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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Elisa the Bunny Stik

about 2 hours ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

August 15, 2020 at 12:08am

#990810

Wait, Isn't Oxygen A GOOD Thing?

Bit of a sensationalist headline here, but I found the article interesting.

http://nautil.us/issue/15/turbulence/fruits-and-vegetables-are-trying-to-kill-yo...

Fruits and Vegetables Are Trying to Kill You
Antioxidant vitamins don’t stress us like plants do—and don’t have their beneficial effect.

I should point out that this article is from 2014. Since then, I'm sure nutrition science has refuted, confirmed, refuted, confirmed, and refuted again the content by now.

You probably try to exercise regularly and eat right.

I did. For a while. In the Before Time.

Perhaps you steer toward “superfoods,” fruits, nuts, and vegetables advertised as “antioxidant,” which combat the nasty effects of oxidation in our bodies.

Nope.

Maybe you take vitamins to protect against “free radicals,” destructive molecules that arise normally as our cells burn fuel for energy, but which may damage DNA and contribute to cancer, dementia, and the gradual meltdown we call aging.

Here's where the refutations come in. Over-the-counter vitamins seem to be, at best, harmless and ineffective; at worst, harmful. But that's today. Wait a few hours.

But that's really the point of the article here.

Warding off the diseases of aging is certainly a worthwhile pursuit. But evidence has mounted to suggest that antioxidant vitamin supplements, long assumed to improve health, are ineffectual.

"Assumed?" No one did any actual, you know... studies?

Fruits and vegetables are indeed healthful but not necessarily because they shield you from oxidative stress. In fact, they may improve health for quite the opposite reason: They stress you.

Then there's the whole "fiber" thing, which I'm pretty sure they have done studies on.

That stress comes courtesy of trace amounts of naturally occurring pesticides and anti-grazing compounds. You already know these substances as the hot flavors in spices, the mouth-puckering tannins in wines, or the stink of Brussels sprouts. They are the antibacterials, antifungals, and grazing deterrents of the plant world. In the right amount, these slightly noxious substances, which help plants survive, may leave you stronger.

I do love me some hot pepper. Brussels sprouts, though? Thanks to science, they don't stink as much as they used to. And do I need to sing the praises of wine here?

Parallel studies, meanwhile, have undercut decades-old assumptions about the dangers of free radicals. Rather than killing us, these volatile molecules, in the right amount, may improve our health. Our quest to neutralize them with antioxidant supplements may be doing more harm than good.

This actually tracks with some other stuff I've read. But that doesn't mean much. As you know by now, when it comes to nutritional science, I don't believe a damn thing.

Oddly, our mitochondria, the energy factories of our cells, emitted ROS naturally. So degenerative disease seemed to stem in part from our own metabolic function: Your mitochondria “burned” fuel, emitted this toxic exhaust, and inadvertently set the limits on your existence. That was the working hypothesis, at any rate.

This is what happens when you take a working hypothesis and try to defend it rather than disprove it.

Observational studies, meanwhile, suggested that people who regularly ate vitamin-laden fruits and vegetables were healthier. So were people with higher levels of vitamins E and C in their blood.

All together now: Correlation is not causation.

Vitamins were strongly antioxidant in test tubes.

And bleach kills cancer cells in test tubes. It also kills other cells.

But if those ROS were so harmful, some scientists asked—and the basic design of our (eukaryote) cells was over 1 billion years old—why hadn’t evolution solved the ROS problem?

Do I need to go into this again? Evolution simply doesn't care. If you live long enough to reproduce, what happens to you after that is entirely immaterial. A worthy object for study, sure -- but we're talking about effects that usually happen in a human's fifties and beyond, well after peak reproductive years.

According to the ROS model of aging, animals that exercised and fasted should have died younger. But they lived longer.

And then there's my philosophy: what's the point of living longer if you have to spend all that extra time exercising, and avoiding one of the few pleasures (eating) that makes life worth living? (Yes, I've reduced my caloric intake and, at least when I could still go to the gym, exercised every day. But not to live longer; just to feel better. Now that exercise isn't as easy for me, I'm not doing it as much, and I find myself sinking once more into depression.)

He had 39 male volunteers exercise regularly over several weeks; half took vitamin supplements before working out. The results, published in 2009, continue to reverberate throughout the field of exercise physiology, and beyond. Volunteers who took large doses of vitamins C and E before training failed to benefit from the workout. Their muscles didn’t become stronger; insulin sensitivity, a measure of metabolic health, didn’t improve; and increases in native antioxidants, such as glutathione, didn’t occur.

Small, non-diverse sample.

But the primary role of vitamins in our body, according to Ristow and others, may not be antioxidant. And the antioxidant content of fruits and veggies does not, he thinks, explain their benefits to our health. So what does?

I think we're about to find out.

Mark Mattson, Chief of the Laboratory of Neurosciences at the National Institute on Aging, has studied how plant chemicals, or phytochemicals, affect our cells (in test tubes) for years. The assumption in the field has long been that, like vitamins, phytochemicals are directly antioxidant. But Mattson and others think they work indirectly. Much like exercise, he’s found, phytochemicals stress our bodies in a way that leaves us stronger.

Again: test tube science, while the body is incredibly complex. Also, again: "assumption."

Plants, Mattson explains, live a stationary life. They cannot respond to pathogens, parasites, and grazers as we might—by moving. To manage the many threats posed by mobile life, as well as heat, drought, and other environmental stresses, they’ve evolved a remarkable number of defensive chemicals.

As I noted recently, every living thing on Earth has been evolving exactly as long as we have. It's silly to assume that, because plants don't have opposable thumbs, mobility, and a central nervous system, that they're not fit for their niche.

Mattson and his colleagues say these plant “biopesticides” work on us like hormetic stressors. Our bodies recognize them as slightly toxic, and we respond with an ancient detoxification process aimed at breaking them down and flushing them out.

Sounds good, but as I said, we're going to need more than test tube science here.

In a study on Type 2 diabetics, broccoli-sprout powder lowered triglyceride levels. High triglycerides, a lipid, are associated with an increased risk of heart disease and stroke. Lowering abnormally elevated triglycerides may lessen the risk of these disorders.

"Powder?" Why not just... you know... eat a fucking vegetable? I don't fully buy in to the whole "eat non-processed foods" fad, nor am I vegetarian, but even I have to admit that an actual goddamned vegetable is probably better for you than powder.

Implicit in the research is a new indictment of the Western diet.

Of course it is. Can't leave well enough alone. Gotta convince people they're Doing It Wrong, so you can sell them the "solution."

Sinclair studies another class of native proteins, called sirtuins, associated with health. They’re triggered by exercise and also, Sinclair contends, a molecule called resveratrol, found in grape skins and other plants. “It’s too coincidental that time and time again these molecules come out of nature that have the surprising multifactorial benefit of tweaking the body just the right way,” Sinclair says.

Coincidental? Pfeh. Again: we evolved together. But to be fair, the article goes on to explain that, albeit in a way unconvincing to me.

In the dance between animals and plants, says Hooper, “I think there’s true mutualism. We’re in this together, the plants and us.”

This, however, rings true.

While xenohormesis is a compelling idea, it remains unproven.

As I noted above, this is from 2014. I suppose more studies have been done since then. I'd be curious to see the results, but I can't be arsed to look for scientific papers that I'm certain will contradict each other.

That critique speaks to a larger problem: It’s often unclear how lab research on simple organisms or cell cultures will translate, if at all, into recommendations or therapies for genetically complex, free-living humans.

One thing I'll say in favor of this article: it's pretty clear what's hypothesis and what's established. The whole reason I linked this is to point out that just because articles like this exist, doesn't mean we should take them at face value. Still, it's an interesting article... so here it is.

3 Like • Comment • View 2 comments

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