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Saturday, Apr. 27, 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/

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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 3, 2024 at 9:37am

#1065466

Revisited: "Sea What I Did There"

Almost two years ago, in May of 2022, I wrote some commentary on an Atlas Obscura article about the Great Lakes: "Sea What I Did There"

Being relatively recent, the article I referenced is still there.

Among other things, it's a good read for learning more about the Great Lakes. So, today, I'll just critique my own entry.

And now, today's burning (or really, drowning) question

I thought that intro was clever at the time. In retrospect, it was just silly. Partly because parts of the Great Lakes indeed have a history of burning, due to the chemicals that apes like to dump into them (I kind of chuckle every time they're described as "fresh water," though I know that's meant to describe lack of sea salt). And also partly because it was a bit insensitive.

(Responding to "Are the Great Lakes Really Inland Seas" headline): That's right up there with "Is a hot dog a sandwich?" in terms of categorization questions. (It's not, by the way. It's a taco.)

I stand by my categorization of hot dog as not-a-sandwich, but since then, I've been wondering about where gyros fit in. They're kind of like tacos, too. A taco is clearly not a gyro, nor is a hot dog (even if the hot dog has been spinning around on one of those convenience store heat roller machines). But is it fair to lump several different cultures' foods into a category based on that of only one of the cultures? Hot dogs are basically derived from Germany. Gyros are, famously, Greek, though I'd extend that to nearby Mediterranean regions. And tacos are, of course, Mexican. Other cultures have the similar idea of folding some sort of carbs around a filling, but not fully (that would be a "wrap" or a burrito. or perhaps one of those meat pies you can get from street vendors in England, though those are probably closer to calzones). So what we need is an overarching category for foods that are not technically sandwiches, because they're not fillings between two hunks of bread, but also not technically wraps or calzones, but something in between the two. A sandwich sandwich, if you like.

It is entirely possible that I spend too much time considering these important philosophical questions.

I think we're all aware of how angry the Lakes can become.

Frankly (see what I did there?), I have no memory of why I made such an idiotic blanket statement. Maybe I assumed everyone's heard Gordon Lightfoot's most famous song, the one about a ship sinking in the Great Lakes. Another fun categorization question: Ship or boat? The Edmund Fitzgerald was a freighter, which is most definitely a ship, because, well, it shipped stuff between ports, or, in its case, between a port and the bottom of Lake Superior.

In any case, I should always know better than stating "we're all aware" of anything. There are always people who don't know, be it from age or geographical distance or lack of exposure to cultural references.

Is Pluto a planet? Depends on definition of "planet." Under current internationally accepted definition, no.

As of this writing, I'm still obsessed with playing Starfield, a video game that features lots of star systems with planets and moons to explore. One of the star systems is our own solar system. Amusingly, they list Pluto as a planet and Charon as its moon, though here in reality, Charon and Pluto orbit a center of mass which doesn't lie within either body, which should make it a binary system. Further, in reality, this double-not-a-planet has other, smaller satellites.

This is, of course, not the only, or even the worst, technical issue in Starfield. But it's a game, so I just take it for what it is.

Virginia and three other US states are technically Commonwealths. We still refer to them as states.

That was not as apt an analogy as I apparently thought it was at the time. There's a difference between the arbitrary boundaries and technical name of a state or country, and the border between bodies of water and land. Except, of course, when there is no difference; Hawai'i, e.g.

The Dead Sea is famously salty as hell, sure, but so is the Great Salt Lake, which is about 7 times bigger than the Dead Sea.

Since then, I've learned that the GSL is shrinking pretty fast, so now, two years later, I'm not sure about that size comparison. The Dead Sea is also shrinking, of course. In any case, my real point was that they're both endorrheic bodies of water, and we call the bigger one a lake and the smaller one a sea.

In the spirit of what I said above about assuming awareness, "endorrheic" describes a watershed that's self-contained and doesn't allow for runoff into an ocean. They're really common in the American West. Hell, the vast majority of Nevada is endorrheic. Why they're usually found in areas we call deserts, I really do think should be obvious.

At one time, though, around the time the dinosaurs bit it, the Rockies were at the bottom of an inland sea, and the Appalachians were much, much higher (and originally extended into Scotland).

You know, when I make a claim like the bit about Scotland, I really should include a reference. At this time, however, I can't find where I learned that little tidbit, though I do remember that it involved the really stupendous age of the mountain range, combined with continental drift. Since I can't locate a reference, even on Wikipedia, you can probably safely ignore that bit of trivia, because it might well be the result of a misunderstanding on my part.

The relative youth of the Rockies and former height of the Appalachians, though, those are well-documented parts of geological history. here,

for one.

All continents can be considered big islands, and there is really only one world-spanning ocean surrounding all of them.

I object to my use of the word "really" in that sentence. Sure, it's another way to look at things, but there are real differences

between the areas we label as "oceans," even if the boundaries between those bodies are fuzzy.

I'll just end with this little tidbit, relevant to this entire discussion, which takes the categorization problem to its logical extreme: The Earth sandwich.

1 Like • Comment • View 3 comments

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