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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:

Kåre Enga in Udon Thani

about 12 hours ago
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

about 12 hours ago
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

about 12 hours ago
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

about 23 hours ago
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

a day ago
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

a day ago
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

a day ago
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

a day ago
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

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

December 12, 2020 at 12:01am

#1000093

Red and Redder

"30-Day Blogging Challenge ON HIATUS" [13+]:
12. Pointsettia Day
According to Mexican folklore, there is a story of a little poor girl who had nothing to bring to church for Christmas. On her way to church, she picked some plants by the side of the road. As she entered the church, the leaves at the tips of the branches turned into bright, brilliant red flowers. You guessed it... Poinsettias.
Write anything you like about Poinsettias.

"JAFBG" [XGC]:
They say blood is thicker than water. How does your family prove this saying wrong?

I think one of the most interesting subgenres of mythology is what Kipling popularized with the "Just So Story," making shit up about how things got to be the way they are. Apparently the technical term is "etiological myth," which I found on Wikipedia. Etiology itself is the study of origins, and one can approach it mythologically or scientifically.

My favorite etiological myth is from the Lenape and involves the turkey vulture (big surprise there) and several other animals. This is the version I could find.

Now look, I'll be the first to admit that I rarely click on links others provide; I just can't be arsed. So maybe I'm being hypocritical to expect you to click on the above link. But I'm asking you to do it, anyway, because I don't want to just copy all the text into here and make this post even longer and more difficult to get through. It is one reason I consider the turkey vulture to be the closest thing I have to a spirit animal.

It is, of course, fiction, a made-up story. Such folklore says nothing at all about how things got to be the way they are in reality, but it speaks absolute volumes about how we humans look at the world -- first by coming up with such legends, and second by how they are passed down.

Science, of course, approaches etiology in a more systematic way. The scientific explanation for the vulture's featherless head has to do with its adaptation to its niche in the ecosystem, not the vulture's ancient self-sacrifice for the good of all. Both stories are fascinating, each to its own purpose. The important thing is to be able to separate reality from fantasy.

Like with the poinsettia. Admittedly, I didn't look very hard for the evolutionary explanation for the red bits -- probably something to do with warding off predators, or maybe attracting certain pollinators; those are the usual "reasons" for any plant coloration that isn't green. But the reason I quit looking is because I found another, more modern myth about the poinsettia.

Honestly, I never paid those plants much attention. To me it was a Christmas thing and Christmas things ward me off like garlic does vampires. But one thing that I heard repeated, over and over: "Watch out for poinsettias if you have children or pets. They're highly poisonous!"

This turns out not to be the case.

Yes, I just included yet another link for you to click on. Summary: Poinsettias aren't edible, but they also aren't toxic.

And yet, this is the first I've heard of that, because the myth is so persistent. Unlike the myth about the little girl going to church, though, people don't know it's a myth, and they repeat it as pure truth. Which, again, it is not. Now, is it a harmful falsehood? Probably not; it's usually better to assume that something is dangerous than to assume it's perfectly safe. And yet it is a lie, a falsehood, a thing with little to no basis in fact.

So it is (SEGUÉ) with "Blood is thicker than water."

A proverb isn't a plant. It has no existence or reality beyond people repeating it; while poinsettias would still be there if humans were to suddenly disappear with all of their records, that phrase would vanish as if it never was. So it's even more a reflection of human ideas than creation myths are.

I've heard it said that the "original" expression was "the blood of the covenant is thicker than the water of the womb," but as with poinsettias' toxicity, that appears not to be the case. No, as far as I can tell, the original meaning was indeed that genetic family ties are stronger than others.

But that's the kind of thing that's only true if you think it's true. And that can be a dangerous fiction. For instance, presumably if you get married, it's to someone who isn't too close a relation, right? Like, I know we're all related to some extent, but at least you don't go with a sibling or first cousin or whatever. So you take an oath, a vow, to support each other. But if you believe in the "blood is thicker than water" myth, then your responsibility to your sibling or parents would supersede responsibility to spouse.

Maybe you do feel that way; I can't say that it's wrong, exactly. But to me, an oath that you enter into willingly is far more binding than one that is a mere accident of relation. Like when my parents adopted me. No close genetic relation there, but they fulfilled all the necessary responsibilities and then some (me, not so much, probably, but then, it wasn't an oath *I* took). Blood? Water? That's all metaphor anyway. What matters is commitment.

And on the other side, if your family is toxic, I don't feel like you should continue to put up with them because of some mythology about "blood" bonds.

Like I said, it's important to discern reality from fantasy, and while that's easier in some cases than others, one shouldn't believe every little saying that comes down the road. Poinsettias aren't poisonous. Family sometimes is.

9 Like • Comment • View 6 comments

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