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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 4 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 4 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 5 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 15 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

a day ago
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

a day ago
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?

February 19, 2021 at 12:07am

#1004786

And the Spiders from Mars

Well, it looks like I didn't get as astronomical as I expected, so here's my usual just-after-midnight post. Last night, after completing my blog entry, I ended up getting completely danchu, sampling from many different bottles. Predictably, I felt like shit all day (worth it). Hence, tonight, I ended up just taking my traditional couple of shots of birthday tequila, and I also had some port with dessert. Consequently, I'm functional right now.

I'm sure you've all heard about the successful Perseverance rover landing on Mars yesterday. Technology works, and we can do some pretty awesome shit when we put our minds to it. By the time we finally land human boots on the fourth planet, there will be a thriving robot society there. They'll probably pass resolutions to send us back home, or maybe put us in concentration camps to discourage further immigration.

So with the success of the landing, which is designed to search for signs of past life on the Red Planet, it's appropriate that this subject came up at random from my stash.

Where is the informed discussion of alien life on the internet?

I'm somewhat sure that there is no alien life on the internet, politicians' tweets notwithstanding. Oh, wait, that prepositional phrase is meant to modify "discussion," not "alien life." English is fun.

Anyway, this blogger, PZ Myers, is one of the few that I follow regularly, because he usually has something interesting to say, even if he does have an obsession with arachnids. Fortunately, spiders are (we think) from Earth so there aren't any in this particular post.

I like speculations about alien life, just as I appreciate the diversity of life on Earth, the different forms of life in the past, and the prospect of evolution in the future, but every time I read about this stuff in astronomy-related journals, I feel like they’re making an effort to reduce my intelligence.

I know the feeling.

The problem is that they have no imagination and no biology, but they’re trying to imagine the nature of alien biology, and all they end up doing is running around in circles trying to figure out why little grey humanoids aren’t landing their flying saucers en masse to march out and shake hands with the president.

I know I've ranted about this sort of thing in here before, so here's another guy's take on the subject.

So, in this post, Myers does what I usually do, which is comment on someone else's article. So here I am commenting on an article that comments on an article. If someone quotes this entry and comments on it, that would be really meta.

This is my problem with the general tenor of these speculations. They all assume that we, that is human-like intelligences, are desirable, inevitable, and the only proper kind of life; they’ve read far too many science-fiction novels prophesying a colonialist destiny led by strong-jawed Anglo-Saxons with glinting eyes and a finger on the trigger of their blaster.

I mean, hey, I love those stories as much as anyone, though I prefer phasers.

They’re always going on and on about the likelihood of finding an intelligent civilization like ours. Why not speculate about finding a planet that has produced kangaroos? Or stomatopods? Or baobab trees? These are all unlikely outcomes of a contingent, complex process that produces immense diversity, and they’re all wondering what the “barrier” is that prevents our kind from winning the cosmic lottery every time. Get over it, we’re not a favored outcome, there’s no direction to evolution, and that’s why there aren’t smarty-pants bipeds tootling about the galaxy stopping by for tea.

I kinda hate that Myers puts this more eloquently than I have in the past. But I agree.

Anyway, I'm not quoting the quotes here, and I'm leaving out a lot of the commentary, so of course I recommend reading the source. Whether you read the article he's ragging on is up to you; I certainly didn't.

Point is, though, that this is a biologist's perspective on the question, but it's similar to one I came up with previously: that if we find life "out there" -- be it living life, or, as is hoped with the Mars robot, evidence of life that existed in the past -- it will, in all likelihood, be something other than a technological civilization. And yet, we've been so conditioned to equate "life" with "intelligent life" (as usual, please refrain from making silly dismissive comments about "intelligence," as the fact that you can make such a comment immediately contradicts it) that it's almost impossible to find someone who doesn't immediately think of bulbous-headed big-eyed gray guys, or Klingons, or other generally humanoid, tool-using, spaceship-flying aliens.

That said, I really hope this Mars rover finds unequivocal evidence of some sort of living, or once-living, organism on Mars. Because that by itself would be an incredibly important discovery, never mind what sorts of things we might learn from it.

Mini-Contest Results!

Thanks for all the mixed drink recipes! I definitely saw a few I'd like to try. Rolling the Virtual Dice for the nine commenters (sorry, one comment came in just after midnight), the result is 3, so the Merit Badge will go to... Cubby~Cheering House Florent!

for the mojito recipe (and I do love mojitos).

Still, I appreciated all the comments and we'll do this again soon!

6 Like • Comment • View 5 comments

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