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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

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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 23 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 23 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 23 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?

November 5, 2022 at 8:25am

#1040284

Listen to the Music

Ever notice that there are very few, if any, tolls to go into New Jersey, but a lot of tolls to get out of New Jersey?

My cousin, who has a beach house in NJ, says it's that way because if you charged people to go into the state, no one would visit.

Funny, but a little unfair. A lot of New Jersey is actually quite nice, including his beach house, which is just south of Asbury Park. No, he's never met Springsteen.

Most of the drive was rather pleasant, absent the perpetual traffic jam in the DC area. My one gripe apart from that is that I was forced to listen to commercial radio.

By which I mean, I have to listen to something, and commercial radio was really the only option.

See, on long-distance trips, I was used to plugging in my iPod Touch, setting my vast music collection on shuffle, and just letting it play. I had enough songs to drive all the way across the country and back without ever hearing a repeat.

And now they've discontinued the iPod, or so I've heard, so if mine breaks, I'm fucked. Besides, I don't see a plug for it. Maybe there is one and it's just hard to find. So I did this trip without the iPod to see what options I had, and I am not pleased. But maybe there's an option I'm missing, so here's what I understand about the current sad state of listening to music while driving.

1. Commercial radio. Emphasis on "commercial." Ads piss me off at the best of times, but the week before an American election? Oh boy. Road rage. According to every single ad, the ad purchaser's opponent punches pregnant women, kicks puppies, eats babies, was once caught jaywalking, is unbearably stupid, and will destroy America. The only thing I can glean from this is that no matter who we vote for, they will destroy America. The only choice, therefore, is to pick the one you think will give us the best bread and circuses during our slide into oblivion. (At least I haven't heard any holiday ads yet; those will definitely take over the airwaves after Tuesday).

1a. Apart from goddamned ads, which I can always switch stations for, again, I like to travel cross-country. There are entire swaths of America where the only options are Christian music stations, Christian talk radio stations, broadcast church services, evangelical emissions, country music, or static. I don't mind country music so much, but too much at one time gets real old real fast, so static it would have to be. Which would put me to sleep, not a good idea while driving.

2. Satellite radio. Advantage: lots of formats to choose from, only drops out in tunnels and maybe some mountainous areas (in which driving is perilous enough that I wouldn't get lulled to sleep). Disadvantage: another monthly subscription to pay for, one that I'd only use like 2-3 times a year, max. Bigger disadvantage: also ads (unless they've changed that, which I doubt), and if there's one thing I hate more than ads, it's paying and still having to put up with ads (this is why I never got cable TV).

3. One of those online music services, like Amazon Music or Pandora or whatever the hell is out there right now (I won't get Spotify under any circumstances) on my mobile, which I think (but I'm not sure) I can use while also using the phone's GPS. Advantage: no ads. Disadvantage: there are entire swaths of America without mobile coverage, during which the streaming stops working. Result: asleep at the wheel again.

4. Downloading some tunes on my smartphone (I will take this opportunity to note that my new car has a feature where if you plug in your phone, it'll transfer the GPS (I use Google Maps) to the car's screen for a totally hands-free experience, which is nice). Advantage: doesn't cut out, ever. Disadvantage: I have limited space on the Android, and that space is even more limited if I download offline Google Maps because, again, entire swaths of the country have no cell coverage; consequently, I could fit maybe two albums' worth of music on there. Repetitive.

5. Bring a big stack of CDs and use the car's CD player. Advantages: no ads, never cuts out. Disadvantage: can't shuffle between CDs. Bigger disadvantage: Come on, it's nearly 2023.

Now, I managed to make it to NJ and back without becoming too enraged by the lack of options. This is because that involves driving through areas with good stations: DC, Baltimore, Philadelphia. Basically anywhere in the Northeast Corridor is fine for radio if I can put up with the occasional ads and lame DJ chatter. But eventually, I'm going to want to drive through the Midwest again, perhaps even next month (we'll see).

So, what do people do for driving music these days? And don't suggest listening to recorded books. I don't have anything inherently against them, but they also take up limited smartphone space. Same goes for podcasts, except I don't like them in general anyway. I want to hear music; I want to hear it in random order (radio station programming counts as random as far as I'm concerned); I want no-to-very few ads; and I want the music to not repeat every few hours. My iPod met all those criteria and also had the advantage that I didn't have to fiddle with it while driving.

Without a good setup, I don't know if I can handle a cross-country trip. So... help?

3 Like • Comment • View 8 comments

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