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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

a day 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

a day 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

a day 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

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?

December 6, 2018 at 12:33am

#946926

Expert Opinion

I wanted to share this, not because I have a stake in the topic being discussed, but because my lack of a horse in the race means I have some tangential thoughts to share.

https://www.nytimes.com/2018/11/27/well/family/the-fallacy-of-the-i-turned-out-f...

The column is ostensibly about child-spanking. To summarize, a soi-disant "parenting expert" (

) from Down Under asserts that spanking should be a criminal offense (or offence or however they spell it in Oz), is nonplussed when a bunch of people disagree with him, and then goes on to demolish what he sees as their arguments.

I'm not overly familiar with the political situation in Australia, but first of all, it seems to me that "this practice is ineffective and probably dangerous" shouldn't automatically translate to "this practice should be a criminal offense." It's dangerous in itself to turn everything that we don't like into ban laws; also, a law of this sort strikes me as being especially easy to abuse. It's a basic idea in a so-called free country that not everything is either "permitted" or "banned." Some bad ideas can be handled with social pressure. For instance, I dare you to speak out against public breast-feeding. Go on. I triple-dog-dare you. There's no law against speaking out against public breast-feeding, nor, by our right to freedom of speech, should there be. But I guarantee that you will face a typhoon of scorn and ridicule that will make you wish you had never been born, let alone breast-fed.

I also wanted to point out that there is a such thing as making things too safe, for kids as well as for adults. Take away all the world's sharp corners, foam-pad all the walls and floors, and rope off anything that might pose even the slightest hazard to people, and that's what people will come to expect. They'll be wired to think that if something might be dangerous, there will be a barricade. And that if there is no barricade, it can't possibly be dangerous. Or, worse - you rope off areas of both minor and major hazard, so people get complacent about ignoring barricades. And that's how we get things like people sliding into the Grand Canyon

or boiling to death at Yellowstone.

Life, in short, is full of risk, and there may be paradoxically more risk in protecting people all the time than in allowing for the occasional bump on the noggin. Or, to stay more on topic, the occasional light swat on the tuchis.

Unfortunately, if you don't foam-pad everything lawsuits happen, so the world just gets more and more covered with the illusion of safety.

Getting back to the article I originally linked, however, I'll just point out one fallacy that the author didn't even consider, which is the survivorship fallacy. "We didn't have car seats when I was a kid and I'm still alive." He covers that as "anecdotal fallacy" but doesn't go far enough. It doesn't take into account the loose kids bouncing around in station wagons who didn't make it to adulthood. Such a logical fallacy doesn't even require hazardous situations. You see it all the time in articles with such titles as "The Five Traits of a Successful Business Leader." The implication is that anyone with these traits will be successful, and completely ignores the vast majority of people with those traits who are standing in line at the food bank.

Well, that's enough for now. But I did want to say, one more time: "parenting expert."

4 Like • Comment • View 2 comments

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