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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 20 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

November 21, 2021 at 12:03am

#1022046

Facing the Music

Music certainly has the capacity to provoke emotion in me. It's one reason I started this blog, and one interpretation of the title.

PROMPT November 21^st

Tonight/today, listen to this beautiful song.

Where does your mind go when listening to this? What emotions surface as you experience the music? If you've seen the movie (Return To Me), tell us how this affected you. Has it changed you at all?

This one, however, doesn't do much for me. Oh, I don't dislike it, not at all. It's technically brilliant, and the particular version linked above is especially well done.

Where does your mind go when listening to this?

There was a particular musical style, popular mainly pre-Beatles, that featured the "crooner." Frank Sinatra was probably the most famous of these, but also Dean Martin, who originally sang the song in question. Maybe Sammy Davis Jr. Almost all of them ended up in Vegas.

A few years ago, I got to see one of them, Wayne Newton, perform there. Amazing vocal talent. You'd never know it from listening to him speak; his ordinary voice is kind of raspy and undisciplined, but as soon as he starts to sing, it's like magic.

Anyway, Tony Bennett was certainly in that illustrious company. People nowadays dismiss it as "lounge" singing, but it's a legitimate art form. So much of music these days emanates from talentless hacks whose only draw is their looks, or maybe their dance moves (not that dance doesn't require talent, but I'm talking about the music itself). Hell, even their songs are autotuned so they don't have to work on their voice, only their display. Me, I prefer to hear songs by people who can actually sing, even if they flub a note or two from time to time; that's part of the charm.

Don't get me wrong. I'm not ragging on "damn kids these days and their music." Some contemporary music is remarkable. There was plenty of talentless tripe when I was growing up, too (but without autotune); it just tends to be forgotten, much as many of today's acts will be forgotten.

Anyway, the point is, I believe that, as with movies, it's very helpful to have a sense of historical context when considering music, be it jazz from the WW2 era, folk from the early 20th century, disco from the 70s... or lounge singers from the 50s.

What emotions surface as you experience the music?

None, really. I have an appreciation for the talent involved, but the songwriting doesn't move me. I especially appreciate the spirit of international cooperation displayed; music has, as I've noted before, the capacity to bring people together.

Mostly, because it's a couple of old guys who are excellent singers, it made me miss Leonard Cohen. Now that's music that affects me emotionally.

If you've seen the movie (Return To Me), tell us how this affected you.

Not only have I not seen that movie, I didn't even know that the movie existed until I saw this prompt.

So of course I read the Cliff's Notes versionWikipedia page. There's no way to get the emotional impact of a movie from reading the plot synopsis (which I didn't mind doing, despite spoilers, because it's in a genre I tend to avoid anyway), but I was surprised to find some really big names involved. Surprised, because as you all know, I'm a movie fan, though my participation has waxed and waned over the years. I've always especially liked Minnie Driver as an actor, and of course David Duchovny is David Duchovny.

No desire to see it, though. The plot seems trite and way too mushy, like many romantic comedies. To each their own, of course.

Has it changed you at all?

I guess you could say yes, because I now know that something (the movie) exists that I didn't know existed before a couple of hours ago. I suppose it's possible I did know the movie, but forgot about it.

Still, I hope no one interprets this entry as "that song sucks and the movie looks like it sucks too." That's not my intention. It's kind of like with beer. I like certain styles of beer. Other people like certain other styles of beer. But that diversity is awesome, because it keeps beer companies in business. Likewise, different genres of movies and music keep people making movies and music; I'm always going to prefer some to others, while other people will have different preferences.

What a boring world this would be if we all liked the same stuff.

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