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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 16 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 16 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 16 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

a day 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

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

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?

October 26, 2019 at 12:01am

#968432

Born in the USA

I pick these links at random from a fairly long list of potential things to talk about. Today's link fortuitously coincides with the release of Western Stars, the Springsteen concert movie. But it's not about new music - it's about something that hit the record stores (for such things existed then) half of Bruce's lifetime ago.

https://www.npr.org/2019/03/26/706566556/bruce-springsteen-born-in-the-usa-ameri...

What Does 'Born In The U.S.A.' Really Mean?

In the summer of '84 I was working long, arduous hours, outdoors, as a surveyor's assistant. This is probably where I got my utter hatred of unenclosed spaces from; it certainly wasn't the farm I spent my childhood on. Getting covered with poison ivy and stung by angry yellow jackets can do that to a kid.

But that was the summer that Born in the USA was released, and I was a Springsteen fan even then. I thought about asking my boss for the day off so I could buy the album - in LP format, of course - but he wasn't the type who would understand. In one of the many lucky coincidences involving me and Springsteen-related things, though, it rained that day so I wasn't working after all.

So I went to the shopping mall (80s, remember), bought the album, and went home to listen to the thing. That's how we did things back then, pre-internet.

If you're listening closely, the lyrics of "Born in the U.S.A." make its subject pretty clear: The 1984 hit by Bruce Springsteen describes a Vietnam War veteran who returns home to desperate circumstances and few options. Listen only to its surging refrain, though, and you could mistake it for an uncomplicated celebration of patriotism. You wouldn't be the only one.

This was about the time I first started realizing that most people don't listen to music the way I do. The big "hit" from that album was "Dancing in the Dark," a decent enough song, but all anyone else could talk about was the music video. I felt it was the weakest song on the album, though - everything else sounded like it was done in two or three takes, but that one song suffered from typical 80s overproduction. I found out later that I was right - the rest of the album was basically live.

But that asshole Reagan - he talked about the title track, all right. And got it entirely wrong.

By playing on the hope, Reagan seemed to overlook the despair.

That's a... generous way to put it.

"After it came out, I read all over the place that nobody knew what it was about," [Springsteen] said before performing "Born in the U.S.A" to a crowd in 1995. "I'm sure that everybody here tonight understood it. If not — if there were any misunderstandings out there — my mother thanks you, my father thanks you and my children thank you, because I've learned that that's where the money is."

See? Still a capitalist.

Maybe the meaning of "Born in the U.S.A." is the distance between the grim verses and the joyous chorus. It's the space between frustrating facts and fierce pride — the demand to push American reality a bit closer to our ideals.

The title of this blog is "Complex Numbers," with the mathematical definition front and center, but it's actually a multileveled play on words - "numbers" also refers to pieces of music, and I'm fascinated by the emergence of complexity from simplicity.

2 Like • Comment • View 2 comments

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