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Thursday, Apr. 25, 2024

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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 17 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 17 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 17 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 6, 2022 at 1:07am

#1040320

Revisited: "Good and Gooder"

Before I get back to posting commentary on articles—I still have a large selection to choose from—I thought I'd try something different today.

This blog is now nearly 16 years old and contains 2142 entries (this one will be #2143). Sure, most of them are from the past few years, and I took a long hiatus in the middle there. But recently I've had occasion to go back to revisit old entries. Some of them made me laugh, because I didn't remember typing them. Others made me cringe because, newsflash, I change over time.

I wanted to revisit some old entries to see how my views might have shifted. True to form, I picked one at random for this purpose. If I do this again, I'll pick a different one at random. I'm excluding anything from the past 12 months.

So here's one from March 19, 2019: "Good and Gooder"

It's based on this article, which, fortunately, is still available. I'll link it here again in my currently favored format:

An Oxford researcher says there are seven moral rules that unite humanity

While I don't have anything to change about that entry, I wanted to do what I didn't back then, and actually list the "seven moral rules" as presented in the article:

1. Help your family
2. Help your group
3. Return favors
4. Be brave
5. Defer to superiors
6. Divide resources fairly
7. Respect others’ property

What strikes me now (and might have struck me at the time, but I didn't mention it) is what's excluded from this list, moral/ethical rules that, if they're not universal, I think should be.

First and probably most obvious, "Don't murder people."

Most societies that I'm aware of draw a distinction between the illegal killing of people, and the legal killing thereof. There are also gray areas in between, as with self-defense. Some examples of legal killing would be warfare or capital punishment. I'm not saying these things are right, mind you; just that they're legal.

But walking up to someone on the street and putting a bullet in their head is generally frowned upon in most societies I'm aware of, though it happens more often than we'd like. I guess I assumed it was the same everywhere, though with its exclusion from this list, I have to wonder if "don't murder people" just isn't a major no-no in other societies. Sometimes, it's okay to kill someone from the out-group (North Sentinel Island comes to mind) but not the in-group; more on this later.

There might be different lines drawn elsewhere, but I've heard of very few cultures that say it's okay to murder someone—though the definition of "murder" can vary.

Second glaring omission: refraining from child abuse. Again, what's considered abuse can vary. For example, there was a big deal made a few years of the practice of female circumcision in some societies, though I haven't seen anyone railing against it recently. This is usually done to children, so from our Western perspective it's child abuse. And then some people noted "what's the difference between that and male child circumcision," a practice strongly rooted in the West, so I think that's one reason the moral panic died out. It's a false equivalence, and most people recognize that.

That's just one example, though. When I was a kid, it was expected that corporeal punishment (not to be confused with corporal punishment) was acceptable to keep kids in line. My elementary school principal even had a paddle with "School Board" on one side (the other side was engraved with my name). In a nod to changing morality, though, the school needed parental permission to use it. My dad gave them blanket permission, because he knew me. My mom, on the other hand, was more on the side of alternative punishments. Nowadays, even thinking about spanking your kid gets you labeled as a child abuser.

Point being that what we define as child abuse changes over time, just as it changes between cultures. Yet I feel like some version of it should be on that list (as a negative), and it is not.

Third omission: don't rape. I don't have much more to say about that except that, again, it should be right there.

I suppose it is possible that the list was constructed to include "positive" morality; that is, "do these things" rather than the "negative" morality of "don't do these things." But in that case, one could turn all three of these things around to "respect the autonomy of others," and the fact that this is not on the list is still worrisome, in that it implies that there are cultures where that's not a moral rule.

The list, however, purports to be about what is, and not what one random blogger thinks should be. If I were writing codes of morality, I'd make some changes. For example:

1 and 2 would be combined into "help humanity." If you just help your family and your group (which I read as tribe, or political affiliation, or extended family, or religion, or whatever sets you apart from the rest of the population), you're part of the problem. Thinking in such small terms appears to be baked into the tribal psychology of humanity, and it's not easy to change. But hell, some people take it even one step further to "help the biosphere."

I don't have a problem with 3, "return favors." This seems like basic reciprocity, which helps smooth social interactions and ties into 1 and 2.

4, "be brave," can be problematic. There's a blurry line between bravery and stupidity. It's fine to respect someone for being brave, but being brave by itself doesn't cut it, in my view. As an example, it would be brave of me to take off all my clothes and walk up the street. That doesn't mean it's a good idea.

5- I don't have superiors. Nor do I have inferiors. In certain circumstances, a person does have these things; a job, for example, or military service, in which case, well, okay. But the idea that one person is inherently superior to another (rather than being in a manager / employee situation) is, again, part of the problem.

6- "Divide resources fairly." By that measure, the US is a highly immoral society. I'm not going to argue against that. But "divide resources fairly" is not what we practice. Why, that would be soshulism!!! Also, who gets to define what's "fair?"

7 seems to contradict 6, anyway. We tend to idolize the Robin Hood types, but they don't respect others' property. One can make the argument that Robin Hood was just correcting economic injustice, and I won't argue there, but stealing is wrong, whether a rich person or a poor person is doing it (I can make moral, if not legal, exceptions for those who steal to alleviate hunger, but our legal system isn't set up for such nuance).

And I'd add the bits about respecting others' autonomy that I discussed above, with "slavery is wrong" added to that mix just to be clear about it.

That's enough for today. Back to regularly scheduled random programming tomorrow, unless something major happens.

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