world. Factory farming, inhumane conditions, and so on. Now, for me, the taste of delicious meat outweighs those considerations. This makes me an asshole and probably a target for certain groups. I'm okay with that. But when I can be arsed to do so, I try to eat ethically-sourced meat. To me, lab-grown meat is the answer to this ethical quandary: I can have delicious meat, and cows can be cows instead of hamburgers.

I would be remiss if I didn't also note that, in the unlikely event that everyone became vegan at once, say, tomorrow, there would still be a lot of dead cows, pigs, chickens, etc. We just wouldn't be eating them. It would be like when horses became mostly obsolete as a means of transportation, only on a much bigger scale because there are a lot more humans now than there were 100 years ago. So all of those meat animals would still be killed, only instead of serving the useful function of being delicious, they'd just become carrion. Huge benefit for vultures; not so much for humans, and definitely not for the poor piggies.

Anyway.

Lab meat is a way around all of these ethical problems. It would be slow to catch on, so meat animals could decline in numbers slowly. People who enjoy meat but don't like the farming issues could eat meat again.

Except, of course, for the people who will have a visceral distaste for the whole idea. And if you don't think those people exist, look up the GMO controversy. I swear there's no pleasing some people. With this many people in the world, GMOs are really the only way to avoid a Malthusian collapse, and yes, I mean even after the paltry 2% drop in population due to COVID-19. And yet people freak out, calling them "frankenfoods," and forcing producers to label their products as to their GMO content or lack thereof.

And then there are religious prohbitions.

Observant Jews still won't eat pork, lab-grown or otherwise; I'm fairly confident of that. And I talked to a practicing Hindu about this once, and he said they probably still wouldn't eat lab-grown beef. The difference between the Jewish prohibition against pork and the Hindu one about beef is that pigs are considered unclean by the former, whereas cows are considered sacred by the latter. Couldn't be more different. And yet, the result is the same: nope.

I find the whole thing fascinating, though frustrating.

And now if you'll excuse me, I have some dead cow to put in my belly.

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