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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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Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Aiva Raine

about 10 hours 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

a day ago
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

a day ago
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

a day ago
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

a day ago
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

a day ago
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

a day ago
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?

Aennaytte: Free & Wild in GoT

a day ago
In response to "Going to the Dogs"

I have never been to New York, but it seems that my assessment years ago that Berlin, Germany and New York have some similarities. Both cities are known for being inhabited by rude people. And both places, as I now found out, are dog obsessed. And they have high rents.
Berliners though, have a habit of letting their dogs run off leash. Maybe New York is a little better than Berlin in that regard.

1 Like

Robert Waltz

Apr 21, 2024 at 1:48pm
In response to "Revisited: "Hack This""

Wordsmitty ✍️, far be it from me to speculate on the motivations of space aliens, but why in Xenu's name would they do such a thing to the universe?

1 Like

Wordsmitty ✍️

Apr 21, 2024 at 1:39pm
In response to "Revisited: "Hack This""

I thought humans got bigger brains because extraterrestrials affected our evolution, and we eat processed chicken and eggs.

1 Like

June 8, 2021 at 12:01am

#1011473

Out of the Valley

Sometimes you just have to issue a correction. Well, not me, obviously. But other people.

Neandertals don't deserve their bad, dim-witted reputation
Our hominin ancestor had bigger brains and probably went extinct with climate change. Who are we to judge?

Just kidding, of course. When I'm wrong about something, I'll admit it when I find out otherwise. But in this case, it's a matter of science doing what science does: being self-correcting.

Unfortunately, lots of people stick with what they first hear and then ignore any evidence to the contrary. Well, here's some contrary evidence.

On March 3, 2021, the governors of Texas and Mississippi announced that they were lifting their respective mask mandates, prompting criticism from President Biden, who called the move "Neanderthal thinking." Biden was implying that lifting the mandates was a primitive act — but this understanding of Neandertals is an outdated stereotype, unsupported by modern research.

That, by the way, is not an invitation to argue about politics or politicians.

Neandertals (Homo neanderthalensis) are an extinct species of hominin — the taxonomic name for a tribe that includes living and extinct humans and their ancestors. They lived throughout Eurasia up until about 40,000 years ago. One of the earliest Neandertal fossils was discovered in the Neander Valley in Germany, which is where the species gets its name. Interestingly, the contemporary word for “valley” in German is “tal,” so anthropologists tend to call the species “Neandertal,” not Neanderthal (dropping the 'h').

Completely irrelevant aside: Apparently, there were once very productive silver mines in another German-named valley, Joachimsthal (using the older "th" construction). Coins minted from there were called "Joachimsthalers" or "thalers." This got Anglicized as "dollars." Or, well, I may have some of the details wrong, but that's the gist of how "dollars" got their name.

This is one of the things I'll correct later if I'm wrong, but I can't be arsed to look it up again now. I'm planning on going to see Cruella on Thursday, and I'm busy delving into the source material (Dalmatians movies), so my time is somewhat limited. As far as I can tell, though, the "dal" in "dalmatian" has no linguistic connection to tal, thal, or dollars. Pounds, maybe, if you're Cruella.

If, as research suggests, Neandertals behaved similarly to humans, we either need to rethink our definition of human, or update our definition of Neandertal.

Given that they're all gone, except for traces of genetic material in most of us "sapiens," this may be a moot point.

Where did the brutish stereotype that plagues Neandertals originate? This association can be traced back to 1911, when French paleontologist Marcellin Boule described a newly discovered Neandertal fossil from the La Chapelle-aux-Saints site in France. Boule wrote that the fossil had primitive characteristics, including a hunched posture, big, divergent toes, and large brow ridges which, he argued, signaled a lack of intelligence.

I'm betting that was a conclusion born of phrenology. Science has had a few dead ends, and phrenology was one of them. Unfortunately, as with homeopathy, its effects linger.

Boule's original interpretation of Neandertals was based on an incorrect reconstruction of their skeletons.

But hey, you gotta start somewhere, right?

The wider build and shorter limbs common in Neandertal skeletons has also been seen as primitive compared to early humans, though researchers now argue these were adaptations to a colder climate.

"Primitive" is an unfortunate word. They were well-adapted to their environment, until they weren't. Calling them primitive is like calling a lion primitive compared to a tiger.

It's clear that incorrect early reports shaped assumptions about the behavioral and anatomical nature of Neandertals.

The brutish, sloped-forhead caveman stereotype, usually wearing skins and carrying a club, persists in cartoons to this day. Because it's a convenient shortcut for pre-agricultural humanity (or whatever), it's going to be hard to shake that image.

Dinosaurs once faced a similar issue: throughout the first half of the twentieth century, the general public believed dinosaurs were sluggish and cold-blooded. It wasn't until the 1960s that a small scientific revolution, sparked by new discoveries and led by paleontologist John Ostrom, suggested that dinosaurs were probably active, warm-blooded animals.

Like I said, science is self-correcting. Eventually.

Perhaps the Neandertals need their own revolution — a Neandertal rebirth, if you will — to help the general and scientific community understand that they were a sophisticated, cultured species who made art, had some similar anatomical features to humans, and who went extinct due to a changing climate, not inferiority.

Yeah, look, I get that the author is trying to make a point about climate change. Thing is, there weren't all that many Neandertals (comparatively speaking) in the first place, and obviously there was some interbreeding going on with Sapiens, who at this point number in the billions. Whatever your personal perception is of climate change, it's unlikely that we'll go extinct because of it. Which is not to say that things won't take a major turn for the worse.

In any case, I'm mostly presenting this in my long-standing crusade to counter outdated popular interpretations of science, not to argue about climate change or politics.

3 Like • Comment • View 5 comments

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