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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

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

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?

March 8, 2024 at 8:25am

#1065851

Law and Ordering Alcohol

Booze laws can be convoluted. You think they exist to promote public health and safety, and maybe some of them do, but mostly, the laws are about who does and does not get to profit from alcohol. (In order, that would be "governments and rich people," and "you.") Leave it to Cracked to find some of the weirdest ones.

5 Perplexing Laws Over Who Gets to Buy Booze

5. In New York, Bartenders Are Forbidden from Not Giving Alcohol to Pregnant Customers

Of all these examples, this one's actually the least weird.

In general, bartenders are allowed to refuse service to customers. In the case of an intoxicated customer, they’re legally bound to refuse service. But they are legally banned from refusing service to a customer because they're pregnant, as this may be “pretext for discrimination or as a way to reinforce traditional gender norms or stereotypes.”

For starters, the header is misleading. Bartenders aren't known for "giving" alcohol. They're known for selling it, and for patiently listening to sob stories in hopes of getting a tip.

Further, while late-stage pregnancy is usually quite visible, from my understanding, some of the worst and most (eventually) visible effects of drinking while pregnant occur during the first trimester, when it's not usually obvious.

And finally, we either need to assert that all adult humans are responsible for their own decisions, or go the way of places like Texas who exert government control over the pregnant ones.

4. The Very First Campaign Finance Law Was About Banning Alcohol Sales

*in the US

In 1811, Maryland banned liquor sales on Election Day, but the goal wasn’t just keeping the electorate sober in general. The goal was to keep politicians from buying voters alcohol — as bribes. As a result, this is remembered as the first campaign finance law.

Well, this one's not that strange, either. If I'm having a hard time choosing between two evil politicians, I would definitely vote for the one who bought me a beer.

3. Happy Hour Bars That Legally Must Give Discounts on Soda

During happy hour, bars slash prices on drinks, maybe selling you two for the price of one.

The laws I'm most familiar with are, obviously, those of my own state. I still don't fully understand them, but to the best of my knowledge, in Virginia, you can't do two-for-one promotions or ever give out free booze. We do get to enjoy discounts, sometimes, and a half-price beer is functionally the same thing without forcing you to buy two drinks.

This drives up business during times that are otherwise dead, and if they’re able to make a profit after halving their prices, that lets you know just how high the markup is on drinks the rest of the time.

Markups can be high, but this is not necessarily the case. Sometimes it's about ensuring the bartender has the opportunity to get tips so they're less likely to run off and work somewhere else.

France has it’s own anti-happiness law. They do have happy hour there (in France, they call it apéro hour), but since 2019, if a bar offers discounts on alcoholic drinks, they also have to offer discounts on soft drinks.

Sadly, the law we need is to fine people for misusing apostrophes.

2. A Special Car for Drunk Drivers

Also in France, if you get caught driving under the influence, they might suspend your license, leaving you unable to operate your car.

I mean, technically, no, not unless they also impound the vehicle and put you on the "no-buy-vehicles" list. But legally, sure.

If a vehicle moves at a top speed of 28 miles per hour or less, and has an engine with an output of no more than 5.4 horsepower, it’s a quadricycle, and in France, it’s known as a voiture sans permis, a VSP. That means you can drive it even if you have no valid license, so long as you’re 14 or older and have a few hours of recorded driving experience.

On the one hand, a top speed of 28 mph (or, as I'm sure they express it over there, 45 kmh) is going to reduce, though not eliminate, the hazards of drunk driving. On the other, you're pretty much limited to city driving, and cities there tend to have bars within walking distance, so what's the point?

1. The U.S. Has No Minimum Drinking Age, Actually

Yeah, technically correct. But effectively not.

Today, all 50 states set a 21 minimum. Even so, nearly every state offers some exceptions. In states like Wisconsin, for example, people of any age can drink so long as they’re with a parent or spouse who’s over 21.

Wisconsin's drinking culture is a truly awesome thing to behold.

In Puerto Rico (which is part of America, despite what that one song from West Side Story left entire confused generations thinking), the minimum drinking age is simply 18. Puerto Rico has highways, but they gladly reject 10 percent of the funds they could get. Some things are more important.

Me? I say lower the drinking age and increase the driving age. That way you get most of the stupid out of your system before you even get behind the wheel of a car.

Better yet, hurry up with the self-driving car thing. As you know, I'm alcohol-positive. But drunk driving? Let's just say I agree more with the French on that one.

6 Like • Comment • View 2 comments

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