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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 17 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 9, 2020 at 12:07am

#990304

Wine, Cheese, and Bread

Today I get to talk about some of my favorite things.

https://www.smithsonianmag.com/science-nature/how-cheese-wheat-and-alcohol-shape...

How Cheese, Wheat and Alcohol Shaped Human Evolution
Over time, diet causes dramatic changes to our anatomy, immune systems and maybe skin color

You aren’t what you eat, exactly. But over many generations, what we eat does shape our evolutionary path. “Diet,” says anthropologist John Hawks, of the University of Wisconsin-Madison, “has been a fundamental story throughout our evolutionary history. Over the last million years there have been changes in human anatomy, teeth and the skull, that we think are probably related to changes in diet.”

Well, that doesn't mean I'm a pizza, no, but at least on a fundamental level, we know that we're made up of atoms taken from the substances we breathe and ingest.

I do have some issues with the way this article so cavalierly uses the word "evolution." I get the feeling an evolutionary biologist would have a cogent argument against it. I'm not one, so I'm just going to go with the flow here, for now.

When mammals are young, they produce an enzyme called lactase to help digest the sugary lactose found in their mothers’ milk. But once most mammals come of age, milk disappears from the menu. That means enzymes to digest it are no longer needed, so adult mammals typically stop producing them.

Thanks to recent evolution, however, some humans defy this trend.

Another interpretation: some of us retain childlike features. As should be apparent if you've been reading my stuff.

Hawks explains why being able to digest milk would have been such a boon in the past...

Speculation without supporting evidence. I mean, if you look at the article, the explanation makes sense, but I shy away from such just-so stories on principle. The facts are what they are; the explanation for the facts is suspect.

The article then goes on to talk about grains.

Since wheat and rye became a staple of human diets, however, we've have had a relatively high frequency of celiac disease. “You look at this and say how did it happen?” asks Hawks. “That's something that natural selection shouldn't have done.”

Translation: "I don't understand how natural selection actually works."

Yet despite the obvious drawbacks of celiac disease, ongoing evolution doesn't seem to be making it less frequent. The genetic variants behind celiac disease seem to be just as common now as they've been since humans began eating wheat.

That's because evolution doesn't care if an individual gets sick or has dietary limitations. Evolution only cares if someone lives long enough to reproduce. Well, I'm speaking metaphorically. Evolution neither cares nor doesn't care. It just is. Point is, unless there's something that kills people at an early age, the genes for it happily pass on. Okay, that's another metaphor. Genes don't feel happiness.

Then the author gets into the discussion of how diet could influence skin color, over time and in populations, which is kind of a minefield these days. Keep in mind it's not promoting any shade of skin as better or worse, in terms of evolution, than another, just describing what could have influenced skin tone changes in humans.

But then you get sentences like:

Mostafavi's genetic research also revealed that some variants that actually shorten human life, like one that prompts smokers to increase their consumption above smoking norms, are still being actively selected against.

Again: evolution is concerned with reproduction. Poor health outcomes from things like smoking or alcoholism don't tend to show up until later in life, after a human has had plenty of opportunity to reproduce (whether such habits influence the portion of the process of evolution called "sexual selection," I leave as an exercise for the reader). Point is, you can have some disease that kills you at, say, 30 or 40, but it doesn't stop you from having reproduced by then, so the genes that affect the expression of said disease aren't selected against.

So if there is indeed research on that subject, I hope it's peer-reviewed, because as it's presented here it certainly doesn't inspire any confidence from me.

Overall, I expected better from Smithsonian Magazine.

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